Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Optimal exploitation of a salmon fishery: a simulation approach Loose, Verne William 1977

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1978_A1 L66.pdf [ 9.93MB ]
Metadata
JSON: 831-1.0100244.json
JSON-LD: 831-1.0100244-ld.json
RDF/XML (Pretty): 831-1.0100244-rdf.xml
RDF/JSON: 831-1.0100244-rdf.json
Turtle: 831-1.0100244-turtle.txt
N-Triples: 831-1.0100244-rdf-ntriples.txt
Original Record: 831-1.0100244-source.json
Full Text
831-1.0100244-fulltext.txt
Citation
831-1.0100244.ris

Full Text

OPTIMAL EXPLOITATION OF A SALMON FISHERY: A SIMULATION APPROACH by VERNE WILLIAM LOOSE B . S . , U n i v e r s i t y of A r i z o n a , 1965 M . A . , U n i v e r s i t y of Colorado, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Economics We accept th i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA - J u l y , 1977 (c) Verne W i l l i a m Loose, 1977 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Economics  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1WS D a t e May 21, 1978 OPTIMAL EXPLOITATION OF A SALMON FISHERY: A SIMULATION APPROACH by Verne W. Loose ABSTRACT The purpose of th i s study i s to inves t iga te optimal e x p l o i t a t i o n of a P a c i f i c salmon f i s h e r y under the assumption that exc lus ive property r i g h t s i n a f i shery have been vested i n a s ing le or so le owner. Based on a review of the e x i s t i n g f i s h e r i e s economics l i t e r a t u r e which focuses l a r g e l y (but not e n t i r e l y ) on long-term steady s tate a n a l y t i c s , i t i s argued that the associated modeling technique does not apply r e a d i l y to P a c i f i c salmon f i s h e r i e s . This l i n e of reasoning i s supported by the development, along the l i n e s of the e x i s t i n g l i t e r a t u r e , of both a within-season and a long-term model of a salmon f i s h e r y . The within-season model, - :which J f inds l l imited^ precedents i n - t h e l l i t e r a t u r e , provides the opportunity to introduce two r e l a t i o n s h i p s important i n anadromous salmon f i s h e r i e s , i . e . , a product ion funct ion with a catch c o e f f i c i e n t v a r i a b l e at the v e s s e l l e v e l and a stock d i s t r i b u t i o n f u n c t i o n , or t ime-of -entry curve. An a n a l y t i c a l general s o l u t i o n i s obtained to the model incorpora t ing these features . The in ter temporal model incorporates a purely compensatory R i c k e r -type spawner-recruit curve to define the in terseasona l r e l a t i o n s h i p between spawning escapement and future r e c r u i t s . Th i s model i s formulated as an optimal c o n t r o l problem and i s solved by using the general form s o l u t i o n to problems of t h i s type which has recent ly been developed i n the l i t e r a t u r e . A numerical s o l u t i o n to opt imal escapement from the Skeena River f i s h e r y i s obtained for parameters and c o e f f i c i e n t s estimated from t h i s f i s h e r y . Both models have s i g n f i c i a n t shortcomings which motivate the development of a more complete model of the f i s h e r y . Though i t r e ta ins the b a s i c features of the within-season and i n t e r -temporal models, t h i s larger model allows concurrent within-season and interseason a n a l y s i s , contains two species i n a j o i n t harves t ing technology, contains an age s t r u c t u r e and a s tochas t i c element, a l l of which are absent i n the two e a r l i e r models. This model i s simulated on a computer i n two formulat ions - - one dea l ing with weekly f l e e t h i r i n g by the sole owner and the other wi th annual f l e e t h i r i n g . The a l t e r n a t i v e f l e e t h i r i n g assumptions are introduced so as to assess the added e f f i c i e n c y which i s provided by more f l e x i b l e f l e e t h i r i n g r u l e s which would be an important feature of coast-wide attempts at r a t i o n a l i z i n g the B r i t i s h Columbia salmon f i shery v i a establishment of property r i g h t s . The major f indings of the study are s e v e r a l . The determination of an opt imal escapement p o l i c y v i a s imulat ion experimentation demonstrates that a computer model of th i s type can s u c c e s s f u l l y be used i n th i s fash ion . The s p e c i f i e d minimum annual escapement which maximizes the present value of net p r o f i t s was 300,000 sockeye and 400,000 pink for annual f l e e t h i r i n g whi le the weekly f l e e t h i r i n g model requ ired the a d d i t i o n a l a p p l i c a t i o n of a weekly minimum escapement of 30% of the a v a i l a b l e t o t a l sockeye and pink stock. Comparing the r e s u l t s of the two f l e e t h i r i n g assumptions i t was concluded that the weekly f l e e t h i r i n g regime r e s u l t e d i n a l arger present value of net p r o f i t s cons ider ing the Skeena River f i s h e r y a lone . In .a comparison of the optimal f i s h i n g e f f o r t ind ica ted by the s imul t ion model (as measured i n vessel -days) with the e f f o r t expended i n the a c t u a l f i s h e r y i t was concluded that a minimum estimate of excess e f f o r t i s approximately .15% for annual f l e e t h i r i n g and 50% for weekly f l e e t h i r i n g . Successful development and a p p l i c a t i o n of t h i s computer-based model i s a s i g n i f i c a n t step towards development of the l arger coast-wide model which can be used to estimate optimal capaci ty for the e n t i r e f l e e t . ACKNOWLEDGEMENTS This research could not have been completed without the continued support and guidance of my research a d v i s o r , Dr . A . D. Sco t t . To him I am deeply indebted both for h i s guidance and support , and i n t e l l e c t u a l l y , for the knowledge I have gained by studying under him over the past s e v e r a l years . To Dr . H . F . Campbell I owe a s p e c i a l debt for h i s i n s i s t e n c e upon a h igh standard of p r e s e n t a t i o n . My hope i s that h i s e f f o r t s have born f r u i t . D r . Gordon Munro acted as a c a t a l y s t i n s e t t i n g the s t ruc ture and content of the t h e s i s and to him I am most g r a t e f u l . D r . Peter H . Pearse contr ibuted to the e a r l y c o n c e p t u a l i z a t i o n of the problem. Other members of the U n i v e r s i t y of B r i t i s h Columbia f a c u l t y who contr ibuted to the success of t h i s e f f o r t are Dr . Peter A . L a r k i n and D r . C o l i n W. C l a r k . D r . L a r k i n provided ideas and i n s p i r a t i o n during t the e a r l y stages of formation of the research problem and method. D r . C l a r k ' s own research forms the bas i s f o r much of Chapter 2. I a l so had the opportuni ty to p a r t i c i p a t e i n o c c a s i o n a l d i scuss ions with Drs . C a r l Wal ters , Randa l l Peterman and Ray H i l b o r n of the I n s t i t u t e of Animal Resource Ecology at the U n i v e r s i t y of B r i t i s h Columbia. Mr. Dick Roberts of the F i s h e r i e s and Marine S e r v i c e , whose work i s important to t h i s s tudy, was h e l p f u l i n a s e r i e s of d i scuss ions on s e t t i n g the scope of the re search . D r . J . M. Kennedy of the U n i v e r s i t y of B r i t i s h Columbia Computing Center and D r . K o i t Teng of the Department of Environment, Marine Hydro-g r a p h i c a l Serv ice k i n d l y paved the way for my use of the computer i n s t a l -l a t i o n of the Marine Hydrographica l Service at P a t r i c i a Bay, B . C . Thi s i v arrangement s u b s t a n t i a l l y eased the burden of computing. Mr . Ed Zyblut and Mr. Dick Roberts , both of the F i s h e r i e s and Marine S e r v i c e , provided data and general background on the f i s h e r y . Mr. K e i t h Wales of the U n i v e r s i t y of B r i t i s h Columbia Computing Center provided computing a s s i s -tance. During the conduct of t h i s re search , i n d i s p e n s i b l e f i n a n c i a l support has been u t i l i z e d a r i s i n g from a research fund o r i g i n a l l y granted i n 1967 and augmented i n 1974, by what i s now the F i s h e r i e s and Marine Service of the Canada Department of the Environment. I must express my thanks for the h e l p f u l and pat i en t a t t i t u d e of the admin i s tra tors of these funds. F i n a l l y , I cannot adequately express the magnitude of the debt I owe to my wife who, p a t i e n t l y and s u p p o r t i v e l y , saw me through the long per iod during which I have been involved i n t h i s research and who, i n so do ing , s a c r i f i c e d a great d e a l . v TABLE OF CONTENTS CHAPTER RAGE GLOSSARY . . . . . . . x i i i 1 REVIEW OF THE LITERATURE 1 In troduct ion and Problem Statement Review of the L i t e r a t u r e Summary 2 A GENERAL MODEL OF A PACIFIC SALMON FISHERY . . . . . . . . 26 In troduct ion The B i o l o g i c a l Growth Funct ion The Harves t ing Technology A S ing le Season Model An Intertemporal Model of a Salmon F i shery T r a n s i t i o n to Simulat ion Approach 3 A COMPUTER SIMULATION MODEL OF THE SKEENA RIVER GILLNET SALMON FISHERY 60 The I n s t i t u t i o n a l and Geographical Se t t ing Development of the Computer S imulat ion Model Within-Season Re la t ionsh ips : D e t a i l e d D e s c r i p t i o n of the S imulat ion Model Interseasonal R e l a t i o n s h i p s : D e t a i l e d D e s c r i p t i o n of the S imulat ion Model V a l i d a t i o n of the Model 4 SIMULATION RESULTS 96 In troduct ion Optimal Escapement—The Case II Model Other Resul ts from Case II Simulat ions Summary of Case II Simulation Results Optimal Escapement—The Case I Model Other Resul ts from Case I Simulat ions Summary of Case I Simulat ion Resul ts Comparison of the Case I and II Results 5 SUMMARY OF FINDINGS, POLICY IMPLICATIONS AND SUGGESTIONS FOR FURTHER RESEARCH . • 153 In troduct ion Review of the Study Findings and P o l i c y Impl icat ions Suggestions for Further Research and Extensions of the Model v i APPENDICES A ESTIMATION OF COEFFICIENTS AND PARAMETERS B PILOT TESTS OF PRODUCTION FUNCTION PARAMETERS . . . . BIBLIOGRAPHY v i i LIST OF TABLES TABLE PAGE I. Tenure Schema . . 4 I I . Parameters Employed i n Solution to Optimal Escapement for Intertemporal Model. 48 I I I . Roots of Equation 2.56 for Various Values of Season Length and Discount Rate and L i f e Cycle Length . . . . . i 49 IV. I n s t i t u t i o n a l Assumptions 67 V. Comparison of Actual and Simulated Catch, Escapement, Recruitment and Fleet Sizes for Skeena River Fishery . . , 93 VI. Optimal Annual Escapement Simulations: Case I I Model. . . . 99 VII. Results of Application of Weekly Escapement Constraint to Optimal Annual Escapement Simulation: Case I I Model. . . 101 VI I I . Optimal Annual Escapement Simulations: Case I Model . . . . 123 IX. Results of Application of Weekly Escapement Constraint to the Optimal Annual Escapement Simulation: Case I Model . 125 X. Actual and Desired Escapement, Recruitment and Present Value of the Net P r o f i t s : Case I Model . 146 XI. Combined Mean Annual Escapement and Recruitment: Case I and Case I I Models and Actual Fishery ,147 XII. Proportions of Sockeye Maturing at Various Ages 165 XIII. Time-of-Entry Coefficients . -166 XIV. Production Function Parameters 175 XV. Data Employed i n the Production Function Estimation . . . . . . 176 XVI. Prices of Net Caught Sockeye and Pink Salmon 177 XVII. Calculation of Weighted Average 1974 Replacement Cost of the Weighted Average Length of Salmon G i l l n e t Vessel constructed i n 1974 181 XVIII. Estimated Daily Net Returns of a Vessel which G i l l n e t t e d for Salmon Only 182 v i i i TABLE PAGE XIX. G i l l n e t Vessel Variable Operating Costs Per Day 183 XX. Calculation of Net Returns Per Vessel Day 184 XXI. Egg Production Factors—Sockeye 185 XXII. Length-Fecundity Relationship for Skeena River Pink Salmon 186 XXIII. Compensatory Mo r t a l i t y Coefficients 187 XXIV. Maximum Daily Discharge from Skeena and Fraser Rivers, 1965-1973 189 XXV. S e n s i t i v i t y Analysis on Percentage Mortality Rates with Varying Parameter Values (after application of Compensatory Mortality) . 190 XXVI. Comparison of Actual and Simulated Catch, Escapement, Stock and Fleet Sizes for the Skeena River Fishery: Case I Model . . . 195 i x LIST OF FIGURES FIGURE PAGE 2-1 The Time-of-Entry Relationship . . „. 35 2-2 Pl o t of Equilibrium Equation (2.56) for Selected Parameter Values from Table IV 50 4-1 Average Weekly Sockeye Escapement Versus Time: Case I I Model, Simulation 6 ..105 4-2 Average Weekly Pink Escapement Versus Time: Case I I Model, Simulation 6 106 4-3 Combined Recruitment of Sockeye and Pink Versus Time: Case I I Model, Simulation 1 . ,108 4-4 Combined Harvest of Sockeye and Pink Versus Time: Case I I Model, Simulation 1 109 4-5 Combined Escapement of Sockeye and Pink Versus Time: Case I I Model, Simulation 1 . . . . . . . . . . . , 110 . 4-6 Combined Recruitment of Sockeye and Pink Versus Time: Case I I Model, Simulation 6 112 4-7 Combined Harvest of Sockeye and Pink Versus Time: Case I I Model, Simulation 6 ,113 4-8 Maximum Vessel-Days Available per Week, Case I I Model, Simulation 6 .115 4-9 Fleet U t i l i z a t i o n During a Typical Season: Case I I Model, Simulation 1 and 6 . . . . . . . . 117 4-10 Weekly Harvest — Available Stock Ratio for a Typical Season, Case I I Model, Simulation 6 119 4-11 Average Weekly Sockeye Escapement Versus Time: Case I Model, Simulation 23 128 4-12 Average Weekly Pink Escapement Versus Time: Case I Model, Simulation 23 130 4-13 Combined Recruitment of Sockeye and Pink Versus Time: Case I Model, Simulation 23 132 4-14 Combined Harvest of Sockeye and Pink Versus Time: Case I Model, Simulation 12 133 x FIGURE 4-15 Combined Escapement of Sockeye and Pink Versus Time: Case I Model, Simulation 12 . . . . . 134 4-16 Combined Recruitment of Sockeye and Pink Versus Time: Case I Model, Simulation 23 136 4-17 Combined Harvest of Sockeye and Pink Versus Time: Case I Model, Simulation 23 .... . 137 4-18 Mean Annual Vessel-Days of Fishing per Week: Case I Model, Simulation 23 139 4-19 Fleet U t i l i z a t i o n During a Typical Season: Case I Model 141 4-20 Weekly Harvest—Available Stock Ratio for a Typical Season, Case I Model, Simulations 12, 23 . 142 4-21 Fleet U t i l i z a t i o n i n a Typical Season, Case I Versus Case I I Models 148 x i LIST OF PLATES SCHEMATIC DIAGRAM OF SIMULATION MODEL DETAIL OF HARVESTING TECHNOLOGY: CASE I DETAIL OF HARVESTING TECHNOLOGY: CASE II x i i GLOSSARY Age at matur i ty /age at r e t u r n : The number of years between the brood year and the year of recru i tment . Brood year : For a p a r t i c u l a r f i s h or a c l a s s of f i s h , the year i n which the spawn was depos i ted . Compensatory n a t u r a l m o r t a l i t y : A s i t u a t i o n i n which the propor-t i o n a l growth r a t e of the stock i s a decreas ing func t ion of stock s i z e . Depensatory n a t u r a l m o r t a l i t y : A s i t u a t i o n i n which the p r o p o r t i o n a l growth r a t e of the s tock i s an i n c r e a s i n g func t ion of the s tock s i z e f o r c e r t a i n ranges of stock s i z e . Escapement: That p o r t i o n of recrui tment which, though subjec t to capture avoids capture and proceeds to the spawning grounds. Recruitment, catch and escapement form an important b i o - t e c h n i c a l i d e n t i t y : Recruitment = Catch + Escapement. Extrapensatory n a t u r a l m o r t a l i t y : The ra te of n a t u r a l m o r t a l i t y experienced by the s tock i s independent of the stock s i z e . G i l l n e t : A net w i th f l o a t s on one edge and weights on;-the other and which s i t s upr ight i n the water. I t i s p laced i n the path of migra t ing f i s h . The mesh of the net i s such as to entangle the g i l l s of the f i s h . Race: A b i o l o g i c a l l y i d e n t i f i a b l e sub-category of any of the f i v e species of P a c i f i c salmon. Recruitment: F i s h of s u f f i c i e n t matur i ty ( s i ze so as to be subject to capture by the re levant gear. T h i s i s the biomass which i s subject to f i s h i n g pressure . Therefore , run and recruitment are synonyms. x i i i 10. Run: Each year a p o r t i o n of the stock matures and begins i t s one-way migrat ion from ocean feeding grounds to freshwater spawning grounds. Th i s p o r t i o n of the stock or biomass rece ives f i s h i n g pressure en route to the spawning grounds. 11. Stock: T h i s term r e f e r s to the biomass of f i s h i n the sea . For the c o a s t a l f i s h e r y under i n v e s t i g a t i o n the stock i s not d i r e c t l y subject to f i s h i n g pressure as i n demersal or p e l a g i c f i s h e r i e s and i t i s therefore d e s i r a b l e to maintain the d i s t i n c t i o n between the stock subject to f i s h i n g pressure and that not subjec t to f i s h i n g p r e s s u r e . x i v 1 CHAPTER 1 REVIEW OF THE LITERATURE 1 1.0 In troduct ion and Problem Statement The l i t e r a t u r e review which fol lows i l l u s t r a t e s that the major focus of research i n f i s h e r i e s economics during the l a s t quarter century has been upon l o n g - p e r i o d , or i n t e r s e a s o n a l , aspects of the economics of f i s h e r y management. The general c lass of models which have come to be termed s t a t i c have, f or the most p a r t , portrayed the l o n g - p e r i o d , steady s ta te e q u i l i b r i u m of the f i s h e r y . To these the dynamic models, with t h e i r e x p l i c i t i n c l u s i o n of t ime, have added the c a p a b i l i t y of i n v e s t i g a t i n g opt imal (and other) approaches to the long-run e q u i l i b r i u m growth path . With very few exceptions ( i n c l u d i n g S c o t t ' s e a r l y a r t i c l e ) 1 t h i s l i t e r a t u r e has d e a l t s o l e l y wi th e i t h e r the in ter temporal problem of opt imal harvest (escapement), i g n o r i n g shor t - run adjustment v i a entry and e x i t of f i r m s , or with s h o r t - r u n adjustment, and so ignored opt imal escapement. This study takes up the Scott suggestion by cons ider ing both the w i t h i n -season and i n t e r s e a s o n a l problems. This composite approach i s p a r t i c u l a r l y important for the p o l i c y problems of the type of f i shery i n v e s t i g a t e d i n 2 t h i s study; i . e . , those of an anadromous P a c i f i c salmon f i s h e r y . I t i s argued that at l e a s t two aspects of the b i o t e c h n i c a l nature of t h i s f i shery necess i ta te such a d i f f e r e n t i a t e d approach. These are the phenomena of endogenous recruitment (addi t ions to the biomass be ing p o s i t i v e l y a f fec ted by the s i ze of spawning escapement), together with the gauntlet form of h a r v e s t i n g o r g a n i z a t i o n . The pat tern of h a r v e s t i n g w i t h i n the season has 2 a most d e f i n i t e effect upon the size and composition of future recruitment and therefore on the economic returns from future e x p l o i t a t i o n . The objective of the study i s to investigate f l e e t capacity and optimal harvest pol i c y i n a model of an actual fishery within a given set of poli c y and i n s t i t u t i o n a l assumptions. The study begins with the develop-ment of a single-season model i n which a sole owner with short-term tenure over the resource attempts to maximize current net p r o f i t s from ex p l o i t i n g the fishery. This one-period model i s developed i n the fashion customary for short-run models of the firm. An e x p l i c i t a n a l y t i c a l solution i s obtained, and i t i s shown that a numerical sol u t i o n to the current net p r o f i t maximizing f l e e t size can be obtained given parameter estimates for a fishery. This model i s useful for i t s stated purposes but i t does not provide any l i n k between current and future harvesting — the essence of the c a p i t a l - t h e o r e t i c view of renewable resource e x p l o i -t a t i o n . To provide t h i s scope an intertemporal model i s necessary. A model having e s s e n t i a l l y the same features as the single-season model i s developed, which i n addition contains a b i o l o g i c a l growth function of the type employed i n P a c i f i c salmon management. This augmented model provides a problem i n optimal control theory. The objective i s to f i n d a control law (harvest policy) which, when followed, maximizes the present value of the fishery. Given a series of simplifying assumptions i t i s possible to obtain an ana l y t i c solution to t h i s model as w e l l . With parameters estimated for the Skeena River fishery a numerical solution i s obtained for t h i s model by finding the root of the equilibrium equation. In spite of i t s greater scope, t h i s model too, i s found lacking i n a number of aspects regarded as important to investigating the management 3 and e x p l o i t a t i o n of a P a c i f i c salmon f i s h e r y . In order to incorporate these features i t has been found necessary to employ a d i f f e r e n t type of modeling technique—that of computer s i m u l a t i o n . While the computer model contains the same b a s i c features as the s i n g l e season and i n t e r -temporal models, i . e . , the harvest product ion funct ion and the b i o l o g i c a l growth f u n c t i o n , i t a l so al lows a more complete modeling of the f i s h e r y than was p o s s i b l e wi th the a n a l y t i c a l approach. I t w i l l be seen that s imula t ion allows i n v e s t i g a t i o n of the wi th in-season , s h o r t - r u n adjustment problem s imultaneously with that of the in t er seasona l opt imal escapement. An a d d i t i o n a l advantage i s that a more complete b i o l o g i c a l growth model i s poss ible—one which incorporates the v a r y i n g types of m o r t a l i t y cohorts of salmon experience at t h e i r var ious l i f e s tages . The populat ion f l u c t u a t i o n s generated by t h i s m o r t a l i t y model are s i m i l a r to the f l u c t u a -t i o n s observed i n a c t u a l salmon s tocks . This f l u c t u a t i o n has , of course , important e f f ec t s on the harves t ing s t ra tegy . Concomitant wi th the more complete populat ion dynamics model i s a more complete, model of the species and age s t r u c t u r e o f the biomass. With t h i s more complete model, var ious escapement p o l i c i e s may be inves t iga ted by repeated s imulat ion i n an attempt to i d e n t i f y that p o l i c y which r e s u l t s i n the l arges t value (the discounted sum of net p r o f i t s ) f or the p a r t i c u l a r f i s h e r y . Each of the three models employs p o l i c y and i n s t i t u t i o n a l assumptions w i t h i n t h e i r s t r u c t u r e . F i r s t , so l e ownership of the f i s h e r y i s assumed to e x i s t i n the case of a l l three models. Second, f l e e t - h i r i n g arrangements vary among the models: the within-season model i m p l i c i t l y assumes that the f l e e t s i z e may be changed during each d i s c r e t e time p e r i o d w i th in the season, whi le the in ter tempora l model assumes constant e f f o r t f o r the season. The s imula t ion model deals separate ly with both cases. T h u s 4 three elements — sole ownership, f l e e t h i r i n g , and escapement requirements provide the basic policy ingredients of the models described i n the following chapters. We now turn to a b r i e f l i t e r a t u r e review which deals s e l e c t i v e l y with some of the assertions made above. 1.1 Review of the Literature Support for the statement that the bulk of the f i s h e r i e s economics l i t e r a t u r e i s comprised of interseasonal models w i l l be found by c l a s s i f y i n g the relevant l i t e r a t u r e according to the matrix displayed i n Table I. The various contributions to the f i s h e r i e s economics l i t e r a t u r e may conveniently be placed i n one of the s i x elements,of Table I: The most s i g n i f i c a n t categorization for purposes of t h i s review i s that across the columns, i . e . , the tenure policy d i s t i n c t i o n s . TABLE I TENURE SCHEMA TENURE FLEET HIRING SHORT-TERM TENURE PERPETUAL TENURE COMMON PROPERTY or modified common property (p very large for future seasons) C (p very large for future seasons) A 0<p< very large B Continuous adjustment over single season I (1) (2) (3) Constant f l e e t a v a i l a b i l i t y over single season I I (4) (5) (6) 5 Short- term tenure i n column A i s intended to contain the models dea l ing with i n t r a s e a s o n a l h a r v e s t i n g o n l y . In these models the so le owner or f i s h e r y manager takes biomass and c a p i t a l equipment as given and optimizes w i t h i n these cons tra in t s for a s i n g l e season. This s i t u a t i o n can a l so be c h a r a c t e r i z e d as one i n which the s o l e owner discounts future harvests at a p r o h i b i t i v e l y h igh r a t e . C l e a r l y , opt imal harves t ing does not subsume the problem of opt imal escapement for future harves t ing a c t i v i t i e s s ince wi th one-year tenure the future warrants l i t t l e or no c o n s i d e r a t i o n . . . Column B , l a b e l e d perpe tua l tenure , contains the models i n which present and future harves t ing are l i n k e d e i t h e r through the b i o l o g i c a l growth func t ion alone or through a combination of the b i o l o g i c a l growth func t ion and economic a l l o c a t i o n over time. His time hor izon presents the so l e owner or f i s h e r y manager wi th a p o t e n t i a l l y wide range of investment or dis investment opportun i t i e s for both the n a t u r a l and man-made c a p i t a l employed i n the harves t ing process . For these models the impl i ed discount ra te might range over a p o t e n t i a l l y wide band of values from zero to a value somewhat l e ss than that of the in traseasona l v e r s i o n s . Column C i s reserved f o r those models or vers ions of models which descr ibe and analyze the f i s h e r y under the property r i g h t s s t r u c t u r e most f requent ly encountered i n the r e a l world—that of common property or some form of modif ied common p r o p e r t y . Development of the common property model t y p i c a l l y forms the bas i s for a comparison of the unregulated bionomic e q u i l i b r i u m with that which would r e s u l t under optimal taxat ion or s p e c i f i c a t i o n of property r i g h t s . Because of the absence of p r i v a t e property r i g h t s the impl i ed discount rate of competit ive fishermen can be presumed to be very h i g h , i . e . , i n the neighborhood of that of the s h o r t -term tenure so l e owner. 6 1.1.1 Intraseasonal Fishery Models A. D. Scott was among the f i r s t authors to address the intraseasonal 3 harvesting problem. In an early a r t i c l e Scott extended the discussion of fishery economics begun by H. Scott Gordon4 by comparing the "... use of a fishery by competing fishermen with the mode of management that would 5 be most profitable to a sole owner of the same fishery." Scott argued that there are two possible bases upon which competitive (Table I, column C ) and sole-ownership (column A) property rights may be compared. One situation is that i n which the sole owner takes over an existing fishery while the other i s that i n which the sole owner has the opportunity to reorganize 6 the fishery i n the most eff i c i e n t way. Scott identifies the former situation as that relevant to the short-run while the latter situation i s really the essence of the long-run optimization problem. Applying these distinctions to the present study we find that in both weekly and annual fleet hiring cases the sole owner i s faced with the short-run problem of identifying that fleet size for which marginal cost equals price. The difference i n fleet hiring assumptions may affect the magnitude of marginal cost but the general resource allocation rule is the same for both. If a change in fleet size from year to year may be construed as a re-organization of the fishery then the annual fleet hiring case i s also similar to Scott's long-term case. However, i t i s apparent that Scott may have had in mind a more global type of reorganization than that considered i n this study.^ For this.study mobile fishing gear i n the form of g i l l n e t vessels i s the only harvesting technology investigated. 7 Bradley too i s concerned with i n v e s t i g a t i o n of i n t r a s e a s o n a l adjustment i n the f i s h e r y s i n c e , "Most f i s h e r i e s are operated on a seasonal bas i s because of movements of f i s h s t o c k s , matur i ty of the f i s h , o r weather f a c t o r s . Thus proposals f o r r e g u l a t i o n der ived from models which dep ic t l ong-run adjustment may be f a u l t y i n a s i t u a t i o n where fishermen must base 8 dec i s ions on the seasonal changes they a c t u a l l y experience ." Thi s r a i s e s a po int of importance f o r the salmon f i s h e r y : seasonal harves t ing pat terns have an e f f e c t on long-run adjustment thus suggest ing that i n v e s t i g a t i o n s i n c o r p o r a t i n g both are appropr ia te . Cont inuing to quote B r a d l e y , "What we attempt to do, there fore , i s to formulate some models which depict both s h o r t - r u n output dec i s ions based on seasonal f a c t o r s and l o n g -er run adjustment through entry or e x i t . " Thi s statement i s important because i t descr ibes i n very general terms what i s attempted i n a p o r t i o n of t h i s s tudy. The model employed f o r t h i s par t of the i n v e s t i g a t i o n — a computer s imula t ion model—is somewhat more complex but i t neverthe less draws support from B r a d l e y ' s work. Fo l lowing other f i s h e r y economists, Bradley employs the mass encounter f i s h i n g technology assumption i n which the propor t ion of the f i s h popula t ion taken by an i n d i v i d u a l v e s s e l i s a constant."^ E f f o r t i s measured i n numbers of homogeneous v e s s e l s . For gaunt le t - type f i s h e r i e s , of which the salmon f i s h e r y i s an example, entry to the f i s h e r y takes p lace as a r e s u l t of the percept ion of excess p r o f i t s on the p a r t of p o t e n t i a l e n t r a n t s . Entry continues u n t i l the Incremental cost of a day's f i s h i n g r i s e s to meet the constant output p r i c e . The reason f o r r i s i n g costs of f i s h i n g apparent ly i s th inn ing of the stock. 1"' 1 Thus, Bradley uses entry and e x i t from the operat ing f l e e t as the mechanism of adjustment to s h o r t - r u n e q u i l i b r i u m . B r a d l e y ' s model f i t s i n t o element (1) o f Table I. 8 Bradley appears to assume that there i s a ready supply of vesse l s and fishermen prepared to enter the a c t i v e f l e e t at the appropriate s i g n a l . A s i m i l a r adjustment mechanism i s used i n par t o f the present study; however, the mechanism i s c o n t r o l l e d by the so le owner who engages i n an i m p l i e d b i d - a n d - o f f e r process wi th v e s s e l owners. Given equiva lent p r i c e s and costs f o r the two s i t u a t i o n s there i s reason to expect that B r a d l e y ' s mechanism and that used i n the present study would not r e s u l t i n the same f l e e t s i z e and net economic r e t u r n s . Both the s tock and crowding e x t e r n a l i t i e s w i l l work through the v a r i a b l e c a t c h a b i l i t y c o e f f i c i e n t and w i l l impinge upon one o b j e c t i v e funct ion i n the present model whi le only the s tock e x t e r n a l i t y w i l l a f f e c t r e s u l t s i n B r a d l e y ' s model. In cons ider ing opt imal e x p l o i t a t i o n of the f i s h e r y wherein two types of i n t e r s e a s o n a l adjustment are possible—changes i n the f l e e t s i z e and s e l e c t i o n of a l t e r n a t i v e steady s ta t e popula t ion e q u i l i b r i a — B r a d l e y uses 12 the regu la ted and so le p r o p r i e t o r s h i p f i s h e r y as a n a l y t i c a l synonyms. C o n t r o l of the length of the season i s the means of r e g u l a t i o n . S e t t i n g the length of the season impl i e s a given escapement l e v e l which i n turn impl ies a steady s ta te s tock l e v e l . S i m i l a r elements are employed i n the present computer model but are u t i l i z e d i n a somewhat d i f f e r e n t way. The s o l e owner i s r a t i o n a l and contro l s the f l e e t to e f f e c t a current h a r v e s t i n g p o l i c y . The length of the season i s a c o n t r o l v a r i a b l e f o r the so l e owner i n the sense that p r o v i s i o n i s made f o r s e l f - r e g u l a t i o n by the impos i t i on of escapement requirements . Genera l ly a good d e a l of precedent f o r the i n s t i t u t i o n a l and p o l i c y assumptions u t i l i z e d i n the present study i s found i n the Scott and Bradley papers . 9 Much of Clive Southey's work in Studies in Fisheries Economics i s - concerned with an investigation of adjustment in the short-run and may appropriately be classified in column A of Table I.-^ Southey's interest i n short-run adjustment follows from his observation that the empirical identification of low earnings normally attributed to the long-^ run bionomic equilibrium may in fact belong to a period of short-run disequilibrium. This prompts his analysis of the s t a b i l i t y of short-run equilibrium and the conclusion emerges that overall short-run disequilibrium in the fishery i s a likelihood, depending upon scale economies, fish prices and costs of fishing. Later i n the review the model of the Halibut fishery developed by Crutchfield and Zellner is identified as a significant and early contribu-tion to the dynamic analysis of fisheries. In this same work Crutchfield and Zellner present a dynamic market model of port pricing i n addition to other aspects of the Halibut fishery.-^ The port pricing model incor-porates both interseasonal effects (inventories) and within-season effects (catch quota, the effect of current catch and current demand (sales) on desired holdings and the duration of the regular season). Furthermore, this model i s tested empirically with data from the Halibut fishery which distinguishes i t further from much of the fishery economics literature which i s largely t h e o r e t i c a l . ^ The within-season distribution of landings among ports, freezing and storage costs and marketing are among the other within-season aspects of this fishery analyzed by Crutchfield and Zellner. The work of the above authors appears to be the total effort expended to date by fishery economists on the analysis of within-season harvesting. The objective has been to indicate how certain elements of these models are important for the present study. We turn now to a brief review of the bulk of the fishery economics literature which concerns long-period analysis, 10 1.1.2 Interseasonal Models of Harvest ing C o l i n C l a r k and Gordon Munro have r e c e n t l y publ ished work i n which they show that s t a t i c models of harves t ing are but a l i m i t i n g case of dynamic models, the l i m i t being the use of a zero percent discount ra te i n the s t a t i c m o d e l . ^ T h i s d i s t i n c t i o n provides a convenient c a t e g o r i z a -t i o n for expos i tory purposes i n a d d i t i o n to p r o v i d i n g i n s i g h t in to the r e l a t i o n s h i p s between the d i f f e r e n t types of models which have appeared. The modern l i t e r a t u r e of f i s h e r y economics traces i t s l ineage to S c o t t ' s a r t i c l e p r e v i o u s l y discussed."^ Scott a n t i c i p a t e d r e l a t i v e l y recent developments by showing that a long-run problem of opt imal harves -t i n g r e q u i r e d a model which e x p l i c i t l y inc luded the time dimension. He a p p l i e d the concept of user cost to the Scott Gordon model and concluded that the so le owner or r a t i o n a l s o c i a l manager would increase landings to the po int at which marginal net revenue from current landings equal led marginal user c o s t . In s p i t e of t h i s breakthrough i n t h i n k i n g , many models continued ( indeed, continue) to be formulated i n s t a t i c terms. C l a r k and Munro suggest that t h i s may w e l l have been due to an i n s u f f i c i e n c y of mathematical . „ 18 instruments . To the e a r l y a r t i c l e s of H . Scott Gordon and A . D. S c o t t , a long with c o n t r i b u t i o n s by James C r u t c h f i e l d and others may be assigned the d e s i g -n a t i o n , 'the t r a d i t i o n a l l i t e r a t u r e . ' Combining t h i s des ignat ion wi th C l a r k and Munro's d i s t i n c t i o n mentioned above one may speak of e i t h e r s t a t i c or dynamic extensions of the t r a d i t i o n a l model when r e f e r r i n g to the subse-quent l i t e r a t u r e . The remainder of t h i s review i s c l a s s i f i e d along the l i n e s of that d i s t i n c t i o n . 11 1.1.2.1 S t a t i c Extensions of the T r a d i t i o n a l Model The s t a t i c models which w i l l be discussed i n t h i s s e c t i o n have severa l features i n common. The most obvious common feature i s that which i d e n t i f i e s the models as s t a t i c ; i . e . , the use of a zero discount r a t e . In a d d i t i o n , a l l of these models d e a l with long-term o r , as designated here , perpe tua l tenure s i t u a t i o n s . R e f e r r i n g to Table I we may thus place a l l of these models i n column B. Frequent ly , authors compared the economic performance of perpetua l tenure so le ownership to the common property case designated i n column C. Thus, a l t e r n a t i v e formulat ions of the models i n any given a r t i c l e would f a l l i n t o columns B or C as a p p r o p r i a t e . Such i s the case, f o r 19 20 9 1 9 9 example, w i th the a r t i c l e s of Smith, B e l l , Agnel lo and Donne l l ey , x>^z-23 ' and Gould . Although i t may not always be e x p l i c i t , most of these authors are d e a l i n g with the annual f l e e t h i r i n g (or f l e e t ownership) assumption and accord ing ly are placed i n element (5) of Table I . A l l of the models c i t e d above emphasize the steady s ta te e q u i l i b r i u m of the f i s h e r y by assuming that the r a t e o f change o f the biomass with respect to time i s equal to zero . A d d i t i o n a l l y , a l l models wi th the excep-24 t i o n of Southey dea l wi th b i o l o g i c a l growth i n an aggregate fa sh ion 25 employing the P e a r l - V e r h u l s t l o g i s t i c equation which Schaefer d i s c u s s e d . With the biomass i n steady s ta te e q u i l i b r i u m , recruitment exac t ly equals the harves t . However, d i f f e r e n t authors handle the i n t e r a c t i o n between harvest and 26 biomass p r o d u c t i v i t y i n d i f f e r e n t ways. Turvey assumes that there i s no 27 i n t e r a c t i o n whi le Smith assumes that the biomass growth r a t e i s a f fec ted 28 by h a r v e s t i n g . In the present study, the b i o l o g i c a l growth f u n c t i o n takes a s i g n i f i c a n t l y d i f f e r e n t approach from these s t a t i c models. This i s because i n the b io- technology of the salmon f i s h e r y , harvest and 12 r e s u l t i n g recruitment are h i g h l y i n t e r a c t i v e , and so must be dea l t with i n a disaggregated manner. Southey's regenerat ive model i s of p a r t i c u l a r i n t e r e s t i n the l a t t e r re spec t . In t h i s model the reproduct ive c h a r a c t e r i s t i c s of the populat ion take precedence. Future biomass s i ze i s a func t ion only o f the s u r v i v a l r a t e i n previous y e a r s . Although both f i s h i n g and n a t u r a l m o r t a l i t y are important , Southey dea l s e x p l i c i t l y only wi th f i s h i n g m o r t a l i t y , and shows that the y i e l d curve may have a p o s i t i v e l y sloped s e c t i o n i n 'biomass-e f f o r t ' space. T h i s has the i m p l i c a t i o n that both catch and biomass may be sus ta inab ly l a r g e r over t h i s p o r t i o n of the curve.^9 Another feature which acts to d i s t i n g u i s h among some of the models i s the handl ing of the concept of f i s h i n g e f f o r t introduced by Scott Gordon and A . D. Scot t . Most researchers employ the assumption of a f i x e d - p r o p o r t i o n s product ion func t ion f o r the v e s s e l and combine t h i s wi th the assumption of homogeneous vesse l s so that an aggregate index of f i s h i n g i n t e n s i t y or e f f o r t can be d e r i v e d . Changes i n the i n t e n s i t y of f i s h i n g e f f o r t can therefore be r e l a t e d d i r e c t l y and only to changes i n the number of ve s se l s f i s h i n g , or i m p l i c i t l y , to the time p e r i o d over which a g iven number of ve s se l s operate . 30 Examples of t h i s view of f i s h i n g e f f o r t are the models of Anderson, Agne l lo and D o n n e l l e y , 3 1 , 3 2 B e l l , 3 4 and T u r v e y . 3 5 While the above authors dispensed wi th the use of v a r i a b l e proport ions at the f i r m or v e s s e l l e v e l there has r e c e n t l y been a resurgence of i n t e r e s t i n p o r t r a y i n g f a c t o r s u b s t i t u t i o n i n order to descr ibe more a c c u r a t e l y the h a r v e s t i n g technology and to b r i n g f i s h e r y economics i n t o accord with standard product ion theory . Gould employs a product ion func t ion which inc ludes among inputs c a p i t a l and labor as w e l l as the biomass and therefore 36 never uses the term f i s h i n g e f f o r t . He concludes that admit t ing the p o s s i b i l i t y of f a c t o r s u b s t i t u t i o n r e s u l t s i n a s i t u a t i o n i n which one 13 cannot determine a p r i o r i that a free-access resource i s n e c e s s a r i l y 37 overexp lo i t ed as compared wi th a Pareto opt imal e x p l o i t a t i o n program. Smith a l so dispenses wi th the concept of f i s h i n g e f f o r t and uses ins tead an operat ing cost equation for the indus try which i s a f u n c t i o n , not of the number of v e s s e l - d a y s , but of the catch r a t e , biomass, mesh o o s i z e and number of u n i t s i n the i n d u s t r y . Anderson extends the Smith model by i n t r o d u c i n g a v e s s e l y i e l d f u n c t i o n based upon the i n d u s t r y y i e l d f u n c t i o n . Cost then i s a func t ion not only of mesh s i z e and the number of ve s se l s i n the f i s h e r y but a l so of the amount of e f f o r t produced by the v e s s e l . E f f o r t i n turn i s a f u n c t i o n of the inputs used by the v e s s e l . Fo l lowing the bu lk of the f i s h e r y economics l i t e r a t u r e the present model employs the assumption of homogeneous ves se l s which are assumed to have no scope f o r f a c t o r s u b s t i t u t i o n . F i s h i n g e f f o r t can change only by a change i n the number of v e s s e l s . Stock and gear e x t e r n a l i t i e s work through the product ion func t ion which, g iven p r i c e , i s a l s o an average revenue f u n c t i o n . Cost i s expressed as a f u n c t i o n of vesse l -days of f i s h i n g . As i n Bradley and Smith, changes i n the f l e e t s i z e are s p e c i f i e d as a f u n c t i o n of the gap between marginal revenue and marginal c o s t . 1 .1 .2 .2 Dynamic Extensions of the T r a d i t i o n a l Model Fo l lowing the 1962 p u b l i c a t i o n of C r u t c h f i e l d and Z e l l n e r ' s dynamic model there was a lapse of n e a r l y a decade before a d d i t i o n a l c o n t r i b u t i o n s to the dynamic a n a l y s i s of f i s h e r i e s were forthcoming. 4"'" In the 1970s, f o l l o w i n g the d i f f u s i o n of opt imal c o n t r o l theory throughout the economics p r o f e s s i o n , a s e r i e s of dynamic economic models of f i s h i n g have appeared. Examples of t h i s work are the a r t i c l e s of Quirk and Smith, P l o u r d e , Brown, 4 ^ C l a r k and M u n r o , 4 ^ Spence, 4 ' ' and B u t l i n et a l . 4 * * Other dynamic models not employing opt imal c o n t r o l theory to t h e i r s o l u t i o n are those of C l a r k ^ ' ^ O a n c i Hannesson. ^ 14 L i k e the s t a t i c models discussed i n the previous s ec t i on the dynamic models dea l s o l e l y wi th the in ter seasona l e x p l o i t a t i o n problem. In terms of Table I the dynamic models would be placed i n element (5); however, as with many of the s t a t i c models, many of the dynamic models compare the performance of opt imal economic organ iza t ion wi th a common property r i g h t s s t r u c t u r e — column C, element ( 6 ) . C l e a r l y the d i s t i n g u i s h i n g feature of these models i s an a c t u a l or impl ied discount r a t e greater than zero . Many of the dynamic models are b u i l t around the P e a r l - V e r h u l s t l o g i s t i c 52 law of biomass growth. Examples of such models are g iven by the papers of C r u t c h f i e l d and Z e l l n e r , 5 3 P l o u r d e , 5 4 ' 5 5 C l a r k , 5 6 C l a r k and M u n r o 5 7 CO and B u t l i n a l . As wi th many of t h e i r s t a t i c counterparts they employ the concept of f i s h i n g e f f o r t and use the assumption of p r o p o r t i o n a l i t y of cost and e f f o r t . Brown, 5 ^ C l a r k and Munro 6 ^ and B u t l i n , et a l . ^ a l l employ these assumptions. In d e r i v i n g the so lut ions to the o p t i m i z a t i o n , however, both the cost f u n c t i o n and the harvest or product ion f u n c t i o n are shown only as a f u n c t i o n of t ime, i . e . , c ( t ) and h ( t ) . U n t i l r e c e n t l y , because the dynamic models were e n t i r e l y t h e o r e t i c a l , they began to r e c e i v e the c r i t i c i s m that they were too complex and therefore n o n - o p e r a t i o n a l , p a r t i c u l a r l y from the po in t of view of e m p i r i c a l e s t imat ion . Two recent a r t i c l e s have shown that dynamic models can be estimated e m p i r i -c a l l y , thus i n d i c a t i n g that r i g o r and s i m p l i c i t y can c o - e x i s t i n a s i n g l e model. One of these a r t i c l e s — that by Spence — report s on the e s t imat ion of a product ion f u n c t i o n and a growth func t ion for A n t a r t i c b lue whales. Spence sets up a c o n t r o l problem i n which the two estimated equations are c o n s t r a i n t s and proceeds to obta in e x p l i c i t numerica l s o l u t i o n s for var ious values of the re l evant parameters. 15 The more recent a r t i c l e by B u t l i n , et a l . t " t per ta ins to a t h e o r e t i c a l and e m p i r i c a l model of the Manx h e r r i n g f i s h e r y . A continuous time opt imal c o n t r o l model i s constructed hut the model which i s tested e m p i r i c a l l y i s a d i s c r e t e time model. Var ious problems are encountered i n es t imat ing the growth equat ion and product ion func t ion but the major po int for present purposes i s the author ' s observations on having completed the e m p i r i c a l e x e r c i s e . "This paper represents one attempt (of which there are s t i l l r e l a t i v e l y few) to assess the regu la tory p o l i c i e s being a p p l i e d to a p a r t i c u l a r f i s h e r y . I t emphasizes that blanket recommendations cannot be made for the r e g u l a t i o n of a l l f i s h e r i e s . The re l evant b i o l o g i c a l and economic data have to be f u l l y assessed for the p a r t i c u l a r f i s h e r y i n quest ion before the most appropr ia te p o l i c y p r e s c r i p t i o n can be f o r m u l a t e d . " 6 5 The present study of a p a r t i c u l a r salmon f i s h e r y can make the same c la im and draws some support from the above quota t ion . 1.1.3 Models of the F i shery Product ion Funct ion In the d i s c u s s i o n of the s t a t i c and dynamic models of the previous two sec t ions s e v e r a l models p e r t a i n i n g to e m p i r i c a l est imations of f i s h e r y product ion funct ions were not d iscussed or were not f u l l y d iscussed i n t h i s re spec t . These models do not a l l f i t i n t o the Table I format. . Most s i g n i f i c a n t among these a r t i c l e s for our purposes i s the product ion func t ion es t imat ion conducted by R o b e r t s . ^ Roberts employs a c a t c h a b i l i t y c o e f f i c i e n t \< which i s f u n c t i o n a l l y r e l a t e d to recrui tment and f l e e t s i z e . The c a t c h a b i l i t y c o e f f i c i e n t i s v a r i a b l e at the ves se l l e v e l . Through the e f f e c t of these l a t t e r independent v a r i a b l e s Roberts i s able to account for the stock e x t e r n a l i t i e s , gear e x t e r n a l i t i e s and crowding e x t e r n a l i t i e s c h a r a c t e r i s t i c of the salmon f i s h e r y . Roberts ' work turns out to be important to the present study s ince h i s product ion func t ion 16 and parameter estimates are employed i n the models developed below. Other models d e a l i n g p r i m a r i l y wi th the product ion func t ion are those of B e l l , 6 7 G o u l d , 6 8 And erson, and Huang and L e e . 7 ^ These l a t t e r models d i scussed i n s e c t i o n 1 .1 .2 .1 above, do not r e q u i r e e l a b o r a t i o n . 1.1.4 Model l ing Technique and P a c i f i c Salmon F i s h e r i e s B i o l o g i s t s have been c r i t i c a l of the a p p l i c a t i o n of the d e t e r m i n i s t i c Schaefer, and Beverton and Holt models to the P a c i f i c salmon f i s h e r i e s . In the words of P a u l i k and Greenough: "The e x i s t i n g d e t e r m i n i s t i c theory has been most s u c c e s s f u l l y a p p l i e d to marine populat ions having enough homogeneity to a l low sweeping aggregat ion. For these f i s h e r i e s , which t y p i c a l l y i n v o l v e a s i n g l e dominant species i n a f a i r l y constant environment, some s o r t of steady s tate c o n d i t i o n makes sense and i s of p r a c t i c a l i n t e r e s t to the management agency. On the other hand, for p e l a g i c f i s h e r i e s i n general and for salmon f i s h e r i e s i n p a r t i c u l a r , the t r a d i t i o n a l approach has been i n e f f e c t i v e . The t r a n s i e n t behavior of these h i g h l y dynamic f i s h e r i e s i s of primary concern to the management agency. I t i s not s u r p r i s i n g that r e g u l a t o r y p o l i c y for salmon f i s h e r i e s has evolved as an e m p i r i c a l a r t and has b e n e f i t t e d l i t t l e from the t r a d i t i o n a l theory of f i s h i n g . " 7 1 T h i s type of t h i n k i n g has l e d to two developments of importance for models of salmon f i s h e r i e s . The f i r s t i s the adaptat ion of the y i e l d concept to render i t more a p p l i c a b l e to the p e c u l i a r b i o l o g y of salmon. 7 2 T h i s reformulated y i e l d concept i s due l a r g e l y to the work of R i c k e r . I t accounts for the f a c t t h a t , i n contras t to demersal and p e l a g i c spec i e s , P a c i f i c salmon are a v a i l a b l e for capture only f o r a short p e r i o d d u r i n g t h e i r l i f e c y c l e , have only one or two year c las ses exp lo i t ed s imultaneously by the f i s h e r y and d ie a f t e r one spawning m i g r a t i o n . Recruitment i s therefore a h i g h l y important phenomenon for salmon f i s h e r i e s and cannot be assumed to be independent of biomass. The Ricker model has been extended 73 and a p p l i e d by a number of b i o l o g i s t s , notably L a r k i n . 17 The second development i s the a p p l i c a t i o n of systems theory and computer technology to s imulat ion of salmon popula t ion dynamics. Examples of such models are those of L a r k i n and M c D o n a l d , 7 4 and L a r k i n and Hourston. Economists have not yet made widespread use of s imula t ion modeling techniques i n f i s h e r i e s re search . However, one important salmon s imula t ion 7fi model i s that developed by Royce, et a l . which incorporated economic v a r i a b l e s . The infrequent use of s imula t ion i n f i s h e r i e s economic s tudies i s l i k e l y not due to lack of p r o f e s s i o n a l support for the method but ra ther to the sheer infrequency of economists' attempts to completely model any f i s h e r y , p a r t i c u l a r l y the salmon f i s h e r y . 7 7 Several eminent economists have expressed support for f i s h e r i e s s imulat ion models. To quote A r n o l d Z e l l n e r , "Given the formulat ion of even a crude s imulat ion mode, much can be learned by p u t t i n g i t on the computer and performing s imulat ion exper iments . . .The r e s u l t s of such experiments may lead to improved models, models whose p r o p e r t i e s and p r e d i c t i o n s should be checked wi th data . I t i s my b e l i e f that analyses l ead ing to models O f t h i s sor t and use of models, coupled with good judgment, w i l l lead to improved p o l i c i e s i n the management of marine r e s o u r c e s . " 7 8 James C r u t c h f i e l d i s s i m i l a r l y impressed wi th the p o t e n t i a l of s imula t ion models as a technique for s tudying i n t e r a c t i o n s and feedback mechanisms 79 c h a r a c t e r i s t i c of salmon resource systems. The model employed i n t h i s study fo l lows these d i r e c t i v e s and therefore draws support from them. 1.2 Summary In support of the a s s e r t i o n that the r e l a t i v e emphasis of the f i s h e r y economics l i t e r a t u r e i s upon the i n t e r s e a s o n a l a l l o c a t i o n problem, the Tenure Schema of Table I was set up to c l a s s i f y the f i s h e r i e s economics l i t e r a t u r e along the l i n e s proposed. Fourteen of the seventeen a r t i c l e s reviewed were placed i n column B of Table I whi le the remaining 18 three were placed i n column A. In the fo l l owing chapter the Tenure Schema w i l l be modif ied and w i l l serve as an o u t l i n e of the p o l i c y and i n s t i t u t i o n a l combinations s p e c i f i c a l l y dea l t with i n t h i s study. While the review has served to i n d i c a t e the precedents which may be marshal led i n the l i t e r a t u r e f o r p a r t i c u l a r features of t h i s s tudy , i t has a l so served to i d e n t i f y features not yet attempted: simultaneous modeling of both the wi th in-season and in ter seasona l harves t ing problems; use of an aggregate f i s h e r y product ion f u n c t i o n based on a c a t c h a b i l i t y c o e f f i c i e n t v a r i a b l e at the v e s s e l l e v e l ; use of a disaggregated model of popu la t ion dynamics which incorporates s t o c h a s t i c v a r i a b i l i t y ; a p p l i c a t i o n of the model to a s p e c i f i c f i s h e r y which thereby renders the conclus ions of the i n v e s t i g a t i o n comparable to an a c t u a l case; employment of a method of a n a l y s i s which shows s i g n i f i c a n t promise but which has not yet rece ived much a t t e n t i o n by f i s h e r y economists. Common features of t h i s wi th other f i s h e r y models are the use of the so le ownership assumption and the comparison of the economic performance of so le ownership with modif ied common property tenure , the use of the concept of homogeneous f i s h i n g e f f o r t , the assumption of a f i x e d proport ions product ion f u n c t i o n at the v e s s e l l e v e l and of the f i x e d p r o p o r t i o n a l i t y of cost and e f f o r t . In a d d i t i o n , the f l e e t adjustment mechanism, the constant r e l a t i v e p r i c e assumption and the j u x t a p o s i t i o n of an economic b e h a v i o r a l model and a n a t u r a l product ion func t ion have a l so been examined i n s e v e r a l previous s t u d i e s . 19 FOOTNOTES 1. A . D. Sco t t , "The Object ives of Sole Ownership," J o u r n a l of P o l i t i c a l  Economy, L X I I I , No. 2 (1955), pp. 116-124. 2. James A . C r u t c h f i e l d and G u i l i o Pontecorvo, The P a c i f i c Salmon  F i s h e r i e s : A Study of I r r a t i o n a l Conservat ion , Resources f o r the F u t u r e , I n c . , (Bal t imore: Johns Hopkins Pres s , 1969), p . 28. C r u t c h f i e l d and Pontecorvo s t res s the importance of i n t r a s e a s o n a l models i n P a c i f i c salmon f i s h e r i e s , "Since the f i s h e r y f o r P a c i f i c salmon i s d i scont inuous , both the indus try and any regu la tory au thor i ty are a l so concerned wi th i n t r a s e a s o n a l y i e l d - e f f o r t r e l a t i o n s ; i . e . , the catch per u n i t of time over the f i n i t e p e r i o d i n which a f i n i t e b l o c k of f i s h are moving through the gear operat ing along the migrat ion p a t h . " 3. S c o t t , "The O b j e c t i v e s , " pp . 116-124. 4. H . Scott Gordon, "The Economic Theory of a Common Property Resource: The F i s h e r y , " J o u r n a l of P o l i t i c a l Economy, L X I I , No. 2 (1954), pp . 124-142. 5. S c o t t , "The O b j e c t i v e s , " p . 117. 6. I b i d . , p . 120. 7. c f . S c o t t ' s comment, "For ins tance , i t has been suggested that on the west cost the so le owner of a salmon f i s h e r y would r e l y more on traps than on v e s s e l s , " I b i d . , p . 121. 8. P a u l G. B r a d l e y , "Some Seasonal Models of the F i s h i n g I n d u s t r y , " i n Economics of F i s h e r i e s Management: A Symposium, ed. by A. D. Scot t , H. R. MacMil lan Lectures i n F i s h e r i e s , (Vancouver: The U n i v e r s i t y of B r i t i s h Columbia, 1970), p . 34. 20 9. I b i d . , p. 34. 10. I b i d . , p . 35. 11. I b i d . , p . 37. 12. I b i d . , p . 37. 13. C l i v e Southey, "Studies i n F i s h e r i e s Economics," (Unpublished Ph .D. d i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, 1969), pp. 1-92. 14. U. S. Department of the I n t e r i o r , F i s h and W i l d l i f e S e r v i c e , Bureau of Commercial F i s h e r i e s , Economic Aspects of the P a c i f i c H a l i b u t  F i s h e r y , by James A . C r u t c h f i e l d and Arnold Z e l l n e r , F i s h e r y I n d u s t r i a l Research, I , No. 1 (Washington, D. C . : Government P r i n t i n g O f f i c e , 1963), pp. 49-71. 15. I b i d . 16. C o l i n W. C l a r k and Gordon R. Munro, "The Economics of F i s h i n g and Modern C a p i t a l Theory: A S i m p l i f i e d Approach," Journa l of  Environmental Economics and Management, I I , (1975). 17. Sco t t , "The O b j e c t i v e s , " p . 121. See a l so J . A . C r u t c h f i e l d , e d . , The F i s h e r i e s : Problems i n Resource Management, S e a t t l e , U n i v e r s i t y of Washington P r e s s , 1965, p. 53-55. 18. C l a r k and Munro, "The Economics of F i s h i n g , " p. 96. 19. Vernon L . Smith, "On Models of Commercial F i s h i n g , " J o u r n a l of  P o l i t i c a l Economy, LXXVII , No. 2 (1967), p. 191. 20. F r e d e r i c k W. B e l l , "Technological E x t e r n a l i t i e s and Common Property Resources: An Emprica l Study of the U. S. Northern Lobster F i s h e r y , " J o u r n a l of P o l i t i c a l Economy, LXXX (1972), pp. 155-156. 21. Richard J . Agnel lo and Lawrence P. Donnel ley , " E x t e r n a l i t i e s and Property Rights i n the F i s h e r i e s , " Land Economics, L I I , No. 2 (1976). pp. 518-529. 21 22. Richard J . Agne l lo and Lawrence P. Donnel ley , "Prices and Property Rights i n the F i s h e r i e s , " Southern Economic J o u r n a l , XXXXII, No. 2 (1975), pp. 253-262. 23. J . R. Gould , " E x t e r n a l i t i e s , Factor Proport ions and the L e v e l of Free Access Resources ," Economica, XXXIX (1972), pp. 383-401. 24. C l i v e Southey, "Po l i cy P r e s c r i p t i o n s i n Bionomic Models: The Case of the F i s h e r y , " J o u r n a l of P o l i t i c a l Economy, LXXX, No. 4 (1972), pp. 769-775. 25. M. B. Schaefer , Some Aspects of the Dynamics of Populat ions Important  to the Management of Commercial Marine F i s h e r i e s , B u i . I , No. 1, (La J o l l a : Interamerican T r o p i c a l Tuna Commission, 1959), pp. 29-35. 26. Ralph Turvey , "Opt imizat ion i n F i s h e r y R e g u l a t i o n , " American Economic  Review, L I V , N o . . 2, Par t I (1964), pp. 64-76. 27. Smith, "On Mode l s ," p . 183. 28. Gordon Munro has pointed out to the w r i t e r that t h i s d i f f e r e n c e i n handl ing r e a l l y r e s u l t s from the use of a Schaefer versus a Beverton-Hol t b i o l o g i c a l model. For a f u l l exp lanat ion of t h i s see C o l i n W. C l a r k , Mathematical Bioeconpmics: The Optimal Management of Renewable Resources.; . (New York: John Wiley and Sons, 1976), p . 218. 29. Southey, "Po l i cy P r e s c r i p t i o n s , " pp. 773-775. 30. Lee G. Anderson, "Analys is of Open-Access Commercial E x p l o i t a t i o n and Maximum Economic Y i e l d i n B i o l o g i c a l l y and T e c h n o l o g i c a l l y Interdependent F i s h e r i e s , " J o u r n a l of the F i s h e r i e s Research Board  of Canada, XXXII, No. 10, (1975. 31. Agne l lo and Donnel ley , " E x t e r n a l i t i e s and Property R i g h t s . " 32. Agne l lo and Donnel ley , " E x t e r n a l i t i e s and Factor P r o p o r t i o n s . " 22 33. Southey, "Pol icy P r e s c r i p t i o n s . " 34. B e l l , "Technologica l E x t e r n a l i t i e s . " 35. Turvey, "Optimizat ion I n . " 36. Gould , " E x t e r n a l i t i e s and Factor P r o p o r t i o n s . " 37. I b i d . , p . 396. 38. Smith, "On Models ," p. 186. 39. I b i d - , p . 182. 40. Anderson, "Relat ionship Between Firms and F i s h e r y . " 41. U . S . Department of the I n t e r i o r , F i s h and W i l d l i f e S e r v i c e , Bureau of Commercial F i s h e r i e s , Economic Aspects of the P a c i f i c H a l i b u t  F i s h e r y , by James A. C r u t c h f i e l d and Arno ld Z e l l n e r , F i s h e r y I n d u s t r i a l Research, I , No. 1 (Washington, D . C . : Government P r i n t i n g O f f i c e , 1963). . 42. James P . Quirk and Vernon L . Smith, "Dynamic Economic Models of F i s h i n g , " i n Economics of F i s h e r i e s Management: A Symposium, ed. by A . D. Scot t , H . R. MacMil lan Lectures i n F i s h e r i e s , (Vancouver: The U n i v e r s i t y of B r i t i s h Columbia, 1970). 43. C. G. P lourde , "A Simple Model of Replenishable Natura l .Resource E x p l o i t a t i o n , " American Economic Review, LX (1970), pp. 518-522. 44. C. G. P lourde , " E x p l o i t a t i o n of Common Property Replenishable N a t u r a l Resources ," Western Economic J o u r n a l , IX, (1971), pp . 256-266. 45. Gardner M. Brown, J r . , "An Optimal Program for Managing Common Property Resources with Congestion E x t e r n a l i t i e s , " J o u r n a l of  P o l i t i c a l Economy, LXXXII (1974). 46. C l a r k and Munro, "The Economics of F i s h i n g . " 47. Michae l Spence, "Blue Whales and Appl i ed C o n t r o l Theory," T e c h n i c a l Report No. 108. I n s t i t u t e of Mathematical Studies i n the S o c i a l Sc iences , (Palo A l t o : Stanford U n i v e r s i t y , 1973). 48, 49, 23 J . A . B u t l i n , G . R. Munro, C . W. C l a r k and J . M. Tomkins, " E m p i r i c a l Es t imat ion and F i s h e r i e s Dynamics: The Manx H e r r i n g F i s h e r y , " mimeo. C o l i n W. C l a r k , "Prof i t Maximization and the E x t i n c t i o n of Animal Spec ies ," J o u r n a l of P o l i t i c a l Economy, LXXXI (1973), pp . 950-961. 50. C o l i n W. C l a r k , "The Economics of O v e r e x p l o i t a t i o n . " Sc ience , CIXC (1973), pp . 630-634. 51. Rognvaldur Hannesson, "Fishery Dynamics: A North A t l a n t i c Cod F i shery" Canadian J o u r n a l of Economics, V I I I , No. 2, (1975), pp. 151-173. 52. C l a r k and Munro, "The Economics of F i s h i n g . " 53. C r u t c h f i e l d and Z e l l n e r , "Economic Aspects of the P a c i f i c H a l i b u t F i s h e r y . " 54. P l o u r d e , "A Simple Model of Replen i shable ." 55. P lourde , " E x p l o i t a t i o n of Common P r o p e r t y . " 56. C l a r k , " P r o f i t Maximizat ion ." 57. C l a r k and Munro, "The Economics of F i s h i n g . " 58. B u t l i n , et a l . , " E m p i r i c a l Es t imat ion and F i s h e r y Dynamics." 59. Brown, "An Optimal Program," p . 165. 60. C l a r k and Munro, "The Economics of F i s h i n g . " 61. B u t l i n , et a l , " E m p i r i c a l Es t imat ion and Fi shery Dynamics." 62. C l a r k and Munro, "The Economics of F i s h i n g . " 63. Spence, "Blue Whales and A p p l i e d C o n t r o l Theory." 64. B u t l i n , et a l , " E m p i r i c a l Est imat ion and Fi shery Dynamics." 65. I b i d . , p . 19. 66. R. F . A . Roberts , "A Commercial F i s h e r i e s Product ion Funct ion: The Skeena River Sockeye Salmon G i l l n e t F i s h e r y , " Unpublished Master 's t h e s i s , (The U n i v e r s i t y of B r i t i s h Columbia, 1971). 67. B e l l , "Technolog ica l E x t e r n a l i t i e s . " 68. Gould, "Ex t e r n a l i t i e s and Factor Proportions." 69. Anderson, "Analysis of Open-Access." 70. David S. Huang and Chae W. Lee, "Toward a General Model of Fishery Production," Southern Economic Journal, XXXXIII, No. 1 (1976), pp. 846-854. 71. G. J . Paulik and J . W. Greenough, J r . , "Management Analysis for a Salmon Resource System," i n Systems Analysis i n Ecology, ed. by K. E. F. Watt, (New York: Academic Press, 1966), p. 216. 72. W. E. Ricker, "Stock and-Recruitment," Journal of the Fisheries  Research Board of Canada, XI, No. 5, (1954), pp. 559-623. 73. See, for example, P. A. Larkin, _et_al, "Some Alternative Premises f o r Constructing Theoretical Reproduction Curves," Journal of the Fisheries  Research Board of Canada, XXI, No. 5, (1964), pp.477. P. A. Larkin and A. S. Hourston, "A Model for Simulation of the Population Biology of P a c i f i c Salmon," Journal of the Fisheries Research Board of  Canada, XXI, No. 5, (1964), p. 1252. 74. P. A. Larkin and J. G. McDonald, "Factors i n the Population Biology of the Sockeye Salmon of the Skeena River," Journal of Animal Ecology, XXXVII, (1968), pp. 229-258. 75. P. A. Larkin and A. S. Hourston, "A Model for Simulation of the Population Biology of P a c i f i c Salmon," Journal of the Fisheries Research  Board of Canada, XXI, No. 5, (1964), p. 1252. 76. William F. Royce, Donald E. Bevan, James A. Crutchfield, G. J. Paulik and Robert L. Fletcher, Salmon Gear Limitation i n Northern Waters, Publications i n Fisheries, New Series, I I , No. 1, (Seattle: University of Washington, 1963). 77. Simulation i s widely used i n other branches of economics. Large scale macroeconomic simulation models are employed as forecasting t o o l s . 25 Microsimulation of markets, ind u s t r i e s , etc., are also becoming important research applications. 78. Arnold Z e l l n e r , "Management of Marine Resources: Some Key Problems Requiring Additional Analysis," i n Economics of Fisheries Management: A Symposium, ed. by A. D. Scott, H. R. MacMillan Lectures i n Fi s h e r i e s , (Vancouver: The University of B r i t i s h Columbia, 1970). 79. Organization for Economic Cooperation and Development, Directorate of Agriculture, Fisheries D i v i s i o n , "Simulation Programmes f o r Selected Fisheries," by James A. Cru t c h f i e l d , Economic Aspects of  Fish Production, International Symposium on Fisheries Economics, (Paris, 1972), pp. 152-162. 26 CHAPTER 2 A GENERAL MODEL OF A PACIFIC SALMON FISHERY 2.0 Introduct ion This chapter develops a d i s c r e t e time model of a P a c i f i c salmon f i shery under s o l e ownership as descr ibed i n chapter 1. The model i s general i n that the harves t product ion func t ion and the b i o l o g i c a l growth func t ion apply to any P a c i f i c salmon f i s h e r y employing mobile harves t ing gear. In order to obta in an e x p l i c i t numerical s o l u t i o n to the model, parameter est imates p e r t a i n i n g to the Skeena River g i l l n e t salmon f i s h e r y w i l l be employed. The p lan of the chapter i s as fo l lows: the f i r s t s ec t i on develops the b i o l o g i c a l growth func t ion a p p l i c a b l e to anadromous salmon f i s h e r i e s . This i s the well-known R i c k e r c u r v e . 1 The product ion funct ion of the harves t ing indus try i s d iscussed i n the fo l lowing s e c t i o n . T h i s product ion funct ion incorporates the h a r v e s t i n g technology appropriate to salmon f i s h e r i e s . I t s i n i t i a l development and a p p l i c a t i o n i n th i s form to a salmon f i s h e r y i s due 2 to Roberts . The harvest product ion funct ion and the b i o l o g i c a l growth func t ion comprise the core of the model. Given c e r t a i n p r i c e and cost assumptions and the so le owner's ob jec t ive funct ion def ined on net p r o f i t , a model of a s i n g l e season i s f i r s t developed. This model corresponds to a s i t u a t i o n i n which the so le owner has tenure over the f i s h e r y for one season o n l y . An e x p l i c i t g e n e r a l s o l u t i o n to t h i s model i s o b t a i n e d . T h i s s o l u t i o n r e q u i r e s the assumption t h a t r e c r u i t m e n t for the season ( i . e . , the number o f f i s h o f a s i z e l a r g e enough f o r c a p t u r e by a g i v e n n e t mesh s i z e ) i s g i v e n exogenously. 27 The f i n a l s e c t i o n of the chapter develops a f u l l i n t e r t e m p o r a l model cast i n the opt imal c o n t r o l t h e o r e t i c framework. This in ter tempora l model corresponds to a s i t u a t i o n i n which the so le owner has perpe tua l tenure over the f i s h e r y . An e x p l i c i t harvest p o l i c y i s obtained as a s o l u t i o n to t h i s model based on work by C l a r k who has shown the general form of 3 s o l u t i o n s to models of t h i s type. The assumptions requ ired to obta in t h i s s o l u t i o n are p a r t i c u l a r l y r e s t r i c t i v e and w i l l be reviewed and assessed i n l i g h t of aspects of the a c t u a l f i s h e r y to which the model i s a p p l i e d , i . e . , the Skeena R i v e r . 2.1 The B i o l o g i c a l Growth Funct ion The spawner-recrui t r e l a t i o n s h i p was developed to dea l wi th the phenomenon of endogenous recruitment rates— an important c h a r a c t e r i s t i c of P a c i f i c salmon b i o l o g y . The r e l a t i o n s h i p was o r i g i n a l l y proposed by Ricker who noted that reproduct ion curves for a number of d i f f e r e n t animal species 4 were dome shaped and asymmetrical . In more recent work on salmon popula t ion b i o l o g y R i c k e r ' s spawner-recrui t r e l a t i o n s h i p has been recas t i n t o a form e a s i l y adapted to the ordinary l e a s t squares method of f i t t i n g curves to observed data . The b a s i c concept remains unchanged. Fo l lowing P a u l i k and Greenough the r e l a t i o n between biomass and r e c r u i t s i s hypothesized to be:"* (2.1) N. . = N. : s, , . . k,r+y k»T k(x,T+y) In (2.1) k i s the race i d e n t i f i c a t i o n s u b s c r i p t , T r e f e r s to the season and y i s the l i f e span of i n d i v i d u a l f i s h . Thus, N, i s the number of f i s h of 28 race k i n season T and s, , . . i s t h e i r s u r v i v a l ra te from season x to T+y. k(T,T+y) y I f E-^j i s the number'of spawning u n i t s . i n the escapement of season T and i f i t i s assumed that egg p r o d u c t i o n , y> i s p r o p o r t i o n a l to the number of spawning u n i t s , ^ the r e l a t i o n s h i p between spawning u n i t s , E , and r e c r u i t s , R, of race k which, we assume, has a f i x e d l i f e span equal to y , i s (2.2) \ = Y k E k s k ( 0 , y ) For P a c i f i c salmon, R^ may be taken to be the number of mature r e c r u i t s to the race k . Since (2.2) i s assumed to ho ld for a l l seasons, the T subscr ipt may be omitted. N a t u r a l m o r t a l i t y i s a h i g h l y complex phenomenon. Assuming f o r the moment, however, that the m o r t a l i t y fac tors a f f e c t i n g the biomass are pure ly compensatory^ ( i . e . , the rate of m o r t a l i t y increases w i th increases i n the biomass) , s^(0,y) may be w r i t t e n as a monotonical ly decreas ing funct ion of E ^ . Some stage of the l i f e - h i s t o r y of a populat ion must e n t a i l compensatory m o r t a l i t y i f the populat ion i s to be n e i t h e r i n c r e a s i n g nor g decreas ing i n d e f i n i t e l y . For P a c i f i c salmon, compensation i s thought to occur c h i e f l y i n the e a r l y stages of the l i f e of a brood. A d u l t s competing f o r a l i m i t e d supply of good spawning s i t e s and eggs competing f o r a l i m i t e d supply of oxygen i n the stream water are examples of fac tors l e a d i n g to compensatory m o r t a l i t y at t h i s e a r l y stage. For sockeye which spend part of the j u v e n i l e stage i n the lake environment, the numbers of f r y may a l so tax the support ive capac i ty of the environment which r e s u l t s i n f u r t h e r compensatory processes." 1^ These events are assumed to be encompassed by a s u r v i v a l funct ion of the form (2.3) s k ( 0 , y ) = A k e 29 where s^(0,y) i s the ra te of s u r v i v a l from 0 to y and where X and 3 are parameters of the s u r v i v a l func t ion s ( 0 , y ) . By s u b s t i t u t i n g (2.3) in to (2.2) we obta in (2.4) R k = Y k X k E k e 11 Equat ion (2.4) i s the R i c k e r curve . I t may be f u r t h e r s i m p l i f i e d by 12 l e t t i n g ^ = X k w ^ l c n y i e l d s ( 2 ' 5 ) \ " a k E k . k^k I f both s ides of equation (2.5) are d i v i d e d by E and i f the r e s u l t i s expressed i n n a t u r a l l o g a r i t h m i c form we obta in . 13 (2.6) In = l n a k ~ P k E k For t h i s form of the R i c k e r curve the r e s t r i c t i o n on the values of a and 14 g are a > 1 and 0 > 0. Equation (2,6) may be estimated by a p p l i c a t i o n of o r d i n a r y l e a s t squares, r e g r e s s i o n techniques-. The es t imat ing equation employed by P a u l i k et a l i s (2.7) l n = l n a. S k E k + «k where £ k i s an e r r o r term which i s assumed to be a random v a r i a b l e wi th a zero mean and a f i n i t e but non-zero v a r i a n c e . 1 5 30 2.2 The Harvest ing technology Most models of commercial f i s h e r i e s assume the mass encounter f i s h i n g technology which postu lates that the catch i s p r o p o r t i o n a l to the p h y s c i a l contact between f i s h and f i s h i n g gear. In the mass encounter technology the c a t c h a b i l i t y c o e f f i c i e n t , i . e . , "the f r a c t i o n of a f i s h stock which i s 16 17 caught by a def ined u n i t of the f i s h i n g e f f o r t , " i s a constant . In the salmon f i s h e r y a v a r i a b l e c a t c h a b i l i t y c o e f f i c i e n t more a c c u r a t e l y descr ibes the a c t u a l technology. For example, the increas ing concentrat ion of f i s h as they near the mouths of r i v e r s creates congestion problems for the v e s s e l s . In a d d i t i o n , the operat ion of the f i s h i n g gear i s a f f ec ted by the s i z e of the run . G i l l n e t s used i n the Skeena River begin to s ink when they conta in more than approximately 200 f i s h . I t i s reasonable to assume 18 that t h i s threshold would be reached sooner, the greater the run s i z e . These product ion phenomena are f requent ly l a b e l l e d gear e x t e r n a l i t i e s and gear s a t u r a t i o n e f f e c t s , r e s p e c t i v e l y . To r e f l e c t t h e i r e f fec t s the product ion func t ion developed below hypothesizes a v a r i a b l e c a t c h a b i l i t y c o e f f i c i e n t . Given the above d e f i n i t i o n , the c a t c h a b i l i t y c o e f f i c i e n t i s . h 1 (2.8) = where q^ i s the c a t c h a b i l i t y c o e f f i c i e n t of v e s s e l i f i s h i n g dur ing i n t e r v a l t , h^ i s the t o t a l harvest of v e s s e l i i n the per iod t and X i s the stock of 19 f i s h a v a i l a b l e for capture during t . Given the above d e f i n i t i o n , the harvest during t of v e s s e l i i s 31 (2.9) h* = q* X f c . With n vesse l s f i s h i n g s imultaneously the t o t a l harvest dur ing t i s given by n (2.10) h f c = I q* Xt . i = l Equat ion (2.10) can be re-expressed as: (2.11) h t = q t V t X t where q f c i s the c a t c h a b i l i t y c o e f f i c i e n t f o r the f l e e t , V t i s the number of vesse l s f i s h i n g s imultaneously and h f c and X f c are as def ined above. To complete the development of the product ion func t ion i t i s necessary to s p e c i f y the v a r i a b l e s to which q t i s f u n c t i o n a l l y r e l a t e d . The s p a t i a l o r g a n i z a t i o n of vesse l s during the harvest of a salmon run i s such that vesse l s are s ta t ioned i n what may be assumed to be a g r i d p a t t e r n across the mouth of a r i v e r or i n l e t forming a gauntlet of f i s h i n g gear through which the salmon must pass on t h e i r spawning migrations, . The product ive e f f i c i e n c y of an i n d i v i d u a l v e s s e l i s a f f ec ted both by the number of other vesse l s f i s h i n g s imultaneously (crowding e x t e r n a l i t y ) and by the s i z e of the run which the gear w i l l encounter during that time i n t e r v a l (gear s a t u r a t i o n ) . Thus, the c a t c h a b i l i t y c o e f f i c i e n t i s hypothesized to be f u n c t i o n a l l y r e l a t e d to the number of vesse l s f i s h i n g and the run s i z e o r , (2.12) q f c = q t ( V t , X t ) 32 where a l l v a r i a b l e s are as def ined above. When (2.12) i s subs t i tu ted in to (2.11) the t o t a l harvest i n time i n t e r v a l t i s (2.13) h t = q t ( v t , x t ) v t x t . Our hypothes is i s that the s p e c i f i c f u n c t i o n a l form of (2.12) i s Cobb-Douglas, b u t , u n l i k e most s p e c i f i c a t i o n s of the Cobb-Douglas f u n c t i o n , the exponents 6^ and 6 2 are hypothesized to be negat ive . Given these pos tu la te s , (2.12) becomes 6 1 6 2 (2.14) q ( V t , X t ) = A V t X ^ , -1 <&1 < 0, -1 < 6 2 < 0. The s ign r e s t r i c t i o n s on 6^  and 5 2 r e f l e c t the e f f ec t s of the crowding and 21 gear e x t e r n a l i t i e s descr ibed above. When (2.14) i s subs t i tu ted i n t o (2.13) the f i s h e r y product ion func t ion i s obta ined. Thus 6.+1 6-+1 (2.15) h = AV X . t t t While the r e s u l t (2.15) could have been postulated at the outset as the product ion func t ion of the salmon f i s h e r y , i t s d e r i v a t i o n from (2.8) l i n k s the more general formulat ion us ing a v a r i a b l e c a t c h a b i l i t y c o e f f i c i e n t to the more customary p r o p o r t i o n a l i t y assumption. In order to estimate the parameters 6^  and 6^ of (2.15) as w e l l as the con-stant A , a l ogar i thmic transformation of (2.15) i s performed wi th the r e s u l t (2.16) l n h = l n A + (6.+1) In V + (6 0 +l) In X t 1 t 2 t Equat ion (2.16) i s l i n e a r i n the logarithms of the v a r i a b l e s h , V and X . ^ t t t Assuming that equation (2.16) and the observed data with which i t i s estimated 33 meet the b a s i c assumptions of the ord inary l e a s t squares m u l t i p l e r e g r e s s i o n •model i t may be w r i t t e n i n est imat ing form and employed to obta in estimates of 22 A , 6 ^ and S ^ - • The es t imat ing equation i s (2.17) In h t = In A + (6+1) In V + (<$2+l) In X f c + ? f c , where £ i s the e r r o r term, which i s assumed to have the same p r o p e r t i e s as that of equation (2 .7 ) . 2.3 A S ing le Season Model 2 .3 .1 Development of the Objec t ive Funct ion The r a t i o n a l e f o r development of the s i n g l e season model i s that i t acts as a b u i l d i n g b lock f o r the models which f o l l o w . I t provides a s i m p l i f i e d con-text f o r i n t r o d u c t i o n of the v a r i a b l e t ime-o f -entry r e l a t i o n and, i n so do ing , contras ts wi th the t ime-o f -entry assumption employed i n the opt imal c o n t r o l model which f o l l o w s . F i n a l l y , i t e s tab l i shes a l i n k between the s ing le - season models i d e n t i f i e d i n chapter 1 and the wi th in-season behavior of the s imula t ion model developed i n the fo l lowing chapter . Given the product ion func t ion developed above and the usua l assumptions of constant r e l a t i v e f i s h p r i c e s and costs per vesse l -day of f i s h i n g i t i s p o s s i b l e to cons truct a model which provides a r u l e to guide e f f i c i e n t e x p l o i t a t i o n of the f i s h e r y dur ing a s i n g l e season with no c o n s i d e r a t i o n for future seasons. The model requ ires for i t s s o l u t i o n the assumption that recrui tment i s g iven by nature and past l e v e l s of harves t ing a c t i v i t y . Thus, (2.18) C t = b V t , C ' = b i s the cost equation and i t i s assumed that b i s the f u l l opportuni ty cost of operat ing a v e s s e l dur ing i n t e r v a l t , a s u b d i v i s i o n of the t o t a l 34 length of the season. For the f i s h e r y under cons idera t ion t = ( 1 , 2 , . . . , 1 5 ) s ince f i f t e e n weeks i s the appropriate average season l e n g t h . The o b j e c t i v e of the s o l e owner i s to maximize p r o f i t s f or the season 23 g iven that the f l e e t s i z e i s the only choice v a r i a b l e . F o r m a l l y , 15 (2.19) n = E pWh. - c t = l t t i s to be maximized subjec t to the product ion f u n c t i o n , (2.20) h t = A V t 6X+1 6 2 +l and 15 (2.21) Z h. = H < R . t = l In (2.19) IT i s net p r o f i t f or the season, p i s the p r i c e of f i s h i n cents per pound, W i s the weight, i n pounds, of a f i s h , and h t and C f c are as def ined e a r l i e r . In (2.21) R i s the t o t a l recrui tment for the season. The harves t f o r each time i n t e r v a l has been hypothesized to depend upon the number of vesse l s f i s h i n g s imultaneously dur ing each i n t e r v a l and the s i z e o f the run encountered by those vesse ls as shown by equation (2.20).. P r e v i o u s l y i t was s tated that for t h i s model, R, the t o t a l recrui tment for the season i s assumed to be g iven. With t o t a l recruitment g iven a procedure i s r equ ired for a l l o c a t i n g t h i s t o t a l to each of the i n t e r v a l s w i t h i n the season. The procedure for performing t h i s a l l o c a t i o n has i t s b a s i s i n the a c t u a l f i s h e r y and i s u s u a l l y r e f e r r e d to as the time-o f - e n t r y r e l a t i o n s h i p . I t s form i s (2.22) X t = R f ( t ) . 35 Based on e m p i r i c a l data the t ime-o f -entry r e l a t i o n s h i p for a t y p i c a l salmon f i s h e r y i s observed to be a polynomial express ion i n t which has the general shape shown i n F igure 2-1 below. The curve descr ibes the accumula 15 t i v e stock o f f i s h such that £ X = R. The system ( 2 .19) - ( 2 . 2 2 ) i s t= l now completely determinate . Thus, from ( 2 .19) the problem i s to maximize ( 2 . 2 3 ) : ( 2 . 2 3 ) n t = P w h t - c t subject to ( 2 . 2 0 ) and to h f c > 0 , X f c = R/15, R > 0 and p > 0 . S u b s t i t u t i n g the product ion f u n c t i o n , ( 2 . 2 0 ) , and the cost equat ion , ( 2 .18) d i r e c t l y in to the o b j e c t i v e f u n c t i o n , (2 .19) , y i e l d s &x+l 6 2 +l (2.24) n t = pWAVt Xt - bVt 25 where W i s the average weight of a f i s h . The f i r s t order c o n d i t i o n for maximizing equation (2.24) i s (2.25) 9 n t 3V~ t (fii+l) V * 6 1 X ^ 2 + 1 - b = 0 FIGURE 2 - 1 THE TIME OF ENTRY RELATIONSHIP f ( t ) R 36 By s o l v i n g (2.25) for V we obta in an input demand equation for The r e s u l t i s (2.26) V* ( 6 + 1 ) fc •|pW(61+i)A x t J _1 6 , * In order to e s t a b l i s h that the va lue V maximizes ra ther than minimizes the t o b j e c t i v e func t ion (2 .23) , (2.25) i s d i f f e r e n t i a t e d a second time wi th respect to V . D i f f e r e n t i a t i o n y i e l d s 321T 6 - 1 6 + 1 (2.27) : § - = :PWA ( 6 + 1 ) 6 V 1 X C 3V 6 r l 6 2+l In (2.27) the terms p , W, A , ( 6 ^ + 1 ) , V and X are a l l greater than zero . 2 2" The only remaining terms, 6 ^ , i s between zero and minus one. Therefore ,8 II/3V jiO * which e s tab l i shes that V maximizes (2 .25) . I t should be emphasized that s ince * V i s a f u n c t i o n of time v i a X(t ) i t w i l l vary through the p e r i o d . The model developed i n t h i s s ec t i on i s r e s t r i c t i v e but nevertheless in format ive . I t i s r e s t r i c t i v e i n that i t considers only one season i n i s o l a t i o n . Manipula t ion of the current harvest to r e f l e c t the investment aspects of biomass management i s not p o s s i b l e i n the contest of t h i s model. I t i s in format ive i n i t s showing that the conc lus ions of standard product ion theory are a l so reached i n a t y p i c a l model of salmon h a r v e s t i n g . More important ly i t i s found that the so le owner who i s concerned only wi th maximization of current p r o f i t s w i l l no t , g iven the p r i c e and cost assumptions, completely l i q u i d a t e the biomass. In the fo l lowing s e c t i o n the s i n g l e season assumption i s re laxed i n favor of an in ter tempora l model which e x p l i c i t l y incorporates the e f f e c t of current harvests on future net p r o f i t s . The ma. :imand i n the in ter temporal 37 model i s the present va lue of the f i s h e r y which requires o p t i m i z a t i o n of current and future harvests over a l l t ime, g iven the so le owner's discount r a t e . 2.4 An Intertemporal Model of a Salmon F i s h e r y 2 .4 .1 Development of the Objec t ive F u n c t i o n a l The in ter tempora l model of a salmon f i s h e r y set up as a problem i n opt imal c o n t r o l theory focuses a t t e n t i o n on the determinat ion of opt imal escapement from the h a r v e s t . Harvest i s the c o n t r o l v a r i a b l e and escapement i s the s ta te v a r i a b l e i n such a problem. T h i s s e c t i o n concerns the develop-ment of an opt imal c o n t r o l model and i t s s o l u t i o n i n e x p l i c i t f u n c t i o n a l form. In a d d i t i o n , a numerical s o l u t i o n to the model i s attempted based 2fi upon parameter values estimated from observed data f o r the Skeena R i v e r . Consider the f o l l o w i n g d e f i n i t i o n s and In terseasona l r e l a t i o n s h i p s for any season, T , of length T . Define as the number of vesse l s f i s h i n g dur ing season x. Assume that t h i s e f f o r t r a t e remains constant throughout the season. For t h i s model i t i s assumed that a l l seasons are i d e n t i c a l with respect to the opt imal escapement program. The durat ion of the season i s d i v i d e d i n t o a s e r i e s of sub-periods such that 0 ;< t ^ T . During any p a r t i c u l a r season, T , i t i s assumed that no n a t u r a l m o r t a l i t y occurs . Thus, f i s h i n g m o r t a l i t y i s the only cause of reduct ion of escapement during the season. As i n the s i n g l e season model l e t X(T) denote the f i s h encountered by the gear i n each of the sub-periods t . Thus, the ra te of dec l ine of the a v a i l a b l e f i s h stock as the season progresses i s equal to the h a r v e s t , o r , 38 (2.28) X = - ^ 1 ^ - = - h ( t ) The product ion funct ion developed for the s i n g l e season model appl ies d i r e c t l y to the in ter tempora l model. The product ion funct ion i s w r i t t e n & +1 6 2 +l (2.29) h ( t ) = AV 1 X t Sign r e s t r i c t i o n s on 6^ and 6 2 a re , -1 < 6^ < 0, -1 < 6 2 < 0, as be fore . A i s the c a t c h a b i l i t y c o e f f i c i e n t for the f l e e t as a whole and i s assumed to be a constant . The l a t t e r assumption i s j u s t i f i e d by the s i m p l i f i c a t i o n i t provides toward the s o l u t i o n of the model. Long-term models of f i s h e r i e s t y p i c a l l y assume that the p r i c e of f i s h i s independent of smal l changes i n the output of the f i s h e r y under a n a l y s i s s imply because the output changes are assumed to be smal l i n r e l a t i o n to the t o t a l quant i ty of f i s h products . This assumption i s adopted i n the present study. L i k e the p r i c e assumptions, the cost assumptions for t h i s model correspond to those employed i n the s i n g l e season model , i . e . , costs per u n i t e f f o r t are constant . T o t a l cost i s given by (2.30) C t = b V x , c x = b Given (2.28) and p r i c e and cost assumptions s tated above i t i s now p o s s i b l e to wr i te the p r o f i t funct ion for a t y p i c a l season. From (2.29) T H = | h ( t ) d t . Thus , 0 39 (2.31) n(R , H ) = pH - bV T T T T The so le owner's ob jec t ive i s to maximize the value of the time stream of net p r o f i t s which requires the maximization of the objec t ive f u n c t i o n a l ( 2 . 3 2 ) 2 7 OO (2.32) J = I a T _ 1 n ( R T , H T ) , a = ( j ^ ) * In (2.32) J i s the sum of discounted net p r o f i t s , a i s the discount f a c t o r , r i s the discount ra te of the so le owner and <j> i s a parameter whose value i s determined by the l i f e span of i n d i v i d u a l f i s h . The i n t e r s e a s o n a l b i o l o g i c a l growth equation may be w r i t t e n i n the general f u n c t i o n a l form (2.33) R = F(R - H ) T+(J> T T 2 8 Fol lowing C l a r k the Hamiltonian f o r t h i s d i s c r e t e opt imal c o n t r o l problem i s (2.34) H = a T - 1 n ( R , H ) + X [ F ( R - H ) - R ] \ / X T T T T T In (2 .34) , X_ i s the a d j o i n t v a r i a b l e . From a p p l i c a t i o n of the d i s c r e t e maximum p r i n c i p l e one obtains the fo l lowing equation for an e q u i l i b r i u m o n s o l u t i o n which, when s a t i s f i e d , maximizes (2.34): jm _m (2.35) F ' ( E ) - 8 R 3 I I 8 H = (1+r)* "9H 40 F(E) i s the f u n c t i o n a l no ta t ion form of the Ricker curve developed above (equation 2 .5 ) ; E re fers to escapement and F 1 ( E ) i s , of course , the d e r i v a t i v e of recruitment wi th respect to escapement. The meaning of the other terms i n (2.35) i s s e l f evident from (2.32) and (2.34). From the b io - technology o f the salmon f i s h e r y by d e f i n i t i o n the t o t a l recrui tment f o r a season i s d i v i d e d between harvest and escapement. Thus the t o t a l quant i ty of f i s h a v a i l a b l e f o r harves t ing at the beginning of the season, i . e . , t=0, i s X(0) = R T whi le the quant i ty remaining at the c lose of the l a s t p e r i o d of the season i s X(T) = E . Since by assumption there i s no n a t u r a l m o r t a l i t y occuring w i t h i n the season, the ra te of change of the f i s h stock i s due s o l e l y to harves t . Equation (2.28) shows t h i s c l e a r l y . Thus the i n t e g r a l of harvest with respect to stock present i n the f i s h e r y , X , must equal the i n t e g r a l of the catch per u n i t e f f o r t times the e f f o r t ra te wi th respect to t ime. Thus, d i v i d i n g both s ides of (2.29.) 6 2+1 by X and i n t e g r a t i n g the r e s u l t , we get: (2.36) rEx , fT 6 + 1 6 + 1 1 dX:= - AV dt = - A T V , 6 + 1 ^ j R X 0 T The object i s to obta in an express ion for V from (2.36) which may be s u b s t i t u t e d i n t o the p r o f i t f u n c t i o n , (2 .31) . By performing the necessary operat ions one obtains I -T f t i h (2.37) V„ T^1 AT E r T 1 - X + T d x R X T 6+1 A l l the components of the p r o f i t func t ion for season T have now been obta ined . By s u b s t i t u t i n g (2.37) and the i n t e g r a l form for the t o t a l 41 harvest during season T into (2.31) one obtains (2.38) n(R ,H ) = p h(t)dt - b T T > 0 1_ AT E f T R 6 2 + 1 The p a r t i a l derivatives of the p r o f i t function, equation (2.38), together with the derivative of the Ricker curve, equation (2.5), form the main components of the optimal equilibrium escapement p o l i c y equation (2.35) which maximizes the Hamiltonian expression, equation (2.34). The process of obtaining the p a r t i a l derivatives and manipulating them i n order to obtain an expression i n the single v a r i a b l e , escapement (E) i s described below. From (.2,3.1) by ignoring the t subscript (since a l l seasons are assumed to be i d e n t i c a l ) one obtains (2.39). n = pH - bV and from (2.37) (2.40) y -. J jr R T X (6 2+1) The expression within the brackets w i l l be referred to as S; therefore, 1 (2.41) v T = S (tSj+D i s equivalent to (2.40). D i f f e r e n t i a t i n g (2.39) with respect to i t s arguments V and X one obtains 911 (2.42) ^ = P 3 H B _ 9 X 42 and (2.43) 3n 3H = P 3V 3H The procedure w i l l be to find an explicit expression in terms of the model developed in chapter 2 for the partial derivatives (2.42) and (2.43). Thus, differentiating (2.40) with respect to X yields, 1 6 +1 1 - 1 (2.44) 3 V = 1 3X (S-,+1) (S) as 3R -6 1 6,+l 1 (6 9+l) (6+1) X (X-H) 1 Similarly, differentiating (2.40).with respect to H yields (2.45) 3V 3H (6^1) (S) as • 3 H s 6 l + 1 i (5-,+D • "AT (6 2 +D (X-H) Furthermore, (6+1) 6 (2.46) . H = (62+l)AV 1 X 2 . The general form equilibrium solution stated by Clark i s as given in equation (2.45)- Substituting the expressions (2.42) and (2.43) into the appropriate locations in (2.35) and ignoring the other members of (2.35) yields 43 p M - b ^ + P - b ^ v (2.47.) ^ §H 3H which s i m p l i f i e s to (2.48) p ( ax* ~ ( 9 x + 3H} 3 H From (2.44) and (2.45) the term on the RHS of the minus s ign i n (2.48) i s (2 49) (3V + 9V) = - S i 6 1 + 1 L__ s . -I AT X (6 2 +l) (X-H) (6 2 + l ) -6 6 j+1 1 1 AT (X-H) 'l+l (6X+1) _1 AT (6 2 +l) : + . LX (62+l) (6"2+D (X-H) (X-H) -6, 1 <i« 1 1 T^+i) AT (6 9 +l) X Z By s u b s t i t u t i n g (2.46) and (2.49) in to (2.35) one obtains 44 6,+l 6 2 p{l + [ (6 ? +l)AV X ] i - b 6,+l S . _ ± 1 . 1 AT (52+1) X (2.5Q) F ' ( E ) - b («!+!) S . 1 AT (X-H) (62+1) (1+i) Completion of the d e r i v a t i o n requ ires only the i n t e g r a t i o n and s i m p l i f i c a t i o n of S and the d i f f e r e n t i a t i o n of the Ricker curve F ( E ) . From (2.40) the i n t e g r a t i o n of S y i e l d s (2.51) AT 1 - ( S 2 + 1 ) l - ( 6 , + l ) l-(6 +1) [R 1 - (R-H) ] Given that X(0) = R. Then 1 6, 1 r 6-(2.52) S A T [ l - ( 6 +1)] R l - ( 6 2+D - E l - ( 6 2+D given that R-H = E for the salmon f i s h e r y . S u b s t i t u t i n g (2.51) and (2.52) in to (2.50) y i e l d s 45 (2.53) F'(E) 1 + [(6 2+l)A (JL \ AT l-(6 2 +D R ) -5. p-b iS.,+1 r AT[l-(6 2+l)] "-5. l-(5 +1) l-(6 +1)1 K R -E . 1 1 1 AT (6+1) E Z T_ f 1 V l + 1 6 1 [ A T[1-(6 2+1)] 1-(S +1) l- ( 6 9 + l ) R - E Z '1 1 AT (5-+1) R p-b (S±+l) /AT[l-(6 2+l)] ' 6 1 + 1 l-(6_+l) [R 1_ 5, l-(5 7 +D E ] _1_ AT ( « 2 + D = (1+r) The form of the Ricker curve employed In this-model i s the same as 30 that for the single season model of chapter 2. That i s (2.54) F(E) = R = | e a ( 1 " E / K> K where R i s recruitment i n stock units and E i s escapement i n numbers of f i s h ; a i s the parameter of productivity and K i s the natural equilibrium recruitment. The derivative of (2.54) i s (2.55) F'(E) = M = I e a d - E / K ) ( l - E a } SE K K Substitution of (2.54) and (2.55) into (2.53) as indicated y i e l d s an expres-sion i n the single variable E. Due to the unwieldy size of the expression the substitutions are not carried out. The re s u l t i n g equation can be 46 expressed for convenience as (2.56) F ' ( E ) (O - (1+r) = 0. The opt imal e q u i l i b r i u m escapement can be determined by t r a n s f e r r i n g the term (1+r)^ to the LHS o f (2.53) and f i n d i n g the roots of the r e s u l t i n g equat ion . Thi s exerc i se was performed for the equation (2 .56) . The process and r e s u l t s are descr ibed i n the fo l lowing s e c t i o n s . 2 .4.2 An Optimal Numerical So lu t ion The parmeters employed to obta in a numerical s o l u t i o n to the opt imal escapement p o l i c y have been estimated from a v a r i e t y of sources . The cost of operat ing a v e s s e l i s taken from a study conducted by the 31 F i s h e r i e s and Marine Serv ice during the per iod 1966 to 1968. Several adjustments were app l i ed to the estimates there in to r e f l e c t more recent trends i n costs and r e t u r n s . These adjustments are d e t a i l e d i n Appendix A. F i s h p r i c e s were estimated from a c t u a l landed p r i c e s pa id i n the B r i t i s h Columbia salmon f i s h e r y over the per iod 1951-1974,^ 2 Average weights f o r f i s h were obtained from b i o l o g i s t s respons ib le for management of the Skeena R i v e r f i s h e r y . The parameters of the product ion f u n c t i o n , i . e . , the va lue of the constant A and the values of 6-^  and &2> w e r e obtained from an es t imat ion of the Skeena R i v e r product ion f u n c t i o n conducted by Roberts . Other parmeters requ ired are the discount r a t e , the length of the l i f e cyc l e of Skeena River sockeye and the parmeters of the Ricker curve . The root of (2.56) has been obtained for two values of the discount r a t e , i . e . , 5% and 10% per annum. The populat ion b io logy of the P a c i f i c salmon of the Skeena River i s discussed i n greater d e t a i l i n chapter 3. 47 In that d i s c u s s i o n i t i s noted that Skeena River sockeye evidence a v a r i a b l e l i f e cyc le ranging from 3 to 6 years i n durat ion from the egg stage to adulthood. However, the intertemporal model developed i n the prev ious s e c t i o n i s capable of handl ing only a l i f e cyc le of constant length.. The root of the equation was obtained for values of <J)=4 and <J>=.5. Two p h i - v a l u e s were used to obta in separate so lu t ions to C2.56) because Skeena River sockeye are approximately evenly d i v i d e d between f i sh , of a four^year and f i v e - y e a r l i f e c y c l e . F i n a l l y - , the parameters of the Ricker curve a p p l i c a b l e to the Skeena R i v e r are an qlpha^value of 2.1 and a K-value of 1.1 m i l l i o n . These est imates were obtained from unpublished work through personal communi-35 c a t i o n with: the researcher who performed the e s t imat ion . A f u l l d i s p l a y of the parmeter values used to obta in the root of (2.56) i s provided by Table II;. 48 TABLE I I PARAMETERS EMPLOYED IN SOLUTION TO OPTIMAL ESCAPEMENT FOR INTERTEMPORAL MODEL PARAMETER ESTIMATE Cost per vessel-day Price per f i s h ($0.6239/lb x 6.2 l b s / f i s h ) Production function estimates Catchability c o e f f i c i e n t , A E l a s t i c i t y of output WRT vessel-days, E l a s t i c i t y of output WRT recruitment, 6 2+l Length of season, T Discount r a t e , r Length of sockeye salmon l i f e cycle, <j> • Ricker curve parameters: Biomass productivity, a Natural equilibrium stock s i z e , K *Also solved for 75 days - see discussion of re s u l t s . **Also solved for 5% per annum - see discussion of r e s u l t s . //Also solved f o r <{>=5 . $250.00 3.87 0.5410 0.3335 0.7788 45 days* 10% per annum** 4 years// 2.1 1.1 m i l l i o n 49 Example p l o t s of equation (2.56) f or two combinations of parameter values d i s p l a y e d i n Table I I I are shown i n F igure 2-2. The two curves shown are f o r a 45-day season and 5% and 10% annual discount rates as i n d i c a t e d . An a p p l i c a t i o n of Newton's method was used to f i n d the root of the equation which , f o r the parameter values employed, i s shown to be approximately 0.33 and 0.36 r e s p e c t i v e l y f o r a 10% and a 5% d iscount r a t e . Table I I I below d i s p l a y s the exact values of the root obtained f o r combinations of season length and discount r a t e . TABLE I I I ROOTS OF EQUATION 2.56 FOR VARIOUS VALUES OF SEASON LENGTH AND DISCOUNT RATE AND LIFE CYCLE LENGTH DISCOUNT RATE/LIFE CYCLE LENGTH  SEASON r=5% r=10% r=5% r=10% LENGTH j>=4 (j>=4 ^5 (j>=5 45 days 0.35864258 0.33461914 0.35258789 0.32133789 (394,507) (368,081) (387,847) . (353,472) 75 days 0.35571289 0.33090820 0.34946289 0.31723633 (391,284) (363,999) (384,409) (349,496) The roots shown i n Table H I are i n stock u n i t s . To convert them to numbers of f i s h simply requires that we m u l t i p l y them by 1.1 m i l l i o n , the n a t u r a l e q u i l i b r i u m stock s i z e , K, estimated for the Skeena R i v e r . The r e s u l t s of performing t h i s operat ion are presented i n parentheses below each s o l u t i o n . For the combinations of parameter values employed i n obta in ing values of the root of equation (2.56), opt imal escapement of sockeye from the i I 51 Skeena River gillnet fishery lies in the range of approximately 350,000 to 395,000 fish. Examination of Table III reveals that the value of the discount rate appears to have the most substantial effect upon optimal escapement of any of the parameter changes considered. A comparison of column 1 to column 2 and column 3 to column 4 shows the effect of a discount rate change. The parameter change with the least effect upon optimal escapement is a change in the length of the season. The length of the l i f e cycle of the fish has a reasonably substantial impact on the optimal escapement which suggests that a model which could handle more than one l i f e cycle simultaneously would constitute a marked improvement over the present model. The escapement figures of -Table III compare to an historical escapement which averaged approximately 600,000 sockeye over the past 35 years. It is tempting to conclude from this comparison that the fishery has historically been managed to allow too much escapement. This conclusion would probably be unwarranted in view of the many simplifying assumptions which have been made. In particular, in the modeled fishery, everything is smooth and deterministic—the recruitment is automatically determined from a given escapement and the vessels operate so as to harvest at the optimal level. In the actual fishery for example, escapement ranged from a minimum of 110,000 sockeye to a maximum of 1.2 million sockeye over the period used to calculate the 600,000 average sockeye escapement. Harvesting is not as finely controlled in the actual fishery as i t is in the model. Finally, while the real objectives of management of the Skeena River are diffuse and diverse i t is safe to assume that is is not managed solely to maximize its value as a productive asset. A host of other factors could be mentioned as possible causes of the difference in the actual and modeled escapement figures. 52 This w i l l be l e f t to the next s e c t i o n of the chapter i n which a review and assessment w i l l be made of the s i m p l i f y i n g assumptions employed to obta in the s o l u t i o n to the opt imal c o n t r o l problem. 2 . 4 . 3 . Assessment of the Intertemporal Model The in ter tempora l model developed and solved i n the previous two sec t ions provides u s e f u l i n s i g h t i n t o the a n a l y t i c s of a s i m p l i f i e d f i s h e r y . However, from the po int of view of p o l i c y and informat ion of i n t e r e s t to the managers of the a c t u a l f i s h e r y , the a n a l y t i c a l s o l u t i o n i s obtained at the cost of a lo s s of a s i g n i f i c a n t degree of s i m i l a r i t y between the a c t u a l f i s h e r y and that which e x i s t s i n the model. In t h i s s e c t i o n i t i s proposed to i d e n t i f y those assumptions which are thought to be p a r t i c u l a r l y r e s t r i c t i v e i n t h i s respec t . A genera l c h a r a c t e r i s t i c of the opt imal c o n t r o l model i s i t s determinism. While t h i s assumption i s necessary for s o l u t i o n of the model i t l i e s q u i t e apart from the character of the a c t u a l f i s h e r y to which the model i s a p p l i e d . One of the most s t r i k i n g features of the salmon f i s h e r y i s the n a t u r a l f l u c t u a t i o n which i s exempl i f ied by the escapement s t a t i s t i c s c i t e d i n the previous s e c t i o n . The a c t u a l f i s h e r y could be more accurate ly descr ibed as s t o c h a s t i c than d e t e r m i n i s t i c and, from the viewpoint of guidance to management, conclusions drawn from a model i n c o r p o r a t i n g s t o c h a s t i c features would have more f o r c e . The outgrowth of the assumption of determinism (or , perhaps, examples of i t ) are the assumptions that a l l seasons are a l i k e i n e q u i l i b r i u m and the constancy of the e f f o r t ra te dur ing a l l seasons. To the contrary , the l o g i c a l extension of s t o c h a s t i c in f luences i s that a l l seasons w i l l not be 53 a l i k e and that n a t u r a l f l u c t u a t i o n s w i l l cause f l u c t u a t i o n s i n the so le owner's e x p l o i t a t i o n a c t i v i t i e s . The determinism of the opt imal c o n t r o l t h e o r e t i c model could be correc ted by the i n t r o d u c t i o n of a s t o c h a s t i c element in to the b i o l o g i c a l model. T h i s s t o c h a s t i c model could then have been solved by the use of the technique of s t o c h a s t i c dynamic programming. T h i s p a r t i c u l a r avenue was not pursued s ince the shortcomings i d e n t i f i e d i n the remainder of t h i s s ec t i on would s t i l l apply to the s t o c h a s t i c model as w e l l . The i n t r a s e a s o n a l b io - technology of the opt imal c o n t r o l model i s p a r t i c u l a r l y s i m p l i f i e d . Equat ion (2.28) s ta tes t h a t , dur ing the season, the r a t e of change of the s tock encountered by the gear i s equal to the harvest r a t e . T h i s means that the annual recrui tment i n t o t a l i s present on day one of the f i s h e r y and remains i n the area (excepting those f i s h harves ted , of course) u n t i l the c lose of the harves t ing season. Those f i s h remaining at t h i s po int escape to spawn. In c o n t r a s t , i n the a c t u a l f i s h e r y , recrui tment takes p lace over the f u l l span o f the season i n a flow p a t t e r n . F i s h enter the h a r v e s t i n g a r e a , remain there f o r a short time and, i f not captured, escape to spawn. Thus, the r a t e o f change o f s tock present i n the h a r v e s t i n g area i s made up o f harves t ing and escapement dur ing each time i n t e r v a l w i t h i n the season ra ther than s imply harves t ing as assumed by the in ter tempora l model. The opt imal c o n t r o l t h e o r e t i c model i s capable of handl ing only one 36' species i n i t s present s ta te of mathematical development. Most P a c i f i c salmon f i s h e r i e s conta in m u l t i p l e species which are both b i o l o g i c a l l y and t e c h n o l o g i c a l l y (and therefore economical ly) interdependent . In a d d i t i o n to the presence of more than one spec i e s , most species conta in an age d i s t r i b u t i o n w i t h i n the biomass. Sockeye salmon are among the species e x h i b i t i n g an age d i s t r i b u t i o n . The opt imal c o n t r o l t h e o r e t i c model conta in ing t h i s l a t t e r feature has not yet been so lved . Given the complexity of even the s i m p l i f i e d model developed above, one can env i s ion the mathematical complexity of a m u l t i - s p e c i e s , age s t ruc ture model. Yet t h i s i s the nature of the f i s h e r y to which the model i s to apply . The f i n a l comment on the assumptions of the c o n t r o l t h e o r e t i c model i s d i r e c t e d towards the b i o l o g i c a l growth equation used i n the model. This funct ion was assumed to encompass compensatory m o r t a l i t y only; i . e . , i t i s a pure ly compensatory model. Two other m o r t a l i t y phenomena have been i d e n t i f i e d wi th salmon popula t ion dynamics both of which are important i n e x p l a i n i n g the l a r g e f l u c t u a t i o n s which take p lace i n observed recrui tment . These phenomena are depensatory and extrapensatory m o r t a l i t y which are def ined r e s p e c t i v e l y as a negat ive r e l a t i o n between m o r t a l i t y rate and biomass and a m o r t a l i t y ra te which i s independent of biomass. A more complete m o r t a l i t y model which generates f l u c t u a t i o n s i n recruitment leads to a model which i s more e a s i l y adapted to ana lys i s of management problems a c t u a l l y encountered. I t i s c l e a r from the above d i scuss ion that the s o l u t i o n to the opt imal c o n t r o l model comes at the cost of s a c r i f i c e d r e a l i s m and therefore i n f o r m a t i o n a l content . T h i s p o t e n t i a l l y ser ious shortcoming suggests that other p o s s i b l e approaches to modeling the f i s h e r y should be cons idered. I t i s p o s s i b l e that cross-comparisons of the r e s u l t s of a l t e r n a t i v e approache to modeling can y i e l d f u r t h e r i n s i g h t in to the system being modeled. 2.5 T r a n s i t i o n to S imulat ion Approach The w i t h i n season and in ter tempora l d e t e r m i n i s t i c , a n a l y t i c a l models developed i n t h i s chapter have served a u s e f u l purpose i n a d d i t i o n to the r e s u l t s provided by t h e i r s o l u t i o n . The reader should now have a reason-55 ably complete grasp of the b io- technology b a s i c to a P a c i f i c salmon f i s h e r y . The approach adopted i n the model descr ibed i n the fo l lowing chapter i s to preserve the c e n t r a l b i o - t e c h n i c a l r e l a t i o n s h i p s employed i n these a n a l y t i c a l models but to b u i l d around these c e n t r a l r e l a t i o n s h i p s a s t r u c t u r e which renders the r e s u l t i n g model more r e a d i l y recognizable as apply ing to a p a r t i c u l a r salmon f i s h e r y . T h i s i s accomplished by tak ing note o f the s i m p l i f y i n g assumptions of the in ter tempora l model i d e n t i f i e d i n the previous s e c t i o n and attempting to prov ide for t h e i r r e l a x a t i o n i n the s i m u l a t i o n model. The r e s u l t i n g model i s more complex and, as a r e s u l t of t h i s complexi ty , i t cannot be solved i n the same sense as the models of t h i s chapter have been s o l v e d . To be sure , numerical va lues are obtained for the c o n t r o l v a r i a b l e s and the s tate v a r i a b l e s ; however, i t i s not p o s s i b l e to f i n d a r u l e or c o n t r o l p o l i c y which may be fol lowed f o r a l l time such as was p o s s i b l e for the opt imal c o n t r o l model. Rather , the model s a c r i f i c e s the opt imal s o l u t i o n i n favor of a more h e u r i s t i c model which w i l l h o p e f u l l y y i e l d more management informat ion than can be provided by, the :opt imal contro l /model of t h i s chapter . 56 FOOTNOTES TO CHAPTER 2 1. W. E . R i c k e r , "Stock and Recruitment," J o u r n a l o f the F i s h e r i e s  Research Board of Canada, X I , No. 5, (1954), pp. 559-623. 2. R. F . A . Rober t s , "A Commercial F i s h e r i e s Product ion F u n c t i o n : The Skeena R i v e r Sockeye Salmon G i l l n e t F i s h e r y , " Unpublished Master ' s Thes is (The U n i v e r s i t y of B r i t i s h Columbia, 1971). 3. C o l i n W. C l a r k , Mathematical Bioeconomics: The Optimal Management of  Renewable Resources, (New York: John Wiley and Sons, I n c . , 1976), pp p p . 245-248. 4. R i c k e r , "Stock and Recruitment," p . 562 and pass im. 5. P . A . L a r k i n , R. F . Rale igh and N. J . Wil imovsky, "Some A l t e r n a t i v e Premises f o r Cons truc t ing T h e o r e t i c a l Reproduction Curves ," J o u r n a l  of the F i s h e r i e s Research Board of Canada, XXI, No. 5 (1964), p . 477. 6. G. J . P a u l i k and J . W. Greenough, J r . , "Management A n a l y s i s f o r a Salmon Resource System," i n Systems A n a l y s i s i n Ecology , ed. by K. E . F . Watt, (New York: Academic P r e s s , 1966), p . 224-233. This development i s due to P a u l i k and Greenough. Notat ion has been changed to correspond to n o t a t i o n used elsewhere i n t h i s s tudy. 7. C l a r k , Mathematical Bioeconomics, p . 17. 8. P . A . L a r k i n and A . S .Hours ton , "A Model for S imulat ion of the Populat ion Biology of P a c i f i c Salmon," J o u r n a l of the F i s h e r i e s Research  Board of Canada, X X I , No. 5, (1964), p . 1252. 9. I b i d . , p . 1253. 10. I b i d . , p . 1253. 11. G. J . P a u l i k , A . S. Hourston and P. A. L a r k i n , " E x p l o i t a t i o n of M u l t i p l e Stocks by a Common F i s h e r y , " J o u r n a l of the F i s h e r i e s Research Board of Canada, XXIV, No. 12 (1967), p . 2530. 57 12. I b i d . , pp. 2530-2531. The notation employed i n various forms of the Ricker curve i s d i f f e r e n t . Careful reading of the d e f i n i t i o n of the variables and parameters w i l l reconcile any s u p e r f i c i a l notational differences. 13. I b i d . , p. 2531. 14. I b i d . , p. 2531. 15. Jan Kmenta, Elements of Econometrics, (New York: Collier-MacMillan Limited, 1971), p. 202. 16. W. E. Ricker, Computation and Interpretation of B i o l o g i c a l S t a t i s t i c s  of Fish Populations, B u l l e t i n of the Fisheries Research Board of Canada, Environment Canada, Fisheries and Marine Service, (Ottawa, 1975), p. 2. 17. Clark, Mathematical Bioeconomics, p. 14. 18. Roberts, "Production Function," pp. 16-17. 19. A problem of terminology can e a s i l y develop here. Total recruitment for the season i n an anticipatory sense i s given by the Ricker curve. However, s t r i c t l y speaking, f i s h are not recruited u n t i l they are within the range of the relevant gear. Thus, for salmon f i s h e r i e s , recruitment takes place throughout the season as f i s h enter the range of the gear. Thus, the stock of f i s h available during any sub-seasonal time period are the r e c r u i t s f o r that period. 20. Roberts, "Production Function," p. 14. 21. Note that i n equation (2.14), q(V ,X ) i s the c a t c h a b i l i t y c o e f f i c i e n t S r l 6 2 and, given the form assumed there, 9q/9Vfc = <^AVt Xfc . This i s not the same as 3h/3Vt where h i s given by (2.15); : i . e . , 6, 5+1 9h/3Vt = (6 1+l)AV t X X . 22. Kmenta, Elements of Econometrics, p. 202. 58 23. The assumptions concerning f l e e t u t i l i z a t i o n of t h i s model are s i m i l a r to those of case I of the s imulat ion model descr ibed i n chapter 3. In a d d i t i o n , the e f f e c t of d i scount ing of p r o f i t s i s ignored for t h i s short per iod model and i s customary i n the l i t e r a t u r e . While t h i s i s not s t r i c t l y c o r r e c t ( s ince annual d i scount ing impl ies d i scount ing for sub-<— d i v i s i o n s of a year and, i n the l i m i t , impl i e s continuous d i s c o u n t i n g ) , . i t i s a s u i t a b l e approximation to accuracy for purposes of t h i s model. 24. Roberts , "Product ion F u n c t i o n , " p .23 . 25. A . C . Chiang, Fundamental Methods of Mathematical Economics, (New York: McGraw-Hi l l Book Company, I n c . , 1967), p. 350. 26. C l a r k , Mathematical Bioeconomics, p.245. C l a r k has shown the d e r i v a t i o n of the e q u i l i b r i u m equation from a general opt imal c o n t r o l problem and has proven that the most r a p i d approach to e q u i l i b r i u m i s the opt imal t r a j e c t o r y . T h i s development fo l lows C l a r k . 27. I b i d . , p . 253. 28. I b i d . 29. I b i d . , p . 256. 30. The form of the Ricker curve employed here i s that employed by H i l b o r n i n obta in ing est imates of a and K; i t can be r e c o n c i l e d wi th equation (2 .5 ) , which i s the form employed by P a u l i k , simply by reviewing the d e f i n i t i o n s of terms for the two forms. For the no ta t ion used i n t h i s equation R i s the number of o f f s p r i n g that w i l l r e t u r n to spawn as adul t s (before h a r v e s t ) ; E i s the t o t a l number of spawners;a i s the parameter of p r o d u c t i v i t y for the stock; K i s the number of spaxmers at which the average number of r e t u r n i n g f i s h per spaxmer i s 1. For the P a u l i k form of the Ricker curve, equation (2 .5) , E i s defined i n stock u n i t s and i s , there fore , equivalent to the r a t i o E / K employed 59 ct here. Also, the a of equation (2.5) i s equivalent to Hilborn's e . Substituting these l a t t e r relationships into (2.5) y i e l d s _ a -aE/K . a(l-E/K) R = e E/K e or, E/K e . 31. Blake Campbell, "Returns from Fishing Vessels i n B r i t i s h Columbia," Canada, Department of Environment, Fisheries and Marine Service, P a c i f i c Region, Vancouver, 1969. 32. B r i t i s h Columbia Catch S t a t i s t i c s , 1951-1974. 33. Personal communication with E.R. Zyblut, B i o l o g i s t , Northern Operations Branch, Fisheries and Marine Service. 34. The estimation of the production function carried out by the author i s reported i n Appendix A. P i l o t tests of these parameters with those estimated by Roberts as inputs to the simulation model of chapter 3 can be found i n Appendix B. 35. Personal communication with Ray Hilborn, I n s t i t u t e of Animal Resource Ecology, University of B r i t i s h Columbia, Vancouver, B.C. 36. I have been informed that some progress i s being made toward con-stru c t i o n of the expanded model but that i t i s an exceedingly d i f f i c u l t problem. Personal communication with G.R. Munro, Department of Economics, University of B r i t i s h Columbia, Vancouver, B.C. 37. Clark, Mathematical Bioeconomics, Clark discusses these issues i n chapter 7. 60 CHAPTER 3 A COMPUTER SIMULATION MODEL OF THE SKEENA RIVER GILLNET SALMON FISHERY 3.0 The I n s t i t u t i o n a l and Geographical S e t t i n g Before d e s c r i b i n g the s imula t ion model, t h i s chapter opens w i th a s e c t i o n developing the i n s t i t u t i o n a l and geographica l s e t t i n g w i t h i n which the s imula t ion ana lys i s takes p l a c e . By b r i e f l y d i s c u s s i n g the present organ iza t ion and management i n s t i t u t i o n s i t w i l l be p o s s i b l e to understand the magnitude of the assumed modi f i ca t ions to the e x i s t i n g a c t u a l f i s h e r y necessary f o r purposes of t h i s p o r t i o n of the a n a l y s i s . A somewhat d e t a i l e d d i s c u s s i o n of the sole-owner i n s t i t u t i o n a l framework, and the reasons f o r s e t t i n g up t h i s framework, w i l l rece ive the b u l k of a t t e n t i o n i n s e c t i o n 3 .0 . T h i s w i l l e s t a b l i s h the bas i s f o r the subsequent broad d e s c r i p t i o n of the wi th in-season ( intraseasonal ) r e l a t i o n s h i p s of the computer s imula t ion model i n s ec t i on 3 .2 , and of the i n t e r s e a s o n a l r e l a t i o n s h i p s i n s e c t i o n 3 .3 . 3 .0 .1 The Perspect ive of the Research The goal of t h i s p o r t i o n of the study i s to shed some l i g h t on the fac tors important i n determination of ' o p t i m a l ' f l e e t ^ s i z e . But the term opt imal has meaning only w i t h i n a s p e c i f i e d context . Th i s subsect ion i s devoted to cons ider ing whether, i n the context of t h i s s tudy , the so le owner's dec i s ions would be i n s o c i e t y ' s i n t e r e s t s . 61 In order to investigate the optimal f l e e t s i z e of an actual salmon fishery the viewpoint of the sole owner of the fishery i s adopted. Cost concepts are defined so as to be commensurate with those normally employed i n the theory of the firm. The sole owner faces salmon market prices as w e l l as input and investment costs over which he i s assumed to have no control. These assumptions are consistent with both the fact that only one of numerous P a c i f i c salmon f i s h e r i e s i s under in v e s t i g a t i o n and with the p a r t i c u l a r f l e e t procurement assumptions made (see section 3.2.4). Using these assumptions does not imply that these private value or cost measures are equivalent to or indicate the s o c i a l costs of those resources. Because of the p o s s i b i l i t y that the prices used are s o c i a l l y non-optimal i t cannot be, and i s not, argued that the f i n a l results that are best for the sole owner are necessarily s o c i a l l y optimal. For example, a high private present value of the net returns stream may leave many fishermen unemployed and demoralized during the f i s h i n g season. T h i s study does not deal with the r e c o n c i l i a t i o n of such s o c i a l and the sole owner's private costs. 3.0.2 The Present Skeena River G i l l n e t Salmon Fishery The Skeena River fishery i s t y p i c a l of most B r i t i s h Columbia salmon f i s h e r i e s . I t i s exploited by i n d i v i d u a l operators, many of whom own t h e i r vessels. The c a p i t a l invested In these vessels varies widely, some being very seaworthy and therefore highly mobile while other are somewhat less mobile. In 1968 the Fisheries and Marine Service (which manages the salmon fishery under the authority of the federal Fisheries Act) i n s t i t u t e d an e n t r y - l i m i t a t i o n program for salmon vessels. This 62 program created an upper l i m i t based on h i s t o r i c a l f l e e t s i z e on the number of vesse l s i n the f i s h e r y . Simultaneously a f l e e t - r e d u c t i o n program was launched, which assigned one out of s e v e r a l c lasses of l i c e n s e to each v e s s e l i n order to determine which were to be r e t i r e d over t ime. Vesse ls wi th low-rated l i c enses were not to be r e p l a c e d . The reduct ion program, depending c h i e f l y on wear and tear f o r re t i rement , was speeded up by an o f f i c i a l buy^in program. However, as e x c l u s i o n , ret irement and b u y - i n reduced the f l e e t s i z e , the rent generated accrued to the remaining fishermen and eventua l ly to t h e i r l i c e n s e d v e s s e l s . The l i censes be ing i n d i v i s i b l e and n o n - t r a n s f e r a b l e (so as to preclude agglomeration) the h igher c a p i t a l i z e d value of the l i c e n s e and v e s s e l manifested i t s e l f i n p r i v a t e replacement investment i n b igger and b e t t e r v e s s e l s . One r e s u l t has been to increase the catching power of the f l e e t . Another has been to reduce the v a r i a t i o n of c a p i t a l i n t e n s i t i e s between v e s s e l s . 3 .0 .3 The Sole Ownership Assumption The l i t e r a t u r e , of f i s h e r i e s economics, and the optimum use of a salmon stream, have a lready been discussed i n chapters 1 and 2 r e s p e c t i v e l y , on the assumption that some s i n g l e opt imiz ing agency had i n t e r n a l i z e d many of the choices (about harves t ing instruments) that are otherwise d i f f u s e d among many fishermen i n a common-property f i s h e r y . In t h i s chapter i t i s proposed to i n v e s t i g a t e some of the c h a r a c t e r i s t i c s of an a c t u a l opera t ing owner. Two sets of c h a r a c t e r i s t i c s w i l l be i d e n t i f i e d ; those that r e l a t e to the so le owner's o b j e c t i v e s ; and those that r e l a t e to the instruments he can use to achieve those o b j e c t i v e s . 63 Throughout this study i t i s assumed that, within constraints, any sole owner w i l l attempt to maximize the present value of the fishery. In a situation in which the sole owner's tenure extends for one season only, this objective reduces to one of maximizing the net revenue from that season's fishing alone. This situation was analyzed by means of.the single season model presented in chapter 2. However, when his tenure i s perpetual he w i l l take the returns from a l l future seasons into account in determining the catch for the current season. This situation was analyzed i n the intertemporal model of chapter 2. In that model the sole owner's objective was postulated to be the maximization of the sum of discounted net returns. So far as the choice of instruments i s concerned, the sole owner of a salmon stream can select among a number of harvesting techniques. Indeed, the biology of salmon i s such that highly e f f i c i e n t , strategically placed fixed gear could be used to exploit sockeye and pink salmon. Such a fishing technique was once applied in the British Columbia salmon fishery. In this study the po s s i b i l i t y of use of this type of gear i s not considered; rather, the harvesting technology i s limited to the present form of gill n e t t i n g . However, from a fishery management perspective i t i s obvious that some form of exclusive property rights does bring the use of fixed gear within the realm of po s s i b i l i t y and perhaps desirability; this question i s not addressed i n this study. Assuming that the sole owner i s restricted to using gillnetters for harvesting the Skeena River salmon i t i s necessary to make some assumptions about the procurement of the fishing f l e e t . Two sets of circumstances are developed below, each based on the assumption that the fleet i s acquired by rental: in case I the vessels are rented on a weekly basis 64 whi le i n case II they are rented on a seasonal b a s i s . The assumption of r e n t a l of the f i s h i n g f l e e t i s not as contr ived as i t may appear. One method of r a t i o n a l i z i n g the salmon f i s h e r y would be to s e l l , l ease or auc t ion so le ownership r i g h t s to one of the present ly in tegrated f i s h processors who i n turn would probably b i d for the serv ices of v e s s e l s , sk ippers and gear. There i s reason to b e l i e v e that such a h i r i n g method would achieve greater e f f i c i e n c y i n the f i s h e r y by i n t e r n a l i z i n g the crowding e x t e r n a l i t y i n the product ion f u n c t i o n , yet would a l low room for i n d i v i d u a l i n i t i a t i v e on the part of f ishermen. However, i n what f o l l o w s , we are more concerned wi th the r e l a t i o n between r e n t i n g and escapement than that between r e n t i n g and catch ing c o n d i t i o n s . 3 . 0 .3 .1 Case I Case I i s the l a b e l g iven to the assumption that the so le owner may rent f i s h i n g v e s s e l s , gear and labor serv ices ( a l l inputs being considered as a u n i t of product ive fac tors ) on a week-to-week b a s i s . Th i s time p e r i o d corresponds exac t ly to the length of the d i s c r e t e time per iod assumed for purposes of the computer model. I t i s assumed that the f l e e t s i z e may be c o s t l e s s l y adjusted from one week to another. 3 .0 .3 .2 Case I I In case I I the so l e owner i s assumed to rent vesse l s much i n the same fash ion as i n case I except that the r e n t a l per iod i s now the e n t i r e season. Thus, i n a d d i t i o n to the necess i ty of tak ing dec i s ions as to how many vesse l s to operate each week, the sole owner must now determine at the beginning of the season how many vesse l s he w i l l rent for the season. In 65 dec id ing on y e a r l y f l e e t s i z e the so l e owner must balance the cost of excess capac i ty during some weeks when the run s i z e i s reasonably smal l against the revenues l o s t from having a d e f i c i e n t f l e e t operat ing dur ing weeks when the run s i z e i s r e l a t i v e l y l a r g e . Under case I assumptions, the r e t u r n to c a p i t a l (normally regarded as a f i x e d cost) i s a p a r t of v a r i a b l e cos t . Under case I I assumptions, on the other hand, the r e t u r n to c a p i t a l i s a f i x e d cost for the season and only the weekly operat ing costs of the v e s s e l are v a r i a b l e cos t . Case I obv ious ly incorporates a greater degree of f l e x i b i l i t y s i n c e the so le owner may make a new f l e e t s i z e d e c i s i o n each week. The d e c i s i o n set of the case II so l e owner i s constra ined by the f l e e t s i z e obtained at the commencement of the season. A s u b s t a n t i a l element of f i x e d cost must be taken i n t o account by the so l e owner i n making h i s d e c i s i o n and he must opt imize w i t h i n a r e s t r i c t e d choice s e t . A p r i o r i reasoning suggests that the case I so l e owner w i l l be able to achieve a h igher l e v e l o f net returns from the f i s h e r y . J u s t i f i c a t i o n f o r cons ider ing the two cases can be found on s e v e r a l l e v e l s . The var ious salmon f i s h e r i e s along the coast peak and wane a t d i f f e r e n t times dur ing a p e r i o d from A p r i l to October each year . As a p o l i c y instrument to r a t i o n a l i z e the coast-wide f i s h e r y the establ ishment of so l e owners on each of these r i v e r s under a requirement of annual f l e e t h i r i n g impl i e s a s u b s t a n t i a l impediment to the most economical e x p l o i t a t i o n of the f i s h e r y . The most economic o r g a n i z a t i o n would take advantage of the d i f f e r e n t i a l t iming of the runs to increase the capac i ty u t i l i z a t i o n rate f o r the f l e e t as. a whole. Th i s type of organ iza t ion i s that p o s s i b l e under the case I assumption of weekly h i r i n g . Under t h i s regime the so l e owner can obta in a f l e e t s i z e j u s t i f i e d by the quant i ty of f i s h 66 a v a i l a b l e f o r capture and can readjust as r e q u i r e d . The s i n g l e season model of chapter 2 a l so incorporated t h i s assumption. The i n v e s t i g a t i o n and comparison of the e f f ec t s of these two assumptions for a model of an a c t u a l f i s h e r y i s the major mot ivat ion behind the assumptions. As a f i r s t step i n the development of a coast-wide model of the salmon f i s h e r y , the model and r e s u l t s developed i n t h i s study should be i n s t r u c t i v e . The aggregate coast-wide model i s not attempted i n th i s s tudy. 3 .0 .3 .3 Escapement Contro ls As s ta ted e a r l i e r i n the general d i s cuss ion of the so l e ownership assumption, two tenure assumptions are employed—short-term or s ing le - season tenure and long-term or perpetua l tenure . For purposes of the computer model t h i s tenure d i s t i n c t i o n i s o p e r a t i o n a l i z e d through the use of annual escapement contro l s which, i n the case of perpetua l tenure , the so l e owner imposes on h imse l f . Thus, short - term tenure i s def ined as a s i t u a t i o n i n which the so le owner's objec t ive i s to maximize the sum of • current net p r o f i t s i r r e s p e c t i v e of the harvest or escapement p o l i c y impl i ed by these a c t i o n s . Long-term tenure, i n contras t , i s def ined as a s i t u a t i o n i n which the so le owner acts to maximize the sum of discounted net p r o f i t s ; i n an e f f o r t to recognize the e f f e c t of current on future h a r v e s t i n g , he imposes escapement cons tra in t s upon h imse l f . Given t h i s p e r p e t u a l tenure arrangement, the so l e owner can vary the l e v e l of escapement contro l s i n order to determine that l e v e l which r e s u l t s i n the l a r g e s t present value for the f i s h e r y . 67 3.0.4 Summary of I n s t i t u t i o n a l Assumptions The tenure and other i n s t i t u t i o n a l assumptions employed on the simulation model can conveniently be summarized by the matrix presented i n Table IV below. . TABLE IV INSTITUTIONAL ASSUMPTIONS SOLE OWNERSHIP WITH VESSEL ONE-YEAR PERPETUAL HIRING TENURE TENURE Weekly (1) (2) Annual r (3) (4) With respect to the elements of the matrix of Table IV, the single season model developed i n chapter 2 employed the i n s t i t u t i o n a l assumptions of element (1) while the intertemporal model employed the assumptions of element (4). The simulation model described i n the remainder of t h i s chapter deals with a l l four elements of the matrix. I t i s to the more detailed description of th i s model that we now turn. 68 3.1 Development of the Computer S imulat ion Model 3.1.1 In troduct ion The s t r u c t u r e of the computer s i m u l a t i o n model descr ibed i n t h i s s e c t i o n , l i k e the models developed i n chapter 2, i s genera l i n that i t could" be a p p l i e d to any P a c i f i c coast salmon f i s h e r y . For t h i s s tudy, parameter estimates are obtained from observed data f o r the Skeena R i v e r f i s h e r y . As i n the d e t e r m i n i s t i c a n a l y t i c a l models developed i n chapter 2, the s imula t ion model r e l i e s on the two key r e l a t i o n s h i p s developed i n the e a r l y p a r t of chapter 2; i . e . , . t h e harvest product ion func t ion and the b i o l o g i c a l growth f u n c t i o n . The l a t t e r i s modif ied so as to incorpora te a more complete m o r t a l i t y model than the simple Ricker curve . The s i m u l a -t i o n model a l s o conta ins a s t o c h a s t i c element i n the b i o l o g i c a l growth f u n c t i o n . The d e s c r i p t i o n of the computer model begins wi th an overview of the f u n c t i o n i n g and operat ion of the model. A more d e t a i l e d d e s c r i p t i o n fol lows the genera l overview. The d e t a i l e d d e s c r i p t i o n progresses i n s e q u e n t i a l , r e a l - t i m e fashion—analogous to the model's operat ion—so as to enhance the reader ' s i n t u i t i o n of the model . For purposes of t h i s d e s c r i p t i o n i t i s assumed that the harves t ing model developed i n chapter 2 i s understood and therefore requires l i t t l e or no a d d i t i o n a l d e s c r i p t i o n . A f t e r the t e c h n o l o g i c a l and b i o l o g i c a l r e l a t i o n s h i p s are developed, p r i c e and cost assumptions are introduced which then lead to statement of the ob jec t ive f u n c t i o n s . This completes the d e s c r i p t i o n of the i n t r a s e a s o n a l r e l a t i o n s h i p s of the model. The i n t e r s e a s o n a l b i o l o g i c a l r e l a t i o n s h i p s are then developed thus completing the d e s c r i p t i o n of the s t r u c t u r e of the 69 model. During the exposition of the model, frequent reference w i l l be _ made to Appendix A which displays the process and results of estimation of the parameters employed in the various relationships. The f i n a l section of this chapter contains a comparison of test runs of the model with h i s t o r i c a l results from the fishery in an effort to validate the model as a descriptive device. Successful validation w i l l establish a firmer basis for the analytical tests and comparisons performed with the model. 3.1.2 Overview Models of the type developed in this chapter as well as the inter-temporal model developed in chapter 2 contain both state variables and control variables. The state variables describe the condition of the model system at any time and are completely endogenous to the system. Control variables give the system i t s impetus. They are normally exogenous to the system and can therefore be changed independently of other system variables. This model, l i k e those of chapter 2, consists of two major sections — an economic sector and a biological sector. The economic sector contains the relationships describing the harvesting optimization process. For the whole simulation, the objective function (equation (3.18)) i s defined on the present value of net pr o f i t s . Due to the complexity of the simulation model i t i s not possible to solve 'the' optimization problem in a 'once-for'all' fashion. Rather, the alternative procedure i s adopted. This procedure provides for the economic sector, i n real-time sequence, to receive an endogenous input — the run of fish — and, based on a given value of the control variable (vessel-days of fishing activity) to produce an output of captured salmon. The within-season objective function i s defined on current net revenue and converts this physical output to gross 70 revenue. T o t a l h a r v e s t i n g cost — a func t ion of vesse l -days of f i s h i n g a c t i v i t y — i s subtracted from gross revenues. Thus, the so le owner chooses c a p i t a l (vesse l -days of f i sh ing) by the week (case I) or by the season (case II) so as to maximize the value of seasonal net r e t u r n s . The wi th in- season o p t i m i z a t i o n i s s e l f - r e g u l a t e d by the so le owner's cho ice of a s p e c i f i e d annual minimum number of escaping sockeye and pink i n conjunc-t i o n wi th the opt ion to spec i fy a minimum weekly escapement (as a percentage of the t o t a l sockeye and pink stock a v a i l a b l e for h a r v e s t ) . The c o n s t r a i n t s are chosen e i t h e r s i n g l y or i n combination to be app l i ed at that l e v e l which r e s u l t s i n the l a r g e s t present va lue of net p r o f i t s . Those salmon not captured i n the harves t ing sector enter the b i o l o g i c a l sector which employs a l i f e - h i s t o r y - s t a g e s model to account for the popula-t i o n dynamics of two species of salmon — sockeye and pink — with two d i s t i n c t races for each spec ies . Un l ike the intertemporal model of chapter 2 which has a pure ly compensatory b i o l o g i c a l growth f u n c t i o n , the s i m u l a t i o n model incorporates a l l three types of m o r t a l i t y phenomena experienced by a salmon p o p u l a t i o n . A f t e r c a l c u l a t i o n of the egg d e p o s i t i o n , based on the assumed sex r a t i o (proport ion of females to t o t a l numbers of escapement), the eggs subsequently hatch in to the f ry stage dur ing which compensatory m o r t a l i t y i s a p p l i e d (see equations (3.19) and (3 .20 ) ) . Compensatory m o r t a l i t y i s def ined as a circumstance i n which the m o r t a l i t y r a t e increases with increases i n the brood s i z e . The progeny s u r v i v i n g the compensation stage then experience the m o r t a l i t y i n f l u e n c i n g e f f ec t s of f l u c t u a t i o n s i n stream flow rates and l e v e l s , water temperature, e tc . These phenomena are s imulated by a random number generator us ing sca led values which encompass m o r t a l i t y e f f ec t s d i f f e r i n g by a fac tor of f i v e at the extremes. This rout ine s imulates extrapensa-tory m o r t a l i t y - - def ined as a m o r t a l i t y in f luence whose ra te i n independent of the biomass s i z e . The output of t h i s m o r t a l i t y stage i s salmon smolts . 71 The f i n a l m o r t a l i t y stage i s depensatory which r e s u l t s i n decreas ing m o r t a l i t y ra tes as the biomass increases (see equations (3.21) and (3 .22) ) . A l l three m o r t a l i t y inf luences are assumed to occur i n the freshwater stages . The salmon smolts remaining a f t er the a p p l i c a t i o n of depensatory n a t u r a l m o r t a l i t y proceed to the marine l i f e s t a g e (a r e a l - t i m e accounting rout ine) to add to the biomass and to r e t u r n one year hence as adul t p ink salmon or three , f o u r , f i v e , or s i x - y e a r - o l d sockeye salmon. 3.2 W i t h i n Season R e l a t i o n s h i p s : D e t a i l e d  D e s c r i p t i o n of the S imulat ion Model P l a t e 3A found on page 72 presents a schematic diagram o u t l i n i n g the general s t r u c t u r e of the s imula t ion model. The f u n c t i o n a l r e l a t i o n s h i p s employed i n the v a r i o u s rout ines are a l so shown together wi th the param-eter values employed i n them. The general s t r u c t u r e of the b i o l o g i c a l model i s s i m i l a r to s i m u l a t i o n models used by b i o l o g i s t s to analyze salmon popula t ion dynamics. The major departure from such models i s embodied i n the h a r v e s t i n g s ec tor . Whereas b i o l o g i s t s t y p i c a l l y assume that the f i s h e r y harvests a constant p r o p o r t i o n of the recru i tment , t h i s model features a more complex harves t ing sector which al lows the techno-l o g i c a l nature of the h a r v e s t i n g process together with output p r i c e s and input costs to j o i n t l y determine the s i z e of the harves t . P la t e s 3B and 3C present schematic views of the r e l a t i o n s h i p s comprising the harves t ing sector under cases I and I I r e s p e c t i v e l y . Thus, i t i s c l e a r that what i s attempted i n t h i s model i s the merging of more s o p h i s t i c a t e d economic and b i o l o g i c a l models than have here to fore been cons tructed . 72 PLATE 3A SCHEMATIC DIAGRAM OF SIMULATION MODEL 'Bank' of Fish i n Che Sea of Various -Species-and- Ages Sockeye: k i k i Assembly of Run of 3,4,5,6 Year-Old Sockeye By Application of Age at Maturity Parameters to 'Bank' - "Assembly, of Run-of Pink Pink: R k T = 8 Dis t r i b u t i o n of Total —-.Annual-Run to. F i f t e e n . Weeks i n Season (Sum over k) . Fishery Catch =-f- (Vessel-Days, Stock) Escapement of' Odd Or Even Pink ~'As Appropriate ~ kt h k t - - f t v t . x t ) ---Escapement- of - Babine And Non-Babine Sockeye By Age Sockeye: V i - ^ \ i — —Determine Age -at Return For Sockeye Based On Age D i s t r i b u t i o n of Escapement Sockeye and Pink Egg Deposit Sockeye: Coeff. Bab. Non-Bab. - - . » ! . 1.75 1.25 a 1.75 1.25 D 0.75 0.75 •Compensatory Stage Z - E e a i a - E ) E<1 Z - (l-D)e" a2 ( E" 1 )+D E>1 Pink: Coeff.. Odd a 1.25 . a 2 .1.25 D 0.75-Even 1.25 1.25 0.75 " Freshwater Extrapensatory Stage Scaled Random Normal Deviate. Applied to Produce Five-fold Ratio of Extremes — 1 — Depensatory Stage : M - CeV C- 1 } G<1 M - C. G>1 a 3 •= 0.5 E't+2-H). 2E t_ 3 Smolts Migrating to Sea Or Lake Environment and Eventually to Sea 73 PLATE 3b DETAIL OF HARVESTING TECHNOLOGY CASE I Distribute Total - Run to Weeks *kt " f k t \ T Fishery Production Function Ln(A)+6 2Ln(V c)+5 2Ln(X |.) Ln (h t) 0.541,6. 0.3335,6, 0.7788 Allocate Catch to Species on-Basis - -of Proportions Each Species Contributes to Total Run. h k £. - X k t/R T-K Calculate Gross Returns h " h k t W k P k Calculate Total Cost C - Vessel-Days-$/V-D Net P r o f i t n - V c t Increment Vessel-Days • * Apply Escapement D i s t r i b u t i o n Constant E > 6X 0 » Any Desired Percentage Log i c a l Check For .Maximum. Net.Pro fit.. - ir* >1,*-! *-.t._. False Record-Weekly Net-Returns — Cumulative Values for Season False Increment* A l l Variables .... With t ... Subscript Apply Total Escapement Constraint kx E, ,;A11 k kt True False Logical Check to Determine End of Season True(t<15) To B i o l o g i c a l Model 7 4 PLATE 3C DETAIL OF HARVESTING TECHNOLOGY CASE-II - - -Read l n I n i t i a l Guess of Seasonal Fleet Size(50 Values); Veslim (50) I — Increment Vessel Limit. Rerun Entire Season Distribute Total Run To-weeks " ~ - \ t - f k t \ T Increment Vessel-Days Fishery Production Function (h t) - -Ln(A)+5-LLn(V-)+o2Ln(Xt) * j _ A - 0.541,6^ = 0.3335,62. =.0.7788 A l l o c a t e Catch to Species on Basis of Proportion Each Species Contributes to Total Run-kt Calculate Gross Returns I Net. P r o f i t " - V c t Calculate-Total-Cost -C t = Vessel-Days-$/V-D Apply Escapement — D i s t r i b u t i o n Constraint-E^0X ; 6 = Any Percentage True Logical Check for Maximum Weekly Net P r o f i t ir* „*-l False Record Weekly Net P r o f i t Cumulative Values for Season Apply Total Escapement Constraint h k t ± E k T = AH k Logical Check to Determine Season End Increment A l l Variables With t Subscript -Calculate Sum of Net P r o f i t For Season Greater Than Logical Check to Determine If Sum of Net P r o f i t For Current Iteration i s Greater Than or Less Than Previous Iter l e s s ^ than Pr i n t Economic Results . To Bio Model T 75 3 .2 .1 I n i t i a l Condit ions The values of c e r t a i n v a r i a b l e s and c o e f f i c i e n t s which are generally-endogenous to the model must be i n i t i a l i z e d i n order to permit the model to func t ion i n the e a r l y years of the s imula t ion experiments. The biomass of sockeye and p ink salmon as w e l l as the proport ions of sockeye maturing at var ious ages are the endogenous v a r i a b l e s i n t h i s model r e q u i r i n g i n i t i a l i z a t i o n . The estimated values of these parameters together with the sources and methods of e s t imat ion can be found i n Appendix A , Table X I I . 3 .2 .2 Formation of the Run The annual c y c l e of the model begins with the formation of the run of salmon which o r i g i n a t e s from the biomass of age-race combinations on the ocean feeding grounds. For sockeye, which e x h i b i t a v a r i a b l e age at r e t u r n , a means must be developed for a s c r i b i n g a p a r t i c u l a r age at matur i ty to i n d i v i d u a l sockeye. One method of handl ing t h i s problem, developed by L a r k i n , l i n k s the determination of age at matur i ty to 2 i n h e r i t a n c e . Simply s t a t e d , t h i s hypothesis holds that f i v e - y e a r - o l d spawners w i l l produce progeny which w i l l mature at age f i v e , f o u r - y e a r - o l d spawners w i l l produce progeny which w i l l mature at age f o u r , and so on. There i s evidence i n the b i o l o g i c a l l i t e r a t u r e which suggests that age at maturi ty i s in f luenced at l ea s t i n par t by inher i tance as suggested by the 3 hypothes i s . Fo l lowing L a r k i n we assume that by n a t u r a l s e l e c t i o n the age composition of the sockeye biomass would s t a b i l i z e i n the absence of 4 a f i s h e r y . This r e f l e c t s the assumption that the greater fecundity of o lder f i s h compensates for t h e i r l e s ser s u r v i v a l to matur i ty . With 76 t h i s assumption, the age composition of the progeny of a brood can be r e l a t e d to the age composition of the parent escapement. For any season, X , and sockeye r a c e , k , the propor t ion maturing at var ious ages i s expressed E k i \ E k i 1 where j r e f e r s to the propor t ion of race k maturing at var ious ages i ( i = 3 , 6 ) . The v a r i a b l e E symbolizes escapement. For the f i r s t four years the proport ions are i n i t i a l i z e d s ince there i s no record of escapement upon which to c a l c u l a t e the maturi ty c o e f f i c i e n t s endogenously. Thereaf ter , the are determined according to (3.1) . The ages at maturi ty i n i t i a l l y employed i n the model can be found i n Appendix A , Table X I I . Un l ike the sockeye salmon, pink salmon have a . f i x e d l i f e c y c l e length and r e t u r n to the n a t a l stream two years a f t e r having been spawned. Thus, the s i z e of the pink salmon run i s determinable d i r e c t l y from the age s t r u c t u r e of the pink salmon biomass on the ocean feeding grounds ( i . e . , the age-spec ies -race accounting rout ine ) w i t h i n the model . Given the above, t o t a l sockeye recruitment for season T i s determined according to the fo l l owing r e l a t i o n : (3.2) \ T = I J k i N k i > k = 1,2- i = 3 , . . , 6 . i R^ i s t o t a l recruitment for season x as def ined i n chapter 2 and i s the sockeye biomass. For purposes of a p p l i c a t i o n of t h i s model to the Skeena R i v e r , the two sockeye races re s ident i n that r i v e r are designated k = 1,2. As noted above, determination of the pink salmon run i s more d i r e c t . Thus, 77 (3.3) = N . k = 3 ,4 , i = 2 The p ink salmon stocks of the Skeena River are designated k = 3 ,4 . R^ and N ± are def ined as f o r (3.2) . Given (3.2) and (3 .3 ) , t o t a l recruitment i n x i s (3.4) R x = I R k x , k k = l 3 .2 .3 The T ime-o f -Entry Re la t ionsh ip The r e l a t i o n s h i p s determining the t o t a l recruitment for a season were o u t l i n e d i n the previous s e c t i o n . Unl ike continuous time models i n which recruitment i s instantaneous , t h i s d i s c r e t e time model, l i k e the w i t h i n -season model of chapter 2, assumes that t o t a l recruitment for a season d i s t r i b u t e s . i t s e l f over time throughout the season. However, un l ike the stock d i s t r i b u t i o n func t ion for the intertemporal model of chapter 2 which assumed X(0) = R^ and X(T) = E ^ , the s tock d i s t r i b u t i o n func t ion (or t ime-o f - en try r e l a t i o n s h i p ) of the s imulat ion model more c l o s e l y p a r a l l e l s that of the s i n g l e season model (see F igure 2-1); that i s , the stock present i n the f i s h e r y e a r l y and l a t e i n the season i s r e l a t i v e l y smal l and that stock present dur ing the middle part of the season i s r e l a t i v e l y grea ter . This r e l a t i o n s h i p more c l o s e l y p a r a l l e l s the nature of the observed f i s h e r y . Thus ( 3 - 5 ) \ t = f k A .T 15. X j , t i s the stock of race k present i n the f i s h e r y during t . The c o e f f i c i e n t s 78 f^ t are given values est imated from observed time d i s t r i b u t i o n s f o r the Skeena R i v e r . The values of the c o e f f i c i e n t s employed i n t h i s model can be found i n Table X I I I of Appendix A . 3 .2 .4 The Harvest ing Procedure The harves t ing sec tor employs the harvest product ion f u n c t i o n d i scussed i n chapter 2 as i t s core r e l a t i o n s h i p . T h i s f u n c t i o n r e q u i r e s as input a value f o r stock s i z e present i n the f i s h e r y during each week, X t > and a value f o r vesse l -days of f i s h i n g each week, V t > i n order to determine the salmon h a r v e s t , h^.. Depending upon the f l e e t h i r i n g assumption (weekly or seasonal ) , a d i f f e r e n t a lgor i thm i s r e q u i r e d f o r determinat ion of the opt imal f l e e t s i z e . Both of the a lgori thms are b r i e f l y descr ibed below. The parameters employed i n the product ion funct ion are presented i n Appendix A , ' Table XIV together with a d i s c u s s i o n of t h e i r e s t imat ion and source . 3 . 2 . 4 . 1 Weekly F l e e t H i r i n g The determinat ion of the opt imal number o f vesse l -days o f f i s h i n g i s c a r r i e d out by the f l e e t c o n t r o l a lgor i thm which employs the product ion funct ion to c a l c u l a t e the t o t a l harvest f o r the endogenously determined stock s i z e present i n the f i s h e r y during a given t and numerous values of vesse l -days of f i s h i n g . For each i t e r a t i o n the harvest i s a l l o c a t e d to spec ies and races on the b a s i s of the p r o p o r t i o n of the t o t a l run c o n t r i b u t e d by each race . This impl ies that the model assumes no d i f f e r e n t i a l gear s e l e c t i v i t y for species or ages. T o t a l revenue i s then c a l c u l a t e d by 79 apply ing the p r i c e s for each species to the a l l o c a t i o n f o r each spec i e s . The cost of the harvest i s determined by m u l t i p l y i n g the number o f v e s s e l -days requ ired to obtain i t by the cost per ve s se l -day . Net returns can then be c a l c u l a t e d . A l i n e a r comparison i s employed to determine whether the net returns from the current i t e r a t i o n are greater than, less than or equal to the net returns from the previous i t e r a t i o n . A 'grea ter than' r e s u l t causes the f l e e t c o n t r o l a lgor i thm to increment vesse l -days and c a l c u l a t e a new value of net re turns as descr ibed above. A ' l e s s than' or ' equal to ' r e s u l t causes the model to record the r e s u l t s from the current i t e r a t i o n and increment to the next f i s h i n g p e r i o d . This process continues u n t i l the model has completed the f u l l f i f t e e n weeks of the season's harves t ing a c t i v i t y . Reference to P l a t e 3B w i l l h i g h l i g h t the flow of l o g i c f o r t h i s a lgor i thm. 3 .2 .4 .1 Annual F l e e t H i r i n g The annual f l e e t h i r i n g assumption requ ires a m o d i f i c a t i o n of the f l e e t c o n t r o l a lgor i thm descr ibed above. P l a t e 3C presents a schematic diagram of t h i s a lgor i thm which can be compared to the schematic f o r the weekly f l e e t h i r i n g assumption. The modif ied a lgor i thm begins by i n i t i a l i z i n g a set of 50 (50-year s imulat ion) values f o r the annual f l e e t s i z e (vesse l -days ) . These values are i n i t i a l lox^ estimates of the opt imal f l e e t s i z e as def ined f o r the s imula t ion model, i . e . , that f l e e t s i z e which maximizes current net r e t u r n s . The assumption that the f l e e t i s procured once annual ly p r i o r to the commencement of the season requires the so le owner to undertake a two-part d e c i s i o n making procedure for determining the opt imal seasonal and weekly f l e e t s i z e . Due to the f l u c t u a t i n g s i z e of the run present each week, 80 the so le owner w i l l not n e c e s s a r i l y wish to operate a l l the vesse l s he has a v a i l a b l e each week. He must choose that annual f l e e t s i z e which r e s u l t s i n that combination of operat ing and i d l e vesse ls ( i f any) which maximizes net returns summed over the e n t i r e season. The procedure for programming t h i s constra ined maximization problem which i s set out more formal ly i n s e c t i o n 3.2.6 i s to regard the i n i t i a l value f o r the seasonal f l e e t s i z e as a b i n d i n g c o n s t r a i n t on the maximum number of vesse l s which could be deployed i n any one week. The f u l l season i s run wi th p r e c i s e l y the same f l e e t c o n t r o l a lgor i thm descr ibed i n the previous s e c t i o n . I f the cons tra in t V f c < prevents the net r e t u r n maximizing f l e e t s i z e from being deployed during any week, a switch r e d i r e c t the flow of l o g i c so that the i n i t i a l guess at the opt imal annual f l e e t s i z e , V , i s incremented. With the new, h igher value for V , the model runs through the same season (same t o t a l recruitment) once aga in . A f t e r completing t h i s i t e r a t i o n a comparison i s made between ( £ ^ )^ and (J ^ t ^ + i t t where i here r e f e r s to an i t e r a t i o n number. I f (Y > (Y Ti_).. V £ t x+1 £ t x x i s incremented once again wi th a new comparison to be made. Otherwise, the net p r o f i t maximizing f l e e t s i z e for the season and i t s week-by-week u t i l i z a t i o n has been determined. 3 .2 .5 Escapement Cons tra in t s The s o l e owner's se l f - imposed escapement cons tra in t s are invoked at the c lose of each week's harves t ing a c t i v i t y . One c o n s t r a i n t ensures that the week's harvest does not exceed a s p e c i f i e d p r o p o r t i o n of the t o t a l number of f i s h present i n the f i s h e r y during that week. Thi s i s the catch d i s t r i b u t i o n c o n s t r a i n t and ensures that the f i s h running dur ing a p a r t i c u l a r week do not rece ive ' exces s ive ' f i s h i n g m o r t a l i t y . The 81 escapement d i s t r i b u t i o n c o n s t r a i n t has the form (3.6) ' . E t > 0X where 0 i s a constant s p e c i f i e d by the so le owner. The other c o n s t r a i n t ensures that the cumulative escapement for the season i s greater than or equal to a so l e owner s p e c i f i e d minimum number of f i s h . This l a t t e r c o n s t r a i n t has the form (3.7) \ T ~[\t> \t where E k T i s the des i red minimum weekly escapement. 3.2.6 Development of the Object ive Functions The wi th in-season r e l a t i o n s h i p s of the model have now been developed and i t i s appropr ia te at t h i s time to develop the ob jec t ive funct ions of the model. The f i r s t step i n t h i s procedure i s statement of the p r i c e and cost assumptions. 3.2.6.1 P r i c e and Cost Assumptions Assumptions concerning the p r i c e of the output of harves t ing for the s i m u l a t i o n model are the same as the p r i c e assumptions employed i n the models of chapter 2. That i s , the r e l a t i v e p r i c e of both sockeye and pink i s assumed to remain constant . The r a t i o n a l e i s that output changes of the magnitude considered here i n order to optimize the f i s h e r y w i l l be smal l i n r e l a t i o n to the t o t a l output from the B r i t i s h Columbia salmon f i s h e r y . A d e s c r i p t i o n of the process of es t imat ion of the p r i c e s used f o r sockeye and pink salmon together with re levant data can be found i n Appendix A. 82 Cost assumptions a l so p a r a l l e l those employed i n the chapter 2 models. The t o t a l cost of e f f o r t i s assumed to be constant with respect to v e s s e l -days. The assumptions of f l e e t h i r i n g neces s i ta t e e x p l i c i t r e c o g n i t i o n of the opportuni ty cost of a v e s s e l , sk ipper and gear to the Skeena R i v e r so le owner. The assumption employed here i s that these resources could always f i n d employment elsewhere i n the harves t ing of one of the many B r i t i s h Columbia c o a s t a l salmon f i s h e r i e s . Therefore , to a t t r a c t the necessary f l e e t capac i ty the Skeena River so le owner must pay no less than the net returns these resources could earn i n an a l t e r n a t i v e f i s h e r y . Th i s i s the equivalent of the assumption that the t iming of the var ious salmon runs along the B r i t i s h Columbia coast i s such as to always provide an a l t e r n a t i v e to the Skeena R i v e r , i . e . , opportuni ty cost i s always p o s i t i v e . This e s tab l i shes a f i x e d cost component which the so l e owner i s ob l iga ted to pay a f t e r he has made h i s weekly or annual f l e e t s i z e d e c i s i o n . The v a r i a b l e cost of operat ing the vesse l s i s an a d d i t i o n a l cost which the so le owner must cons ider . V a r i a b l e cost covers such items as f u e l , minor maintenance on the v e s s e l and gear, b a i t cos t , d e p r e c i a t i o n , and so on . Given these assumptions the t o t a l cost f o r case I and case I I , the weekly and annual f l e e t h i r i n g assumptions a r e , r e s p e c t i v e l y , (3.8) C f c = [D t (m + Q ) ] y t , and Q V (3.9) C t D t m V t . In (3.8) and (3.9) C t denotes t o t a l cost whi le m, Q and D f c denote operat ing cost per ves se l -day , f i x e d cost per vesse l -day and the number of days f i s h i n g for week t . In (3.9) the express ion Q V /N (where N i s the 83 number of weeks i n the season) a l l o c a t e s the t o t a l r e n t a l cost for the season Q T V T to weeks w i t h i n the season. Given the wi th in-season r e l a t i o n s h i p s s ta ted above as w e l l as the p r i c e and cost assumptions, i t i s now p o s s i b l e to s ta te the o b j e c t i v e f u n c t i o n s . 3.2.6.2. The Objec t ive Funct ions Gross revenues r e s u l t i n g from f i s h i n g a c t i v i t i e s for each i n t r a s e a o n a l time i n t e r v a l are (3.10) I t - E h k t • W k • p k . The v a r i a b l e h k f c , harvest of race k at t , i s expressed i n numbers of f i s h and the p r i c e parameter p k i s expressed i n cents per pound. The W k are i / average weights per i n d i v i d u a l f i s h . Since i t has been assumed that costs are constant wi th respect to vesse l -days (3.11) C t = b V t , c ' ( V t ) = b Net p r o f i t i n t , H t , then i s (3.12) n t = i t - c t which, when summed over the e n t i r e season i s (3.13) n - z n = E ( ^ - C f ) T t t t t t The short - term tenure so le owner wishes to maximize (3.13) subject to the product ion f u n c t i o n , (3 .14) , and a harvest no greater than a c t u a l recruitment for the season, (3 .15): 6 ,+1 8+1 (3.14) h = AV X„ t t t 84 (3.15) E h t < R T . In contras t the perpe tua l tenure s o l e owner who, w i t h i n the season acts to maximize the value of current net r e t u r n s , a l so wishes to c o r r e c t f o r t h i s myopia by us ing se l f - imposed escapement c o n s t r a i n t s to f i n d that harves t ing - p o l i c y which r e s u l t s i n the l a r g e s t va lue of h i s a s se t , the f i s h e r y . Thus, dur ing the season, the perpetua l tenure so l e owner wishes to maximize (3.13) subject to the same c o n s t r a i n t s (3.14) and (3.15) and, i n a d d i t i o n , the c o n s t r a i n t s (3.16) and (3.17); (3.16) \ T - | \ t > \ x (3.17) E t > ex However, the s o l e owner recognizes that maximization of current net re turns has i m p l i c a t i o n s for future season's f i s h i n g a c t i v i t y . Therefore , the so l e owner experiments w i th v a r i o u s l e v e l s of se l f - imposed escapement c o n t r o l s i n order to determine that l e v e l of E and 6 which results.'! i n the l a r g e s t discounted present va lue of the net income stream. Thus, e x p e r i -ments are performed so as to maximize (3.18) E n a T - l , a = -1-x x 1+r subject to (3.14) through (3117). 8 5 3 . 3 In terseasonal R e l a t i o n s h i p s : D e t a i l e d D e s c r i p t i o n  of the S imulat ion Model In contras t to the b i o l o g i c a l growth funct ion employed i n the opt imal c o n t r o l model of chapter 2, the b i o l o g i c a l growth model employed i n the s imula t ion model u t i l i z e s a disaggregated approach to salmon popu la t ion dynamics. The more complete approach of the s imulat ion model invo lves segregat ing the l i f e cyc l e i n t o a s e r i e s of s tages . Based on research publ i shed i n the b i o l o g i c a l l i t e r a t u r e , i t i s pos s ib l e to a t t r i b u t e d i f f e r e n t types of m o r t a l i t y phenomena to d i f f e r e n t l i f e stages and, of course , to the flow regimes of p a r t i c u l a r r i v e r systems. In p a r t i c u l a r , research has been conducted on Skeena R i v e r salmon stocks ( e s p e c i a l l y sockeye) which has been used to develop s imula t ion models of t h e i r p o p u l a -t i o n dynamics. The b i o l o g i c a l model employed i n t h i s s i m u l a t i o n i s pat terned a f t e r the models of L a r k i n and McDonald 5 and L a r k i n and H o u r s t o n . 6 The model cons i s t s of four stages—egg d e p o s i t i o n , compensatory m o r t a l i t y , depensatory m o r t a l i t y and extrapensatory m o r t a l i t y . The m o r t a l i t y funct ions a p p l i e d at each of the l i f e stages are d iscussed below. 3 . 3.1 Egg Depos i t ion Upon completion of the h a r v e s t i n g a c t i v i t y i n each season the accumulated escapement by race enters the b i o l o g i c a l phase of the model, the f i r s t stage of which i s the egg depos i t i on stage. The age d i s t r i b u t i o n of the sockeye escapement a f f e c t s the s i z e of the egg d e p o s i t i o n s ince more mature f i s h are of l a r g e r s i z e and produce more eggs per i n d i v i d u a l . A d d i t i o n a l l y , males form a l a r g e r percentage of the younger f i s h . To account 86 for these phenomena, the egg depos i t ion assoc ia ted wi th each sockeye age group i s adjusted by an egg product ion f a c t o r . The fac tors are chosen so that the sum of the products of the proport ions of sockeye maturing at var ious ages and the egg product ion fac tors equals u n i t y . Th i s r e f l e c t s the e q u i l i b r i u m c o n d i t i o n that the race must be j u s t capable o f reproducing i t s e l f . • In contras t to sockeye, p i n k salmon do not e x h i b i t a v a r i a b l e age at r e t u r n . While there i s d e f i n i t e l y a s i z e - d i s t r i b u t i o n of r e t u r n i n g p ink and whi le the number of eggs produced w i l l vary p o s i t i v e l y wi th the s i z e of the female, f o r p r a c t i c a l purposes i t w i l l be assumed that each female deposi ts the same quant i ty of eggs on average. The a c t u a l egg product ion fac tors employed i n the model together wi th the sources and methods of es t imat ion can be found i n Appendix A,. 3 .3 .2 Compensatory N a t u r a l M o r t a l i t y The stages of l i f e through which the new brood must pass on the path to a d u l t h o o d — a l e v i n , f r y and smolt—are a l l subject to var ious types o f mortal i ty—compensatory, depensatory and extrapensatory . The sequence i n which these m o r t a l i t y types a f f ec t the var ious l i f e stages may vary w i th both the race and the freshwater environment. I t i s genera l ly thought that compensation occurs among salmon i n the egg s tage . The compet i t ion among adul ts f o r a l i m i t e d supply of good spawning s i t e s and the compet i t ion among eggs f o r oxygen are thought to be s trong reasons to suspect compensation at t h i s s tage , given freshwater environmental c o n d i t i o n s . For sockeye, which spend at l e a s t one year i n the freshwater s tage , the l i m i t e d r e a r i n g capac i ty of t h i s environment i s a f u r t h e r reason to suspect compensation to occur dur ing the j u v e n i l e s tages . 87 To s imulate compensatory m o r t a l i t y , two equations are employed—one app ly ing to brood d e n s i t i e s greater than one un i t and the other apply ing to brood d e n s i t i e s less than one u n i t . Brood s i ze i n numbers i s converted to d e n s i t y — r e l a t i v e magnitudes—before a p p l i c a t i o n of n a t u r a l m o r t a l i t y . The two-equation approach i s u t i l i z e d to avoid the u n r e a l i s t i c a l l y high compensation impl i ed by the Ricker equation for dens i ty dependent predat ion (appl ied to brood d e n s i t i e s less than one un i t ) given below: a (1-E) (3.19) Z = Ee , E < 1.0 where Z re f er s 'to the product ion of progeny from E adults and a^ i s the compensation c o e f f i c i e n t . For brood dens i t i e s greater than one u n i t the modif ied Ricker equation employed i s - a ( E - l ) (3.20) Z = ( l -D)e . + D , E > 1.0 Here D i s the asymptote of brood u n i t s beyond which there i s no compensation. Z and E have the same d e f i n i t i o n s as above. Th i s sequence and procedure f o r implementing compensation app l i e s to both the sockeye and p i n k spec i e s . The parameters employed i n these equations are d isp layed a long w i th a d i s c u s s i o n of the sources and methods of es t imat ion i n Appendix A. 3.3.3 F l u c t u a t i n g Freshwater Environmental Inf luence F l u c t u a t i o n s i n the q u a l i t y of the freshwater and marine environments can have a s u b s t a n t i a l e f f ec t on salmon s tocks . These f l u c t u a t i o n s have been observed h i s t o r i c a l l y i n wide swings i n the s i z e of the r e t u r n i n g 88 adul t run . The magnitude of such environmental e f fec t s i s u n r e l a t e d to the s i z e of the biomass and for t h i s reason i s termed extrapensatory m o r t a l i t y . To s imulate these e f f ec t s i n t h i s model a procedure developed by Ricker and r e f i n e d by L a r k i n and Hourston i s employed. This procedure involves the a p p l i c a t i o n of sca led random normal deviates . The random number generator employed has a normal d i s t r i b u t i o n wi th a mean of zero and a standard d e v i a t i o n of 1.0. The i n i t i a l i z i n g value for entry to the generator i s se t at the value of the computer system c l o c k . The value of the deviates produced by t h i s procedure are augmented i n absolute value by 1.0 and are used as m u l t i p l i e r s i f the s ign of the deviate i s p o s i t i v e and as d i v i s o r s i f the s ign i s negat ive . To achieve an order of magnitude r a t i o of extremes f o r environmental . f l u c t u a t i o n , , s c a l i n g f a c t o r s are appl ied before the deviate i s augmented i n absolute va lue . The value of the s c a l i n g f a c t o r s employed i s 0.61803 which L a r k i n and McDonald employed i n t h e i r model of the Skeena R i v e r sockeye to produce a f i v e - f o l d r a t i o of extreme v a l u e s . 7 In t h i s model we employ the extrapensatory n a t u r a l m o r t a l i t y rout ine a f t e r the compensatory n a t u r a l m o r t a l i t y stage i n order to s imulate the biomass f l u c t u a t i o n s caused by freshwater environmental v a r i a t i o n . There i s no marine extrapensatory n a t u r a l m o r t a l i t y i n t h i s model. 3 .3 .4 Depensatory Natura l M o r t a l i t y The f i n a l stage of m o r t a l i t y i n the b i o l o g i c a l model i s c r i t i c a l for c r e a t i o n of cyc les i n the s i ze of the spawning p o p u l a t i o n . The mechanism employed to apply depensatory m o r t a l i t y must generate an i n c r e a s i n g rate of m o r t a l i t y for smal ler f r y popula t ions . The r a t i o n a l e behind t h i s type 89 of m o r t a l i t y phenomenon i s to s imulate , f or example, the s i t u a t i o n i n which a predator takes a l a r g e r propor t ion of a s m a l l popu la t ion than of a l a r g e r one. Another type of b i o l o g i c a l s i t u a t i o n i n which depensation operates i s that i n which m o r t a l i t y i s a funct ion of the dens i ty o f prey g i n previous y e a r s . L a r k i n and Hourston hypothesized that predators prospered on eggs and f r y produced by previous broods and that the most appropr ia te s i n g l e index o f the a v a i l a b i l i t y of t h i s food supply was the s i z e of the 9 spawning escapement which produced i t . Thus the value of the c o e f f i c i e n t of depensation was made a funct ion of the previous three years ' spawning . escapements whose in f luence on the value of the c o e f f i c i e n t was hypothesized to d e c l i n e wi th the 'age' of the spawning escapement. L a r k i n and McDonald a l so fol lowed t h i s same procedure. The procedure employed i n t h i s model i s the same as that of L a r k i n and McDonald. Smolt product ion i s expressed as fol lows when the f r y populat ion i s l ess than one s tock u n i t : (3.21) M = G e a 3 ( G _ 1 ) , G < 1.0. . ' . In (3.21) M i s the number of smolts produced i n brood u n i t s , G i s the number of f ry remaining a f t e r the extrapensatory in f luence and a^ i s the c o e f f i c i e n t of depensation. When G >_ 1.0, the a l t e r n a t i v e r e l a t i o n (3.22) i s invoked: (3.22) M = G , G >_ 1.0 For both the Babine and non-Babine sockeye races the va lue of the c o e f f i c i e n t of depensation i s expressed as a funct ion of the s i z e of the previous three y e a r s ' spawning escapements. Thus, 90 (3.23) a „ = 0.5 E _ + 0.3 E „ + 0.2 E . The output of the depensatory m o r t a l i t y rout ine i s a smolt popula t ion which w i l l grow to matur i ty and re turn as spawning adul ts i n subsequent years . In the n a t u r a l s i t u a t i o n some of these smolts w i l l migrate to freshwater r e a r i n g grounds for s e v e r a l years whi le others w i l l proceed d i r e c t l y to the ocean feeding grounds. To s imulate t h i s v a r i a b i l i t y i n l i f e h i s t o r y , an account ing rout ine has been developed. This rout ine advances the b i o l o g i c a l system one year at the c lose of the m o r t a l i t y r o u t i n e . As the salmon 'mature' through t h i s process they eventua l ly become subject to the formation of the run rout ine descr ibed above. This c loses the b i o l o g i c a l l i f e cyc le and completes the d e s c r i p t i o n of the s t r u c t u r e and operat ion of the f u l l computer s i m u l a t i o n model. 3.4 V a l i d a t i o n of the Model One means of t e s t i n g a s imula t ion model i s to compare the time s e r i e s of values of endogenous v a r i a b l e s generated by the model wi th the time s e r i e s for the same v a r i a b l e s observed i n the r e a l system. C l e a r l y , the randomised s imula t ion model cannot d u p l i c a t e the h i s t o r i c a l sequence—that i s not i t s purpose. However, i t should be capable of mimicking other important c h a r a c t e r i s t i c s of the r e a l system by generating s tock , harvest and escape-ment time ser i e s values that are orders of magnitude s i m i l a r to those of the r e a l s e r i e s . '1 For the Skeena R i v e r salmon f i s h e r y time s e r i e s observat ions .on f l e e t s i z e , harves t , escapement and recruitment are a v a i l a b l e . For each of these v a r i a b l e s a comparison w i l l be drawn between the range (minimum and maximum) 91 and mean o f h i s t o r i c a l values wi th the same s t a t i s t i c s for s imulated r e s u l t s . Since the s imulat ions cover a 50-year p e r i o d , the range and mean could r e f e r to the e n t i r e s imula t ion p e r i o d or to one season w i t h i n the s imulated p e r i o d . For Table V which summarizes the comparisons, the terms minimum and maximum are seasonal minima and maxima for an e n t i r e s i m u l a t i o n . The mean values are season means c a l c u l a t e d over the e n t i r e s i m u l a t i o n . Table V summarizes the r e s u l t s of two s imulat ions and a l s o provides - 11 comparable h i s t o r i c a l s t a t i s t i c s f o r the Skeena R i v e r g i l l n e t f i s h e r y . The summary s t a t i s t i c s i n the center column represent a s imula t ion i n which sockeye and p i n k regulated escapement were set of a minimum of 300,000 and 400,000 f i s h , r e s p e c t i v e l y . In a d d i t i o n the weekly d i s t r i b u t i o n of escapement c o n s t r a i n t was set at 40% of the a v a i l a b l e s tock for each week. In c o n t r a s t , the s imula t ion represented by the r e s u l t s i n the r i g h t hand column contained minimal escapement constraints—10,000 and 20j000 r e s p e c t i v e l y , f or sockeye and p i n k and no weekly minimum escapement requirement. A comparison of the h i s t o r i c a l r e s u l t s wi th the r e s u l t s of s imula t ion A revea l s a reasonably c lose correspondence between the mean and range of harves t , escapement and t o t a l recruitment as w e l l as the sockeye-pink breakdown for each of these endogenous v a r i a b l e s . A dramatic comparison e x i s t s between the h i s t o r i c a l f l e e t s i z e and that of s imula t ion A . Severa l fac tors combine to produce t h i s r e s u l t . The t r a d i t i o n a l theory of open-access f i s h e r i e s has as one o f i t s major hypotheses that excess ive entry i s induced by fishermen who react to average (not marginal) costs and r e t u r n s . The obverse of t h i s i s that the cost per vesse l -day i n the model inc ludes opportuni ty costs which increases the cost per vesse l -day and would, other things remaining e q u a l , tend to reduce the number of vesse ls employed i n the 92 s imula t ion model. In c o n t r a s t , some fishermen wi th free access to the f i s h e r y cons ider only out -o f -pocket cost i n t h e i r d e c i s i o n s . A f u r t h e r d i f f e r e n c e i s that the s imula t ion model represents a c o n t r o l l e d environment i n which a d d i t i o n a l vesse l s are deployed only i f t h e i r returns exceed t h e i r cos t s . The congest ion e x t e r n a l i t y encompassed w i t h i n the aggregate f i s h e r y product ion funct ion i s i n t e r n a l i z e d to the so l e owner's o b j e c t i v e f u n c t i o n . Thus , ' the so l e owner may use fewer v e s s e l s , with, lower t o t a l costs and yet maintain harvests of the same order of magnitude as have been obtained h i s t o r i c a l l y . In the a c t u a l f i s h e r y , uncerta inty as to the a c t u a l s i z e of recrui tment may r e s u l t i n comparatively large f l e e t s i z e s which are not j u s t i f i e d by a c t u a l recruitment and p o t e n t i a l catch r a t e s . Some time may pass before informat ion can increase so as to cause adjustment of the f l e e t s i z e to more appropr ia te l e v e l s . F i n a l l y , the f l e e t s i zes for the s imula t ion r e s u l t s were c a l c u l a t e d on the bas i s of a 5 -days - f i sh ing week, whereas the h i s t o r i c a l average has been s l i g h t l y less than 3 days f i s h i n g per week. S imulat ion B represents a b i o l o g i c a l l y unconstrained h a r v e s t i n g s i t u a t i o n and as such provides a marked contras t to the r e s u l t s of both the h i s t o r i c a l f i s h e r y and s imula t ion A . Each season, the so le owner f i shes u n t i l current net re turns reach t h e i r maximum. Although the r e s u l t of t h i s f i s h i n g pa t t ern i s important i n the fo l l owing season, the current behavior i s not modif ied by i t s future e f f e c t s . ! The i n t e n s i t y of f i s h i n g i s so great i n the e a r l y p a r t of the s imulat ion that the stocks are reduced to very low l e v e l s as compared to the h i s t o r i c a l f i s h e r y . While the amplitude of f l u c t u a t i o n s i n harves t , escapement and recruitment i s s i m i l a r to both s imula t ion A and the h i s t o r i c a l r e s u l t s , the peaks and troughs are at s i g n i f i c a n t l y lower l e v e l s as shown by Table V . In s p i t e of the lower average recruitment and harves t , a l a r g e r average f l e e t s i z e i s employed i n s imula t ion B than i n A. This r e s u l t s from two fac tors important i n the cn : I f j • '•! • • i TABLE V COMPARISON'OF ACTUAL AND SIMULATED CATCH, ESCAPEMENT, RECRUITMENT :; j• , ! ::i : i i ' , j ! \ ! I ! AND FLEET SIZES FOR SKEENA RIVER FISHERY Historical-Seasonal (X 103); Simulation A -Seasonal Simulation B -Seasonal Minimum Maximum Sockeye harvest' Pink harvest 1" I I ! • I Total catch ! S o c k e y e Escapement j Pink Escapement ' : ; ; , I' • . i Total Escapement j Sockeye Recruitment j Pink Recruitment ! ! | j I Total Recruitment I 142: 281 423 l i o 261 371 285 : 841 1,499 2,410 3,909 I • 1,147 1,753 2,900 2,599 3,380 Average Minimum Maximum Average ' Minimum iMaximum Average : 786 j 1,000; 203 11 2,073 2,331 822 8i7 23 13 1,207 1,763 276 410 1,786 : 214 : 4,404 I • ! i 1,659 36 ; 2,970 686 '• 639 I i 269 1,202 i. 658 • .13' 48 32 998j 326 1,539 572 . i i 15 ' 494 137 1,637j 1 595 2,741 1,230 | , 28 . 542 , 169 1,426 ! • 485 3,586 1,39:4 . 36 1,235 303 1,979 ; 406 3,871 1,494 : 30 : 1,935 552 Fleet Size*? (X 10°) 1,126 5,979 3,405 891 , 7,457 2,889 66 3,170 855 34 j 947 369-414 2 95 .25 2 290 122 a. Over the H i s t o r i c a l Period 1940 - 1974. I ! 1 1 •' .1 • : \\ ' \ i . • : • ] • b. Over the H i s t o r i c a l Period 1954-1974; Both odd and even stocks included. i 1 1 ' i I • I i ! 11 ! i 1 . i ; '• ' j 1 i c. Over the period 1971-1975, inclusive 1. The range of the average fleet size i s due to variation in the i number :of vessels present on the fishing grounds during any given week.I Assumes a 5-day'fishing week. I ; ; : 11 I ' j ' i d. Sockeye price = 0.6239; Pink price =!0,2344;! Variable vessel operating costs = $118.00/day; Fixed cost | per vessel-week = $660.00; Production function parameters: Constant term = 0.541; E l a s t i c i t y of output ; wrt vessel-days = 0.3335; E l a s t i c i t y j o f output wrt stock = 0.7788; No b i o l o g i c a l constraints; 5 fishing days per week'. • ! i • j I ' i ' ! ; !' • • . ! : ! • • : ! I : : I li i l ! ' < : : e. Same parameters as for simulation 1 with the exceptions: Biological contraintsj annual minimum escapement 300J000 s o j c k e y e , 400,OCJO p i n k , w e k l y escapement d i s t r i b u t i o n = ! o . 4 t i m e s t o t a l r e c r u i t m e n t . i ! i -I 94 model and i n the two s imula t ions . F i r s t , the stock s i z e present i n the f i s h e r y i s a determinant of the harves t . Larger stock s i ze s r e s u l t i n l a r g e r h a r v e s t s , c e t e r i s p a r i b u s . Second, the process of i t e r a t i o n to a current net r e t u r n maximizing f l e e t s i z e was i n t e r r u p t e d p r i o r to reaching such a f l e e t s i z e i n s imulat ion A by the weekly minimum escapement requirement, whereas i n s imula t ion B the search for a maximum current net r e t u r n was completed. Hence, f l e e t s i zes tend to be l a r g e r i n s imula t ion B than i n s i m u l a t i o n A. The purpose of t h i s s e c t i o n has been to show that the in tegrated bioeconomic model descr ibed i n t h i s chapter i s capable of i m i t a t i n g the h i s t o r i c a l f i s h e r y to a reasonable approximation. For t h i s purpose the focus has been l a r g e l y on the p h y s i c a l v a r i a b l e s c a t c h , escapement and stock on the reasoning that establ ishment of the b i o l o g i c a l c r e d i b i l i t y of the model i s necessary before i t can be used as an a n a l y t i c a l t o o l to search for a bioeconomic optimum. On balance based on a comparison of h i s t o r i c a l r e s u l t s with the r e s u l t s of two t y p i c a l s i m u l a t i o n s , one may conclude that the model i s capable of generat ing time s e r i e s which are a reasonable f a c s i m i l e of the h i s t o r i c a l l y observed data s e r i e s for s e l ec t ed important endogenous v a r i a b l e s . In conjunct ion wi th t h i s comparison, data were presented on the h i s t o r i c a l f l e e t s i z e and the f l e e t s i z e r e s u l t i n g from the t y p i c a l s i m u l a t i o n s . T h i s comparison showed that so le ownership of the f i s h e r y over a h i s t o r i c a l p e r i o d would have r e s u l t e d i n the u t i l i z a t i o n of a much smal ler f l e e t (with presumably lower costs) than was present h i s t o r i c a l l y . Due to the aggregated nature of the data presented and the d i f f i c u l t y of obta in ing accurate v e s s e l counts i n the a c t u a l f i s h e r y i t i s not p o s s i b l e to draw conclus ions as to the exact magnitude of the p o t e n t i a l reduct ion i n the f l e e t . However, i t i s c l e a r th it s i g n i f i c a n t excess capac i ty e x i s t s . A more concrete f l e e t s i z e comparison i s developed i n chapter 4. 95 FOOTNOTES TO CHAPTER 3 1. For a more complete ana lys i s of the e f f ec t s of the entry l i m i t a t i o n program see G. R. Munro, "Canada and F i s h e r i e s Management w i th Extended J u r i s d i c t i o n : A Pre l iminary View," i n Economic Impacts of  Extended J u r i s d i c t i o n , ed. by Lee G. Anderson, (Ann A r b o r : Ann Arbor Science P u b l i s h e r s , 1977), pp. 29-50. See a l s o , Organ iza t ion for Economic Cooperation and Development, " R a t i o n a l i z a t i o n of Canada's West Coast F i s h e r i e s , " by P. H . Pearse, Economic Aspects of F i s h  P r o d u c t i o n , I n t e r n a t i o n a l Symposium on F i s h e r i e s Economics ( P a r i s , 1972). 2. P . A . L a r k i n and A . S. Hourston, "A Model for S imulat ion of the Populat ion Bio logy of P a c i f i c Salmon," Journa l of the F i s h e r i e s Research Board of  Canada, XXI; No. 5, (1964). 3. H. Godfrey, "Comparisons of the Index of Return for Severa l Stock of B r i t i s h Columbia Salmon to Study V a r i a t i o n s i n S u r v i v a l , " J o u r n a l of  the F i s h e r i e s Research Board of Canada, XV, No. 5 (1958), pp. 891-908. 4. L a r k i n and Hourston, "A Model for S i m u l a t i o n . " 5. P. A . L a r k i n and J . G . McDonald, "Factors i n the Popula t ion Bio logy of the Sockeye Salmon of the Skeena R i v e r , " Journa l of Animal Eco logy , XXXVII, (1968), p . 251. 6. L a r k i n and Hourston, "A Model for S i m u l a t i o n , " p. 1254. 7. L a r k i n and McDonald, "Fac tors ," p . 251. 8. L a r k i n and Hourston, "A Model for S imula t ion ," p. 1254. 9. I b i d . 10. L a r k i n and McDonald, " F a c t o r s , " p . 251. 11. Both s imulat ions employed the parameter estimates obtained by Roberts . For a d i s c u s s i o n and p i l o t test us ing the Roberts parameters and those obtained by t h i s author , see Appendix B. 96 CHAPTER 4 SIMULATION RESULTS 4.0 Introduct ion Thi s chapter concerns a report and ana lys i s of the r e s u l t s of repeated s imulat ions us ing the model descr ibed i n d e t a i l i n the previous chapter and Appendix. The f i r s t s e c t i o n of the chapter w i l l descr ibe the process and r e s u l t s of an exerc i s e i n which the s imula t ion model was used to determine the opt imal escapement for the f i shery . ' This a n a l y s i s p a r a l l e l s that which was undertaken wi th the in ter tempora l model developed i n chapter 2. However, given the nature of the s imula t ion model, the opt imal harves t ing , p o l i c y must be d iscovered by a t r i a l - a n d - e r r o r process us ing repeated s imulat ions each w i th d i f f e r e n t escapement requirements and moni tor ing the r e s u l t u s ing the net present va lue of the f i s h e r y . Embedded i n the opt imal harves t ing p o l i c y w i l l be a great dea l more i n f o r m a t i o n a l content than y i e l d e d by the opt imal c o n t r o l model of chapter 2 about the e f f e c t s of the h a r v e s t i n g p o l i c y on the wi th in-season e x p l o i t a t i o n program, the f l e e t s i z e requ ired to o b t a i n the opt imal harvest and the e f f e c t of the e x p l o i t a t i o n program on long-term h a r v e s t , escapement and s tock l e v e l s as w e l l as f l u c t u a t i o n s i n these v a r i a b l e s both w i t h i n the season and between seasons. A great d e a l more knowledge of the behavior of the system under var ious escapement requirements w i l l r e s u l t from t h i s a n a l y t i c a l process . In order to p a r a l l e l as c l o s e l y as p o s s i b l e the in ter tempora l model of chapter 2, the case II v e r s i o n (annual f l e e t h i r i n g ) o f the s imula t ion model w i l l be u t i l i z e d . 97 The case II model provides an ana lys i s of the f i shery i n i s o l a t i o n i n the sense that the f l e e t , once committed to the f i s h e r y , must remain i n that f i s h e r y for the e n t i r e season, whether r e q u i r e d during any week for harves t ing or not . However, i f the Skeena River were considered w i t h i n the l a r g e r context of the B r i t i s h Columbia c o a s t a l salmon f i s h e r y , one can see that o p t i m i z i n g the coast-wide f i s h e r y would r e q u i r e greater f l e x i b i l i t y i n f l e e t u t i l i z a t i o n than provided by case I I . This more f l e x i b l e f l e e t u t i l i z a t i o n regime i s modeled by the c h a r a c t e r i s t i c s of case I which assumes weekly f l e e t h i r i n g . The case I model i s a l so used i n t r i a l - a n d -e r r o r experimentation to f i n d that set of escapement requirements which r e s u l t s i n the l a r g e s t net present value of the f i s h e r y . The r e s u l t s of t h i s process w i l l form a second major s e c t i o n of th i s chapter . Fo l lowing each of the above major sect ions w i l l be a presentat ion and a n a l y s i s l ead ing to other conclusions which can be drawn from the a n a l y s i s , p a r t i c u l a r l y r e s u l t s from s imulat ions which contain e s s e n t i a l l y no escapement requirements . 4.1 Optimal Escapement - The Case I I Model 4 .1 .1 Features of the Case I I Model A review of the features of the case I I model w i l l be h e l p f u l i n understanding the process described below as w e l l as i n i n t e r p r e t i n g the r e s u l t s . The o b j e c t i v e funct ion of the case II model i s def ined on the current season's net returns which are to be maximized. As ide from the economic and t e c h n o l o g i c a l cons tra int s which are i n t e r n a l to th i s maximiza-t i o n , escapement cons tra in t s are app l i ed at a constant l e v e l throughout 98 each s i m u l a t i o n . These escapement requirements take the form of minimum des i red numbers of sockeye and pink salmon escapement during any season, and/or a minimum d e s i r e d percentage of the t o t a l sockeye and pink stock escaping each week of any season. (During the i n i t i a l s imulat ions reported below, the weekly escapement contra in t was omitted.) For each 50-year s i m u l a t i o n the present va lue of the net p r o f i t stream i s c a l c u l a t e d assuming a 10% rate of discount for the so l e owner. The procedure i s to employ var ious l e v e l s of escapement constra ints to search for that l e v e l which r e s u l t s i n the l a r g e s t v a l u e of discounted net p r o f i t s . The b i o l o g i c a l model i s as descr ibed i n the previous chapter and contains three d i s t i n c t m o r t a l i t y in f luences which a f f e c t the biomass of various l i f e stages. The contras ts between t h i s model and the in ter temporal model of chapter 2 should be ev ident . 4 .1.2 Simulat ion Results Table VI presents the r e s u l t s of the f i r s t set of s imulat ions us ing the case I I model. The only escapement requirement invoked for these runs i s the t o t a l seasonal escapement of sockeye and pink salmon. The weekly escapement d i s t r i b u t i o n cons tra in t was not invoked for purposes of these s i m u l a t i o n s . The procedure for programming the cons tra int s i s important to an understanding of the r e s u l t s shown i n Table VI . Since des ired escapement inc ludes both a des i red sockeye and pink escapement the cons tra in t was programmed so that f i s h i n g for the season would cease i f e i t h e r the sockeye or pink escapement f e l l below the des ired l e v e l for that spec ies . Given 99 TABLE VI OPTIMAL ANNUAL ESCAPEMENT SIMULATIONS: CASE I I MODEL SIMULATION ESCAPEMENT (xlO 3) PRESENT VALUE OF NET PROFITS (xlO 3) NUMBER SOCKEYE PINK 6% 10% 16% 1 10 20 $31,926 $22,429 $15,972 2 100 200 37,840 25,470 17,222 3 150 400 44,859 29,623 19,378 4 200 400 47,238 30,494 19,450 5 250 400 52,361 33,046 21,125 6 300 400 54,410 34,495 21,706 7 350 400 51,367 32,722 21,147 the r e l a t i v e economic importance of sockeye due to i t s r e l a t i v e l y high p r i c e , the sockeye escapement was regarded as the key control i n i n i t i a l simulations. The desired pink escapement was set so that i t would not l i k e l y be vi o l a t e d and would therefore not constrain the search for the optimum sockeye escapement. The res u l t s of the tentative optimum simulations were checked to insure that the desired pink escapement did not cause f i s h i n g to cease during any season. I f the desired pink escapement constraint did cause closure of the season, i t s value was reduced i n repeated runs using the same desired sockeye escapement u n t i l either the present value of net p r o f i t s f e l l below that of the tentative optimum simulation or u n t i l the desired pink escapement constraint did not cause season closure. The f i r s t seven simulations reported i n Table VI indicate that optimal sockeye escapement l i e s i n the neighborhood of 300,000 f i s h . The present value of net p r o f i t s of the harvest associated with t h i s escapement i s $34,495 m i l l i o n when discounted at 10%. The same conclusion i s reached 100 when the net r e t u r n stream i s discounted at 6% and 16%. The range of present va lue of net re turns f o r the seven s imulat ions narrows cons iderab ly when discounted at 16%. I t would not be s u r p r i s i n g to f i n d that at some higher r a t e of discount the opt imal escapement p o l i c y would change. Indeed, t h i s would be expected. But , f or a reasonable range of discount r a t e s , the opt imal p o l i c y i s i n v a r i a n t . Two a d d i t i o n a l s imulat ions were run us ing sockeye escapement values which bracketed the apparent optimum of 300,000 sockeye, i . e . , 250^000 and 350,000 sockeye. The r e s u l t i n g present va lue of net p r o f i t s of $33,036 m i l l i o n and $32,722 m i l l i o n for the l a t t e r two s imulat ions confirms that the opt imal sockeye escapement i s approximately 300,000 f i s h , g iven the parameters employed i n the model. The d e t a i l e d r e s u l t s of s imula t ion conta in ing the t e n t a t i v e opt imal escapement were examined to determine whether the des i red p ink escapement c o n s t r a i n t of 400,000 caused f i s h i n g to cease during any of the 50 seasons. The r e s u l t s r e v e a l that the des i red pink escapement d i d , i n f a c t , cause f i s h i n g to cease i n 20 of the 50 seasons. Thus, an a d d i t i o n a l s i m u l a t i o n was run wi th a des i red sockeye escapement of 300,000 and a reduced des ired p ink escapement of 300,000. The net present value of the harvest for t h i s s i m u l a t i o n was $33,929 m i l l i o n , s l i g h t l y l e s s than that for s i m u l a t i o n 6. In s p i t e of the f a c t that i n that l a t t e r s imula t ion the d e s i r e d pink escapement c o n s t r a i n t caused f i s h i n g to cease i n 9 of 50 seasons, the reduced present va lue of the f i s h e r y ind ica tes that fur ther reduct ions i n the des i red p ink escapement would not be economically j u s t i f i e d . Thus, i t i s apparent that the combined opt imal escapement of sockeye and pink i s approximately 300,000 and 400,000 f i s h , r e s p e c t i v e l y . An examination of the d e t a i l e d output for s imulat ion 6 of Table VI revealed that the des i red minimum escapement c o n s t r a i n t s s p e c i f i e d for 101 sockeye and pink caused the f i s h e r y to c lose before the end of the salmon-run season i n a l l of the 50 seasons. Indeed, i n no year d i d the season's f i s h i n g a c t i v i t y extend beyond week 12 and i n most years f i s h i n g a c t i v i t y ceased i n week 10 or 11. Given the f l e e t h i r i n g assumptions used for model 2, i . e . , annual f l e e t h i r i n g , i t would appear that forced extension of the season f o r a longer p e r i o d v i a weekly escapement c o n s t r a i n t s would have been p r o f i t a b l e . The s o l e owner would then have had a means of c o n s t r a i n i n g f i s h i n g a c t i v i t y i n each week v i a the weekly minimum escapement d i s t r i b u t i o n which could be any s p e c i f i e d percentage of the t o t a l a v a i l a b l e stock (sockeye p lus pink) present i n the f i s h e r y during each week. To t e s t t h i s , s e v e r a l s i m u l a t i o n experiments were run us ing the opt imal sockeye and pink escapement va lues of 300,000 and 400,000, r e s p e c t i v e l y under annual and weekly r e g u l a t i o n . The r e s u l t s of these s imulat ions are reported i n Table VII below. TABLE VII RESULTS OF APPLICATION OF WEEKLY ESCAPEMENT CONSTRAINT TO OPTIMAL ANNUAL ESCAPEMENT SIMULATION CASE II MODEL SIMULATION NUMBER MINIMUM WEEKLY PERCENTAGE ESCAPEMENT DESIRED ANNUAL ESCAPEMENT (xlO 3) SOCKEYE PINK PRESENT VALUE OF NET PROFIT (xlO 3) 6% 10% 16% NUMBER OF FULL SEASONS COMPLETED 6* 300 400 $54,410 $34,495 $21,706 0 8 40% 300 400 49,168 31,534 20,343 33 9 30% 300 400 49,071 31,382 20,095 17 10 50% 300 400 52,269 32,897 20,497 47 11 60% 300 400 43,291 27,428 17,182 47 *from Table VI. 102 I t i s apparent from the s imulat ion r e s u l t s d i sp layed i n Table VII that i t i s not p o s s i b l e to increase the p r o f i t a b i l i t y of the f i s h e r y by c o n s t r a i n i n g the weekly catch of the case II model. As compared to the opt imal annual escapement s imula t ion number 6 which contained no cons tra in t on weekly catch (with the exception of the obvious cons tra int that h f c < Xj_), a c o n s t r a i n t r e q u i r i n g 40% escapement r e s u l t s i n 33 more seasons f u l l y completed but a dec l ine i n present value of net p r o f i t s of approximately 10% to $31,534 m i l l i o n . A reduct ion of the cons tra in t to 30% escapement has only a smal l e f f e c t on the present value of net p r o f i t s but r e s u l t s i n l e ss than h a l f the number of f u l l seasons completed than the 40% escapement c o n s t r a i n t . An increase i n the weekly cons tra in t to 50% markedly increases the number of f u l l seasons completed and a l so increases the present value of net p r o f i t s by approximatly 4% above the 40% c o n s t r a i n t . Further increases i n the value of the cons tra in t (to 60% of the t o t a l weekly stock) r e s u l t i n a reduct ion of present value of net p r o f i t s and no change i n the number of f u l l seasons completed. The i n a b i l i t y to improve upon the p r o f i t a b i l i t y of the opt imal escapement s imula t ion (number 6 i n Table VI ) by f o r c i n g an extension of the season length i s , o n r e f l e c t i o n , the d i r e c t r e s u l t of the assumptions of case I I which lead to the formulat ion of the objec t ive func t ion f o r t h i s model. F i r s t , the maximizing rout ine considers the e n t i r e season as a u n i t . Season net p r o f i t i s c a l c u l a t e d on the bas i s of a maximum number of vesse l -days a v a i l a b l e for the season and a weekly u t i l i z a t i o n p a t t e r n for the a v a i l a b l e f l e e t s i z e . From the product ion f u n c t i o n , equation (3.14) we know that with a given number of vessels more f i s h w i l l be harves ted , the more f i s h a v a i l a b l e for h a r v e s t i n g , i . e . , the d e r i v a t i v e of the marginal product of vesse l -days wi th respect to a v a i l a b l e stock s i z e i s p o s i t i v e . 103 That i s (4.1) and (4.2) ( 6 1 + l ) ( 6 2 + l ) V t X t 2 Since a l l the terms i n (4.2) are p o s i t i v e , 8 h t / 3 V t 8 X t > 0. A l s o given the assumption of constant cost of e f f o r t , a vesse l -day of e f f o r t w i l l produce l a r g e r net re turns i f used dur ing weeks when the a v a i l a b l e stock i s r e l a t i v e l y large than when i t i s r e l a t i v e l y s m a l l . Hence, s h i f t i n g vesse l -days from mid-season when the run i s peaking to the l a t t e r p o r t i o n of the season when i t i s waning, which i s the e f f e c t of a weekly escapement c o n s t r a i n t , can only reduce net r e t u r n s , other things remaining equa l . Although t h i s conc lus ion may not seem s u r p r i s i n g , i t i s i n contras t to that which w i l l be drawn f o r the case I model d iscussed below. The opt imal sockeye escapement determined wi th the use of the computer model (300,000 f i s h ) compares very favorably to the opt imal sockeye escapement as determined by the opt imal c o n t r o l model of chapter 2 (349,000 - 395,000 f i s h ) , i n an order-of-magnitude comparison. The r e s u l t s of the computer s i m u l a t i o n . i n d i c a t e , however, that the opt imal des i red minimum s i z e of escapement i s - s m a l l e r , by an amount v a r y i n g from 50,000 to 95,000 sockeye than f o r the o p t i m a l - c o n t r o l model. However, the des i red escapement was set as a minimum escapement i n the computer s imula t ion experiments . In order to f i n d the a c t u a l escapement which r e s u l t e d i n the l a r g e s t present value of net p r o f i t s i t i s necessary to examine the 104 d e t a i l e d output from s imula t ion number 6. F igure 4-1 d i sp lays a p l o t of the mean weekly sockeye escapement f o r the 50-year s i m u l a t i o n , number 6. Th i s p l o t i n d i c a t e s tha t , wh i l e i n the major i ty of seasons (35 of 50) sockeye escapement was at or c lose to the des i red minimum of 300,000 f i s h (mul t ip ly y - a x i s values by 15) , escapement d i d f l u c t u a t e and i n 6 of 50 seasons was s i g n i f i c a n t l y above the des i red minimum. The a c t u a l mean annual sockeye escapement f o r t h i s s i m u l a t i o n i s 376,000 f i s h . The years during which a c t u a l sockeye escapement was s i g n i f i c a n t l y l a r g e r than the des i red minimum are years i n which the d e s i r e d minimum p ink escapement requirement caused f i s h i n g to cease too e a r l y i n the season to al low the sockeye harvest (stock - harvest = escapement) to reach i t s maximum. Thi s f i n d i n g i l l u s t r a t e s an important f ea ture of the f i s h e r y , and the s imula t ion model. The j o i n t h a r v e s t i n g of two spec ie s , wi th imperfect gear s e l e c t i v i t y , forces the chosen escapement p o l i c y to r e f l e c t the presence of both spec ie s . While the s imulat ion model inc ludes both spec i e s , the opt imal c o n t r o l model, i t w i l l be r e c a l l e d , was only capable of i n c o r p o r a t i n g a s i n g l e spec ie s . Thus, as i n d i c a t e d i n Table VI». i t i s most appropr ia te to r e f e r to a combined des ired minimum escapement of sockeye and p i n k . F igure 4-2 presents a p l o t of a c t u a l mean weekly p ink escapement. L i k e the equivalent p l o t f o r sockeye, that f o r p ink i n d i c a t e s that i n the major i ty of seasons (26 of 50) the a c t u a l p ink escapement was at or c lose to i t s des i red minimum (mult ip ly y - a x i s values by 15). A c t u a l mean annual p ink escapement for s imulat ion 6 i s 476,000 f i s h as compared to the des i red minimum of 400,000. The d e v i a t i o n of a c t u a l from des ired escapement f o r p ink i s caused by the sockeye escapement c o n s t r a i n t which r e s u l t s i n the c losure of the season p r i o r to the pink harvest reaching i t s maximum. The i n t e r a c t i o n between the cons tra int s i s qui te complex as w i l l be c l e a r from the d i s cuss ion of the fo l lowing s e c t i o n . 105 FIGURE 4-1 106 FIGURE 4-2 2 ^ 0 . 0 3 2 0 . 0 _ 1 RVERRGE E S C R P . PJNK 4 3 0 . 0 4 0 0 . 0 5 6 0 . 0 6 4 0 . 0 l I I 1 IXJO3 t V 2 0 . 0 _] 8 0 0 . 0 _ J BBO.D i 9 6 0 _ l 107 4.2 Other Results from Case II Simulat ions The previous s e c t i o n showed that the computer model can be used i n an experimental fash ion to search for and determine the l e v e l of des i red escapement which, as a minimum, r e s u l t s i n the maximization of the present value of p r o f i t s . The computer model achieves these r e s u l t s i n a two-spec i e s , j o i n t h a r v e s t i n g s i t u a t i o n and, i n these re spec t s , performs very favorably as compared wi th the in ter tempora l model of chapter 2. In the process of f i n d i n g the opt imal escapement minimum by the t r i a l - a n d - e r r o r process , the s imula t ion r e s u l t s generated r e v e a l a great dea l about the dynamical behavior of the f i s h e r y . I t i s the purpose of t h i s s e c t i o n to i n v e s t i g a t e the s i m u l a t i o n r e s u l t s i n greater d e t a i l . One po in t which emerges immediately from an examination of Table VI i s that r e l a t i v e l y unres tra ined harves t ing does not r e s u l t i n the l a r g e s t present value of the f i s h e r y . The d e t a i l e d output for s imula t ion 1 shows that e x p l o i t a t i o n i n the e a r l y years of the s imula t ion was qu i te in tense . The mean annual percentage of t o t a l recruitment harvested was 80% f o r the f i r s t 6 years and continued to be qui te h igh for the e n t i r e s i m u l a t i o n , averaging 78% f or the f u l l 50-year s i m u l a t i o n . The r e s u l t of t h i s i n t e n s i v e e x p l o i t a t i o n i s a reduct ion of the recruitment to low l e v e l s a f t e r the f i r s t 6 years of i n i t i a l condi t ions and a mainta in ing of low l e v e l s of recruitment through susta ined high harves t recruitment r a t i o s . Figures 4-3, 4-4 and 4-5 r e v e a l the e f f ec t s of t h i s temporal e x p l o i t a t i o n p a t t e r n . F igure 4-3 shows the t o t a l combined sockeye and p ink recru i tment . R e l a t i v e l y high i n e a r l y y e a r s , i t f a l l s to a much lower susta ined l e v e l f o r the remainder of the s i m u l a t i o n . The time s e r i e s of harvest and escapement v a r i a b l e s i n Figures 4-4 and 4-5, r e s p e c t i v e l y , show e s s e n t i a l l y the same 108 FIGURE 4-3 D TOTRL STOCK IXJ0" ) 0 . 0 4 0 . 0 eU.D 120.0 J6D.0 2 0 0 . 0 2 4 0 . 0 2 0 0 . 0 3 2 0 . 0 3 6 0 . 0 1 1 I I I I I I I 3!-" m o -< m m 109 FIGURE 4-4 I l l p a t t e r n . Th i s f i s h i n g p a t t e r n could r e s u l t i n a present value maximum (as compared to the other s imula t ions summarized i n Table V I ) , but only at a very h igh discount r a t e . Reference to Figures 4-6 and 4-7 which d i s p l a y p l o t s of annual recrui tment and annual harvest against time f o r s imula t ion 6, the opt imal escapement s i m u l a t i o n , r e in forces the p o i n t . For t h i s s imula t ion t o t a l recruitment f luc tuates around a f a i r l y constant trend as i n d i c a t e d by F igure 4-6. S i m i l a r comments may be made about the trend of harvest as shown i n F igure 4-7. These r e s u l t s suggest that investment i n the biomass w i l l pay o f f i n a b i o l o g i c a l sense, i . e . , r e s t r a i n i n g current harvest w i l l r e s u l t i n l a r g e r future recru i tment . Investment i n the biomass may pay o f f b i o l o g i c a l l y but may or may not pay o f f economica l ly .1 For example, s imula t ion 1 discussed above employed minimum d e s i r e d escapements of 10,000 and 20,000 r e s p e c t i v e l y for sockeye and p i n k . A c t u a l mean annual escapement f o r t h i s s imulat ion was 212,000 (sockeye and p ink combined) whi le annual recrui tment averaged 1.037 m i l l i o n . This s i m u l a t i o n produced a present value of net p r o f i t o f $22,429 m i l l i o n . In comparison, s imula t ion 6 which r e s u l t e d i n the maximum present value of net p r o f i t employed d e s i r e d escapement cons tra in t s of 300,000 and 400,000 for sockeye and pink r e s p e c t i v e l y . Mean annual t o t a l escapement for t h i s s i m u l a t i o n was 966,000 (sockeye and pink combined) whi le mean annual recruitment f o r the two spec ies was 2.756 m i l l i o n . The b i o l o g i c a l investment of approximately 754,000 f i s h (pn an annual basis) therefore p a i d o f f both i n b i o l o g i c a l and economic terms. Mean recruitment for s imula t ion 6 was l a r g e r than s i m u l a t i o n 1 by 1.72 m i l l i o n f i s h . Most important , the present va lue of net p r o f i t s for s imulat ion 6 was l a r g e r by $12.1 m i l l i o n than that for s i m u l a t i o n 1. 112 113 114 In c o n t r a s t , a s i m u l a t i o n i n which the des i red minimum escapements f o r sockeye and p ink were 400,000 and 600,000 r e s p e c t i v e l y r e s u l t e d i n a c t u a l mean annual recrui tment of 3.047 m i l l i o n . Thus, the added d e s i r e d escapement of 100,000 sockeye and 200,000 pink (above s i m u l a t i o n 6 c o n s t r a i n t s ) p a i d o f f i n the b i o l o g i c a l sense because mean annual recruitment increased by 320,000 f i s h . However, the added escapement d i d not pay o f f economical ly as the present value of net p r o f i t s f o r t h i s s imula t ion was $3,728 m i l l i o n l e s s than that f o r s imula t ion 6. Summarizing these r e s u l t s one would conclude that the model e x h i b i t s d imin i sh ing marginal re turns to escapement. The word ' r e t u r n s ' could be i n t e r p r e t e d i n both a b i o l o g i c a l and an economic sense. That i s , increased escapement leads to increased recruitment but at a decreas ing r a t e ; or , increased escapement leads to increased present value of net p r o f i t s , but a l so at a decreasing r a t e . This suggests that at some l e v e l of escapement, f u r t h e r increases might even lead to absolute dec l ines i n r e t u r n s . For economic r e t u r n s , that escapement l e v e l has been found as has been i n d i c a t e d by Table VI . For b i o l o g i c a l r e t u r n s , i . e . , r ecru i tment , no s imulat ions were run with escapement cons tra int s l arge enough to cause absolute reduct ions i n recru i tment . For the net present value maximizing s imulat ion (number 6) there i s , of course , a time s e r i e s of maximum a v a i l a b l e f l e e t s i zes whose use r e s u l t e d i n the harvest that maximizes the present value of net p r o f i t s . F igure 4-8 d i sp lays the maximum number of vessel -days a v a i l a b l e per week f o r each of the 50 seasons s imulated . Unl ike the in ter tempora l model of chapter 2 which assumed constant e f f o r t throughout the season (and with the f u r t h e r assumption that a l l seasons are a l i k e , e f f o r t was i m p l i c i t l y assumed to be constant for a l l t ime) , t h i s model shows that e f f o r t must f l u c t u a t e i n response to 116 e in how the size of total seasonal recruitment in order for the present value of net profits to be maximized. A requirement that fleet sizes remain constant throughout the simulated period would markedly decrease the present value of net profits. Season-to-season fluctuations in the optimal quantity of ffort are characteristic of the salmon fishery just as are fluctuations the size of recruitment. However, the model provides information on just' wide the fluctuations in optimal fleet size could potentially be for this fishery. The mean number of maximum vessel-days available per week for the 50-year simulation is 322. Assuming 5 days of fishing per week this converts to a fleet of 64 vessels. The largest number of available vessel-days per week for any of the seasons of simulation 6 was 760. This is the equivalent of 152 vessels fishing for 5 days. The smallest number of vessel-days available during any season was 40 vessel-days per week which is the equivalent of 8 vessels fishing for 5 days each week. These are great year-to-year differences. Most of the discussion to this juncture has been concerned with interseasonal analysis using seasonal aggregate measures to describe and compare the results of the various simulation experiments. However, one of the useful features of the model developed for this study is that i t also provides for within-season analysis. Essentially the same information available in seasonal aggregate form is also available on a weekly basis for each season. Of this information the within-season fleet utilization pattern is of st interest to this study. Figure 4-9 presents a graph of a typical 's weekly fleet utilization pattern (for two simulations— 1 and 6). It is evident that less than the maximum available fleet i s utilized in the early weeks of the season but that as the available stock size increases, the utilized fleet rises quickly to the maximum available. The utilized fleet remains at its maximum available value until escapement falls to its mos seas on 117 FIGURE 4-9 _. FLEET UTILIZATION DURING A TYPICAL SEASON - - - ••- CASE II 118 minimum desired level and the season's fishing activity ceases. If the season were to continue for the f u l l 15-week period, fleet utilization would drop very sharply in the latter 3-4 weeks. The dotted line in Figure 4-9 represents a typical fleet utilization pattern taken from simulation 1. This shows the pattern of declining utilization in the late weeks of the season. The pattern of exploitation which this fleet utilization pattern produces is analyzed because i t wil l be contrasted to that generated by the case I model which will be discussed below. Figure 4-10 shows the ratio of weekly harvest to weekly available stock size for the same season which generated the fleet utilization in Figure 4-9. Virtually a l l the available stock is harvested in the early weeks of the season. However, after the fleet utilization reaches the maximum available, further increases in the harvest can come only from increases in the stock available. Thus, the harvest-stock ratio declines as mid-season approaches. As the stock begins to decline in the late weeks of the season the harvest-stock ratio again begins to increase. This pattern of exploitation will also be contrasted to that which occurs with the case I model. 4.3 Summary of Case II Simulation Results It was stated in the early part of this chapter that the case II form of the simulation model employing annual fleet hiring was the closest parallel of the two simulation models to the optimal control model of chapter 2, with respect to assumptions, particularly fleet hiring. The constant effort assumption of the optimal control model was likened to the assumption employed in the simulation model, that the sole owner was 119 FIGURE 4-10' WEEKLY HARVEST-AVAILABLE STOCK RATIO FOR —- -_J A TYPICAL SEASON ' " - -CASE'II SIMULATION 6 _ " iJD 0.1 0.1 f-o 1 S L 120 r e q u i r e d to commit h imse l f to a f l e e t s i z e for the season. In examining the r e s u l t s of var ious s imula t ion experiments based on t h i s assumption we found tha t , whi le the s o l e owner was requ ired to pay the f u l l costs of the f l e e t to which he had committed h imse l f , i t was of ten economical for the so le owner not to operate the f u l l f l e e t which was a v a i l a b l e . The r e s u l t was that o p t i m i z a t i o n required that the f l e e t u t i l i z a t i o n v a r i e d both over the season and from season-to-season. Thi s f l u c t u a t i o n was not s u r p r i s i n g , g iven the season-to-season f l u c t u a t i o n s i n recrui tment and the v a r y i n g s tock s i z e a v a i l a b l e from week-to-week wi th in the season. I t w i l l be r e c a l l e d that f l u c t u a t i o n s In the recruitment were not a feature of the opt imal c o n t r o l model of chapter 2. For the case I I model we found that the combined minimum des ired escape-ment requirements of 300,000 sockeye and 400,000 pink r e s u l t e d i n the l a r g e s t present va lue of net p r o f i t s . However, mean a c t u a l escapement of sockeye for the s i m u l a t i o n experiment which produced the maximum present va lue of net p r o f i t s was approximately 375,000 f i s h annual ly . In comparison, Table IV of chapter 2 revea l s a range of opt imal escapement for sockeye of 349,000-395,000 f i s h , depending upon the set of parameter values assumed. While the r e s u l t s of the s imula t ion and opt imal c o n t r o l models for sockeye appear to be of s i m i l a r magnitude, the s i m i l a r i t y i s more apparent than r e a l . For the opt imal c o n t r o l model with i t s d e t e r m i n i s t i c r e l a t i o n s h i p s , the opt imal escapement i s unchanging from y e a r - t o - y e a r . In c o n t r a s t , the a c t u a l time s er i e s of sockeye escapements which maximized the present value of net p r o f i t s for the s i m u l a t i o n model requ ired a v a r i a b l e , not l e v e l , escapement as revea led by F igure 4-1. While the mean sockeye escapement i s 375,000, a c t u a l sockeye escapement ranged over a s e r i e s of values between approximately 210,000 and 625,000 sockeye annua l ly . 121 Perhaps more important than the above i s the fac t that the salmon f i s h e r y i s , i n r e a l i t y , a j o i n t h a r v e s t i n g problem based on m u l t i p l e species with an age s t r u c t u r e which causes recruitment to f l u c t u a t e on an annual b a s i s (qu i te apart from and i n a d d i t i o n to the s t o c h a s t i c in f luence which a l so contr ibutes to recrui tment f l u c t u a t i o n s ) . Because the s i m u l a t i o n model i n t e r n a l i z e s these important features of the salmon f i s h e r y i t i s cons idered a s u p e r i o r p lanning t o o l v i s a v i s the a l t e r n a t i v e s t o c h a s t i c opt imal c o n t r o l approach. The l a t t e r would continue to su f f er from the s i n g l e spec ie s , s i n g l e c y c l e shortcoming a l so a t t r i b u t e d to the de termi -n i s t i c opt imal c o n t r o l model developed e a r l i e r . To round out the d i s c u s s i o n o f the r e s u l t s o f the case I I model var ious s t a t i s t i c a l representa t ions of t y p i c a l wi th in-season r e s u l t s were d i s p l a y e d i n order to show the c a p a b i l i t y of the s imula t ion model to support a n a l y s i s of wi th in-season e x p l o i t a t i o n of the f i s h e r y . Both the opt imal c o n t r o l model and the case I I s imula t ion model are representa t ions of f i s h e r i e s more or l e s s i n i s o l a t i o n . E f f o r t i s constant i n the opt imal c o n t r o l model and, i n the case I I s i m u l a t i o n model, ves se l s are t i e d to. a p a r t i c u l a r f i s h e r y for the e n t i r e season. . In the l a t t e r model, v e s se l s have an opportuni ty cost p r i o r to t h e i r becoming committed to a p a r t i c u l a r f i s h e r y . However, once committed to the Skeena R i v e r s o l e owner t h e i r opportuni ty cost i s i n t e r n a l o n l y . I f the Skeena R i v e r f i s h e r y i s p laced i n the l a r g e r context of the t o t a l B r i t i s h Columbia Coast which i s dotted wi th P a c i f i c salmon f i s h e r i e s , the mer i t of a more f l e x i b l e f l e e t h i r i n g procedure becomes apparent . Given that the t iming of these var ious f i s h e r i e s i s sequenced so that some runs are peaking whi le others wane, oppor tun i t i e s for i n t e r - f i s h e r y f l e e t u t i l i -z a t i o n and r e s u l t i n g o p t i m i z a t i o n present themselves. The case I f l e e t h i r i n g assumptions were designed to i n v e s t i g a t e the e f fec t s of t h i s more 122 f l e x i b l e f l e e t u t i l i z a t i o n scheme. We turn now to a presenta t ion and i n t e r p r e t a t i o n of the r e s u l t s of s imula t ion experiments us ing the case I model. 4.4 Optimal Escapement - The Case I Model The sequence of presenta t ion of r e s u l t s f o r the case I model w i l l p a r a l l e l that f o r the case I I model presented i n the previous s e c t i o n s . The f i r s t s e c t i o n w i l l present the process and r e s u l t s of the experimental , t r i a l - a n d - e r r o r search f o r an opt imal escapement p o l i c y f o r sockeye and pink j u s t as was performed for the case I I model. Fol lowing t h i s w i l l be a presenta t ion of seasonal aggregate r e s u l t s of p a r t i c u l a r i n t e r e s t and, f i n a l l y , a presenta t ion of wi th in-season r e s u l t s w i l l set the b a s i s f o r a comparison of the case I and case I I models. The case I model contains the c a p a b i l i t y f o r s p e c i f i c a t i o n of d e s i r e d minimum l e v e l s of escapement for sockeye and pink j u s t as d i d the case I I model. In the same fash ion as f o r the case I I model, repeated s imulat ions were run wi th var ious l e v e l s of sockeye and p ink escapement. The r e s u l t s of the i n i t i a l series; of, s imula t ion are presented i n Table V I I I below. The s i m u l a t i o n r e s u l t s presented i n Table V I I I i n d i c a t e that the present va lue of net p r o f i t s i s maximized by s p e c i f y i n g a d e s i r e d minimum escapement of 300,000 sockeye and 800,000 p i n k . ^ The present va lue as soc ia ted with these escapement cons tra in t s i s $29,756 m i l l i o n when discounted at 10% (see Table V I I I , s imula t ion 19). Two a d d i t i o n a l s imula -t ions were run wi th d e s i r e d sockeye escapements of 50,000 f i s h l e s s than and 50,000 f i s h more than the t e n t a t i v e op t imiz ing s imula t ion 15. These s i m u l a t i o n s , 20 and 21, generated present va lue of net p r o f i t f i g u r e s of $26,193 m i l l i o n and $28,423 m i l l i o n , r e s p e c t i v e l y when discounted at 10%. Given that the present va lue of net p r o f i t s d e c l i n e d i n both cases we conclude that 300,000 sockeye i s the minimum des i red escapement. 123 . ' • TABLE V I I I OPTIMAL ANNUAL ESCAPEMENT SIMULATIONS: CASE I MODEL SIMULATION NUMBER ESCAPEMENT SOCKEYE ( x l O 3 ) PINK PRESENT 1 6% VALUE OF NET 10% PROFITS (xlO 16% 12 10 20 $23,602 $18,481 $14,605 13 50 100 26,810 19,896 15,103 14 100 200 31,003 21,932 15,814 15 200 400 36,147 24,382 16,737 16 400 800 38,086 24,732 15,924 17 300 300 37,519 24,223 16,178 18 300 400 41,317 27,139 18,039 19 300 800 44,788 29,756 19,497 20 250 800 40,182 26,193 17,114 21 350 800 43,742 28,423 18,721 The present va lue of net p r o f i t s of the s imula t ions shown i n Tab le V I I I a r e , f o r some s i m u l a t i o n s , lower than the present va lue of net p r o f i t s f o r the case I I s imula t ions shown i n Table V I I . T h i s r e s u l t i s on ly apparent ly c o u n t e r i n t u i t i v e . S ince the case I r u l e s are more f l e x i b l e one would except case I to generate l a r g e r present va lue of net p r o f i t s . However, the cause of the l a r g e r present va lue - f i gures for some case I I s imula t ions i s a d d i t i o n a l unexplo i ted o p p o r t u n i t i e s for wi th in-season o p t i m i z a t i o n i n the case I s i m u l a t i o n s . Table IX r e s u l t s show t h i s c l e a r l y and a l s o show that the case I and case I I comparison i s cons i s tent wi th one's i n t u i t i o n . The present va lue of net p r o f i t s f o r s imulat ions 12 through 21 d i s -counted at 6% and 16% show a number o f l o c a l maxima. However, the present 124 value of net p r o f i t s i s maximized by the same desired escapement combination for a l l three discount rates. Examination of the present value of net p r o f i t s of simulations 16 through 19 i n Table VI I I appears to indicate that the desired pink escape-ment minimum has l i t t l e or no ef f e c t upon the present value of net p r o f i t s . The explanation for t h i s i s found i n the f l e e t h i r i n g assumption used i n the case I model and i n the within-season f i s h i n g pattern which r e s u l t s from t h i s assumption. More detailed information on the within-season res u l t s of the case I model w i l l be presented below. Here i t i s s u f f i c i e n t to state that the cause of the apparent independence of the minimum desired pink escapement and the present value of net p r o f i t s i s the fact that the sockeye escapement constraint i s reached early i n the season owing to very intensive harvesting. Consequently, the majority of the pink run has not yet entered the f i s h e r y . Thus, actual pink escapement w i l l run w e l l above the desired minimum. Given the overriding importance of the sockeye constraint i n t h i s respect, i t does not matter a great deal whether the pink minimum i s set at 400,000 or 800,000. Examination of actual escape-ment of pink for these four simulations w i l l support these statements. The mean actual annual pink escapement for simulations 16 through 19 i s 1.4 m i l l i o n , 1.2 m i l l i o n , 1.1 m i l l i o n and 1.3 m i l l i o n , respectively. Furthermore, an examination of the detailed r e s u l t s revealed that no lasted longer than 9 weeks for any of the four simulations (and most seasons the f i s h i n g ended af t e r week 7 or 8). Given that the pink run does not peak u n t i l week 10, i t i s clear why the actual escape-ment diverged so sub s t a n t i a l l y from the desired minimum and why i t apparently was of no consequence that the desired minimum pink escape-ment was set either at 400,000 or 800,000. Given that the season's f i s h i n g a c t i v i t y was suspended so early i n the season, i t again seemed possible that gains could be made by season x n 125 c o n s t r a i n i n g f i s h i n g i n the e a r l y weeks of the season. This would f o r e -s t a l l the i n v o c a t i o n of the minimum annual escapement c o n s t r a i n t , and so a l low the season to extend f o r a longer p e r i o d . The reader w i l l r e c a l l that t h i s s t ra tegy turned out to be u n p r o f i t a b l e i n case I I . For purposes of the case I s imulat ions which were run under an escapement d i s t r i b u t i o n c o n s t r a i n t , the minimum des i red pink escapement was reduced to 400,000. Resul t s of the s imulat ions employing the weekly escapement d i s t r i b u t i o n are presented i n Table IX. ^ ' TABLE IX RESULTS OF APPLICATION OF WEEKLY ESCAPEMENT CONSTRAINT TO THE OPTIMAL ANNUAL ESCAPEMENT SIMULATION CASE I MODEL SIMULATION NUMBER MINIMUM WEEKLY PERCENTAGE ESCAPEMENT DESIRED ANNUAL ESCAPEMENT (xlO 3) PRESENT VALUE OF NET PROFIT (xlO 3) NUMBER OF FULL SEASC COMPLETED SOCKEYE PINK 6% 10% 16% 19* 0% 300 800 $44,788 $29,756 $19,487 0 22 20% 300 400 55,452 35,629 23,088 0 23 30% . 300 400 66,666 37,477 27,126 15 24 40% 300 400 57,161 36,391 23,104 29 25 40% 300 300 49,581 31,727 20,516 45 *from Table VIII. I t i s apparent from Table IX tha t , u n l i k e the case II model, a forced extension of the season by c o n s t r a i n i n g weekly f i s h i n g a c t i v i t y does pay o f f economica l ly . The reason why t h i s i s so i s apparent from the product ion f u n c t i o n . For the s imulat ions i n which f i s h i n g during the week i s unconstra ined, a d d i t i o n a l vesse l -days of f i s h i n g dur ing a p a r t i c u l a r 126 week resu l t s i n diminishing marginal returns. Given that a maximum harvest i s s p e c i f i e d through the cumulative escapement constraint, i t i s obviously-best to take the harvest at least cost. I f some of the vessel-days employed during weeks i n which marginal returns are decreasing can be reallocated to weeks for which marginal returns are increasing, opportunity for a better resource r e a l l o c a t i o n e x i s t s . This i s precisely what has transpired as the r e s u l t of constraining weekly harvest to 70% of the available stock for that week. The opportunity for improving resource a l l o c a t i o n exists only up to a point, however, as indicated by the res u l t s of Table IX. Increasing the weekly minimum escapement from 20% to 30% of the available stock increases the present value of net p r o f i t s by $1,848 m i l l i o n (evaluated at a 10% discount r a t e ) , while a further increase i n the minimum weekly escapement constraint to 40% of the availab l e stock reduces the present value of net p r o f i t s by $1,086 m i l l i o n . This i s c l e a r l y what one would expect. What has been achieved by use of the 30% weekly escapement minimum i s an equalization of the marginal net p r o f i t of a vessel-day for a l l weeks i n a p a r t i c u l a r season, given the cumulative escapement constraints of 300,000 sockeye and 400,000 pink. I t i s worthwhile returning to the case I I model for a moment and considering again why application of the weekly escapement constraint did not improve economic returns for that model since what has j u s t been shown for the case I model provides further explanation for the case I I r e s u l t . Since the case I I sole owner's decision procedure internalized both the p r o f i t maximizing seasonal t o t a l harvest (within the constraints) and the optimal a l l o c a t i o n of the t o t a l seasonal harvest to weeks within the season, there was no further opportunity for optimizing. In contrast, the case I sole owner maximizes only for the week within the given con-t r a i n t s and therefore has remaining opportunities for economical seasonal s i r e a l l o c a t i o n . 127 Simulation 25 was run with the same minimum weekly escapement constraint and minimum annual sockeye escapement as simulation 24 but with a reduced minimum annual pink escapement. This reduction i n the pink minimum escape-ment reduced the present value of net p r o f i t s by approximately $4.7 m i l l i o n below that f o r simulation 24. On the basis of the results presented to th i s juncture i t i s apparent that the optimal s p e c i f i e d minimum sockeye and pink escapement constraints are 300,000 and 400,000 respectively. In addition, optimization requires the imposition by the sole owner of a minimum weekly escapement of 30% of the available combined stock of sockeye and pink. As with the case I I model the escapement constraints for the case I model are sp e c i f i e d minima. This means that actual escapement may not necessarily always equal the specified minima. Figure 4-11 presents a p l o t of actual mean weekly sockeye escapement f o r the optimal escapement simulation 23. This plot indicates that i n 22 of the 50 seasons, sockeye escapement was at or close to i t s desired minimum (multiply y-axis values by 15). In 18 of the 22 seasons i n which sockeye escapement i s at or close to i t s desired minimum, escapement i s actually less than the desired minimum. This occurs as a r e s u l t of the fact that the annual constraint i s applied a f t e r the week i s completed combined with the fact that a l l of these 18 seasons had f i s h i n g suspended when the sockeye stock was peaking, implying large weekly harvests. Clearly from Figure 4-11, actual escapement fluctuates and i n 16 of 50 seasons was s i g n i f i c a n t l y above.the desired minimum. The actual mean annual sockeye escapement for simulation 23 i s 396,000 which exceeds the desired minimum by 96,000 f i s h on a mean annual basis. The mean actual annual sockeye escapement exceeds the desired minimum by a s l i g h t l y greater amount than the case I I optimal escapement simulation (number 6) annual mean actual escapement FIGURE 4-11 129 of 376,000 sockeye exceeded its minimum escapement constraint of 300,000. It is not apparent that any conclusions may be drawn from this result. Figure 4-12 displays a plot of the actual mean weekly pink escapement of simulation 23. Unlike the equivalent plot for sockeye, that for pink indicates that in the majority of seasons (36 out of 50) the actual pink escapement was at or close to its desired minimum (multiply y-axis values by 15). In only 11 seasons does the actual minimum pink escapement appear to be below the desired minimum for the same reason as stated above for sockeye. Actual mean annual pink escapement for this simulation is 578,000 which, like the sockeye, exceeds the minimum specified escapement by an amount slightly larger than the case II model actual annual pink escapement exceeded its specified minimum. The fluctuation in pink escapement appears to be somewhat less than that of sockeye. The deviation of actual escapement from its desired minimum is caused by the interplay of the annual escapement constraints for the two species as well as the weekly escapement distribution constraint and the differential timing of the two runs. These four factors combine to create a very complex optimization problem, but one which may be solved in the fashion described in this section. In the following section we turn to other results of the case I model which are of interest and which support conclusions about the comparative effects of the various escapement constraints. 4.6 Other Results from Case I Simulations Like the case II model, the case I model has performed very effectively in the t r i a l and error search for an escapement policy which maximizes the present value of net profits for the fishery. In the process of finding the optimal escapement a variety of simulations were generated 130 FIGURE 4-12 1 131 whose detailed r e s u l t s w i l l reveal important features of the behavior of the model and, by implication, of the fishery. A point which emerges very c l e a r l y from Table VIII reinforces a si m i l a r conclusion drawn for the case I I simulation. This i s that intensive harvesting i n the early portion of the simulation does not r e s u l t i n the maximum present value of the fishery.. This conclusion r e s u l t s from a comparison of simulations 12 and 23 and the detailed output f o r both. The detailed ouptut for simulation 12 indicates that e x p l o i t a t i o n i n the early years of the simulation was very intense. The mean annual percentage of t o t a l recruitment harvested was 91% for the f i r s t 6 years and continued to be quite high for the entire simulation, averaging 78% for the 50-year period. This early intensive exploitation reduced recruitment to low leve l s after the f i r s t 6 years of i n i t i a l conditions (with specified recruitment); continuous intensive exploitation maintained low recruitment l e v e l s , though the harvest-recruitment r a t i o remained high. Figures 4-13, 4-14 and 4-15 describe the effects of th i s e x p l o i t a t i o n pattern. Figure 4-13 displays the combined annual sockeye and pink recruitment. The substantial decline after year 5 and the eventual p a r t i a l recovery of the recruitment to s l i g h t l y larger l e v e l s i s apparent from the p l o t . The mean annual combined recruitment for t h i s simulation i s 855,000 f i s h . Figures 4-14 and 4-15 display the combined annual harvest of sockeye and pink as w e l l as the combined annual escapement for the two species. The pattern of escapement i s quite e r r a t i c and appears to s h i f t between the minimum desired escapement and an escapement considerably i n escess of the minimum desired on an alternating annual basis. Mean combined annual harvest and escapement are 686,000 f i s h and 169,000 f i s h , respectively. Reference to Figures 4-16 and 4-17 which display plots of annula recruitment and annual harvest against time for simulation 23, the 132 FIGURE 4-13 134 i i FIGURE 4-15 ! 135 opt imiz ing s i m u l a t i o n , w i l l make the po int . In Figure 4-16 i t i s c l e a r that there i s no trend i n recrui tment . Rather, i t f luc tuates around a t r e n d . The t o t a l harvest v a r i a b l e performs s i m i l a r l y as i s confirmed by Figure 4-17. As wi th the case I I model and i t s s i m i l a r r e s u l t s , i t i s p o s s i b l e to conclude that investment i n the f i s h s tock , i . e . , r e s t r i c t i n g current h a r v e s t , w i l l r e s u l t i n l a r g e r future recrui tment . Investment i n the biomass has been demonstrated to pay d iv idends b i o l o g i c a l l y for the case I model. The degree to which fur ther investment pays economic re turns f o r t h i s model remains to be i n v e s t i g a t e d . S imula-t i o n 12 descr ibed above contained des ired minimum annual sockeye and pink escapement c o n s t r a i n t s of 10,000 and 20,000 r e s p e c t i v e l y . A c t u a l mean annual escapement f o r t h i s s imula t ion was 169,000 sockeye and p ink whi le annual recrui tment averaged 855,000 sockeye and p ink . This s i m u l a t i o n produced a present va lue of net p r o f i t s of $18,481 m i l l i o n as shown i n Table V I I I . In c o n t r a s t , s imulat ion 23 which produced the maximum present value of net p r o f i t s of $37,477 m i l l i o n employed c o n s t r a i n t s of 300,000 and 400,000 for sockeye and pink r e s p e c t i v e l y i n a d d i t i o n to the weekly d i s t r i b u t i o n c o n s t r a i n t s . Mean annual t o t a l escapement for the opt imal s imulat ion was 975 t000 sockeye and pink combined whi le mean annual recruitment for both species was 2.611 m i l l i o n . The b i o l o g i c a l investment of approximately 840,000 f i s h on a mean annual bas i s pa id o f f both i n b i o l o g i c a l and economic terms. Mean recruitment for s i m u l a t i o n 23 was l a r g e r than that f o r s imula t ion 12 by 1.753 m i l l i o n f i s h . More important from an economist's po int of view i s that the present va lue of net p r o f i t s of s imulat ion 23 exceed that for s imulat ion 12 by $18,996 m i l l i o n . However, from the economist's point of view i t i s p o s s i b l e to c a r r y the b i o l o g i c a l investment too f a r . This i s i l l u s t r a t e d by s i m u l a t i o n 16 which 136 t } 138 employed sockeye and pink des i red minimum c o n s t r a i n t s of 400,000 and 800,000 r e s p e c t i v e l y and generated a c t u a l escapement of 1.839 m i l l i o n sockeye and pink combined and a mean annual recrui tment of 2.911 m i l l i o n f i s h . Thus, the added des i red escapement of 100,000 sockeye and 400,000 pink as compared wi th s imula t ion 23 paid o f f i n the b i o l o g i c a l sense because mean annual recrui tment increased from 2.611 m i l l i o n f i s h for s imula t ion 23 to 2.911 m i l l i o n f i s h for s imula t ion 16. But , the present va lue of net p r o f i t s dropped from $37,477 m i l l i o n to $24,732 m i l l i o n . The opt imal escapement program for the case I model has assoc ia ted with i t an opt imal f l e e t capac i ty j u s t as d id the case I I model. F i g u r e 4-18 d i s p l a y s the time s e r i e s of mean seasonal vesse l -days of f i s h i n g per week f o r each of the 50 years of s imula t ion 23. L i k e the case I I model, the case I r e s u l t s show c l e a r l y that to o b t a i n the opt imal harves t , f l e e t s i z e must f l u c t u a t e q u i t e markedly from season-to-season e s s e n t i a l l y i n response to n a t u r a l f l u c t u a t i o n s i n recru i tment . Given that these season-to-season f l u c t u a t i o n s are c h a r a c t e r i s t i c of the salmon f i s h e r y , the so le owner must i n t e r n a l i z e these phenomena i n t o h i s d e c i s i o n making procedure. There fore , i t i s u s e f u l to have a model which can generate t h i s type of in format ion . The mean number of vesse l -days used per week averaged over the e n t i r e s imula t ion i s 187 which converts to 38 ves se l s f i s h i n g for 5 days. The l a r g e s t number of vesse l -days f i shed during any week of the 50-season s imula t ion was 690 which, assuming a 5-day f i s h i n g week i s the equiva lent of 138 v e s s e l s f i s h i n g for the f u l l 5 days. The smal les t number of vesse l -days employed dur ing any week was 10 which i s the equiva lent of 2 ve s se l s f i s h i n g for 5 days. While the aggregate r e s u l t s for f l e e t u t i l i z a t i o n on an in t er seasona l bas i s are of i n t e r e s t , the model provides wi th in-season f l e e t u t i l i z a t i o n 139 FIGURE 4-18 i ! 140 in format ion which i s a l so u s e f u l . A t y p i c a l weekly f l e e t u t i l i z a t i o n p a t t e r n i s presented i n F i g u r e 4-19 which shows the very marked peaking i n the number of vesse l -days employed w i t h i n each week. The wi th in- season f l e e t u t i l i z a t i o n pa t t ern i s shown f o r the s imulat ion which r e s u l t e d i n the l a r g e s t present va lue of net p r o f i t s ( s imulat ion 23, Table IX) and i n the smal lest present va lue of net p r o f i t s ( s imulat ion 12 of Table V I I I ) . F i g u r e 4-19 h i g h l i g h t s a very i n t e r e s t i n g feature of the s i m u l a t i o n r e s u l t s for t h i s model. The t o t a l number of vesse l -days of f i s h i n g for the two t y p i c a l seasons of s imulat ions 12 and 23 are , r e s p e c t i v e l y , 2,415 and 3,225—25% more vesse l -days i n s imula t ion 23. The harvest for the t y p i c a l season of s imula t ion 12 i s 619,000. For the t y p i c a l season of s imula t ion 23, the harvest i s 2.657 m i l l i o n f i s h out of a t o t a l recrui tment of 3.784 m i l l i o n . The harvest f o r the t y p i c a l season of s imula t ion 12 exceeded that for the t y p i c a l season of s imula t ion 12 by 77% w h i l e , i n c o n t r a s t , the recruitment for the same season of s i m u l a t i o n 23 exceeded that f o r s imula t ion 12 by 82%. C l e a r l y the s i z e of recrui tment has a great d e a l to do with the net re turns from f i s h i n g . F a i r l y low l e v e l s of recrui tment w i l l j u s t i f y the use of f a i r l y s i z a b l e f l e e t s . For l a r g e r recru i tments , the harvest i s p r a c t i c a l l y a l l p r o f i t f or the so le owner. The wi th in-season pat t ern of e x p l o i t a t i o n generated by t h i s f l e e t u t i l i z a t i o n p a t t e r n i s presented i n F igure 4-20 for both s imula t ions 12 and 23. The unbroken l i n e i n F igure 4-20 which represents the h a r v e s t -recrui tment r a t i o for a t y p i c a l season of s imulat ion 23 which produced the l a r g e s t present va lue of net p r o f i t s r e f l e c t s the ac t ion of the weekly escapement d i s t r i b u t i o n c o n s t r a i n t . The harvest remained at or above 70% of the stock i n each week except week 13 which can be explained by the f a c t that the c o n s t r a i n t i s chekced a f t e r each i t e r a t i o n through the week's f i s h i n g a c t i v i t y . The p a t t e r n of e x p l o i t a t i o n for s imula t ion 12 i s somewhat 142 FIGURE 4-20 "WEEKLY HARVEST-AVAILABLE STOCK RATIO FOR A TYPICAL SEASON —CASE-"lr SIMULATIONS" 12, 23 " + u--0,f • • --r— Simulation 12 i 4J I cn - • i , n, <— Simulation 23 O 1 ' a "3 A 143 more complex, thus r e q u i r i n g more d e t a i l e d i n t e r p r e t a t i o n . The e x p l o i t a t i o n rate (harvest -recrui tment r a t i o ) drops from week 1 to 2 e s s e n t i a l l y because the r e q u i r e d number of i t e r a t i o n s through the week for week 1 uses 20 vesse l -days which i s s u f f i c i e n t to harvest p r a c t i c a l l y a l l the a v a i l a b l e s tock . The e x p l o i t a t i o n rate then r i s e s between weeks 2 and 6 because the sockeye stock i s i n c r e a s i n g i n s i z e j u s t i f y i n g the use of more and more vesse l s u n t i l the sockeye stock peaks i n weeks 7 through 9, again wi th a p r a c t i c a l l y complete harvest of a l l the s tock . As the sockeye s tock dec l ines a f t e r week 9, the number of vesse l -days j u s t i f i e d drops s i g n i f i c -a n t l y , r e s u l t i n g i n lower e x p l o i t a t i o n of the then peaking p ink r u n . The increase i n the e x p l o i t a t i o n ra te i n the c l o s i n g weeks of the season i s the r e s u l t of the same cause of h igh e x p l o i t a t i o n i n the f i r s t 2 weeks of the season. C l e a r l y , the e f f ec t s of seemingly minor changes i n c o n t r o l v a r i a b l e s and c o n s t r a i n t s can have s i g n i f i c a n t e f fec t s on the harves t ing p a t t e r n . 144 4.5 Summary of Case I S imulat ion Results The case I s imula t ion model employing the assumption of weekly f l e e t h i r i n g was employed i n the same fash ion as the case I I model to determine the opt imal annual minimum escapement of sockeye and p ink salmon from the Skeena R i v e r g i l l n e t f i s h e r y . I t was discovered that the minimum des ired escapement of sockeye and p i n k , r e s p e c t i v e l y , was 300,000 and 400,000 f i s h annua l ly , the same escapement p o l i c y assoc ia ted wi th the case I I model. However, u n l i k e the case I I model, the opt imal s p e c i f i e d minima for sockeye and p ink requ ired the simultaneous use of the weekly minimum escapement c o n s t r a i n t which , when s p e c i f i e d at 30% of the a v a i l a b l e stock per week, maximized the present value of net p r o f i t s . However, l i k e the a c t u a l escapement of the case I I model, that of the case I model diverged from the s p e c i f i e d minimum. A c t u a l mean annual escapement of sockeye and p ink salmon was 396,000 and 578,000. These a c t u a l escapement f igures diverged more s i g n i f i c a n t l y from t h e i r associated s p e c i f i e d minima than those f o r the case I I model although i t does not appear that any conclus ions can be drawn from t h i s observat ion . In a d d i t i o n to the annual aggregate a n a l y s i s c a r r i e d out wi th the case I model, wi th in-season r e s u l t s of a t y p i c a l season's harves t ing and f l e e t u t i l i z a t i o n pat terns were a lso presented and analyzed . These r e s u l t s w i l l be fur ther analyzed i n the f o l l o w i n g s e c t i o n which presents a comparison of the r e s u l t s of the case I and case I I models. I t i s to t h i s task which we now t u r n . 4.6 Comparison of the Case I and Case I I Results One of the most important f ind ings of the s imula t ion experiments conducted wi th the case I and II models (aside from the determinat ion of 145 the optimal escapement policy) i s that both models reveal a profitable opportunity for investment in the biomass beyond that which would be engaged in by a myopic sole owner who behaves so as to maximize the current season's net returns. While this myopic behavior did not result i n the complete 'mining' of the biomass, i t did result in a biomass reduced to a sufficiently low level so that economic returns from the fishery were reduced significantly from their maximum level. This clearly corroborates the external management authority's attempt to control escapement via gear and area closures in the actual fishery. There i s , however, an outstanding question of the degree to which additional escapement represents an optimal policy for management of the fishery. Economists would argue that additional escapement i s j u s t i f i a b l e only as that escapement adds to the present value of net profits. Biologists and fishery managers would be more inclined to measure optimality by the degree to which additional current escapement leads to larger long term recruitment. Table X presents results for the . case I simulations which allow comparison of the response of recruitment and present value of net profits to increased escapement. Increasing levels of combined desired escapement result in increasing levels of mean annual actual escapement as shown by columns 2 and 3 of Table X. Mean annual ••; recruitment does not appear to respond in linear fashion to increases in escapement and, in fact, reaches a peak at a specified escapement of 1.150 million f i s h . A further, increase of both desired and actual escapement leads to a decline, in-mean annual recruitment. -Economic returns to increased escapement peak at a somewhat lower level of actual and desired escapement, indicating that escapement policies based on the biologist's and economist's measure would diverge. From results not presented i t can be shown that the case II formulation of the model would lead to the same conclusion. 146 TABLE X ACTUAL AND DESIRED ESCAPEMENT, RECRUITMENT AND PRESENT VALUE OF NET PROFITS: CASE I MODEL COMBINED MEAN COMBINED COMBINED DESIRED ANNUAL ACTUAL MEAN ANNUAL PRESENT VALUE SIMULATION ANNUAL ESCAPEMENT ESCAPEMENT** RECRUITMENT OF NET PROFITS NUMBER* x 10 3 x 10 3 x 10 3 @ 10% x 10 3  12 30 169 855 $18,481 13 150 466 1,591 19,896 14 300 968 2,160 21,932 15 600 1,439 2,680 24,382 17 600 1,368 2,522 24,223 18 700 1,424 2,678 27,139 20 1,050 1,370 2,616 26,193 19 1,100 1,530 2,719 29,756 21 1,150 1,70.7 2,989 28,423 16 1,200 1,839 2,911 24,732 * from tab le V I I I * * these s i m u l a t i o n r e s u l t s were obtained from the case I model wi th no weekly escapement d i s t r i b u t i o n c o n s t r a i n t . A comparison of the combined mean annual escapement and recruitment r e s u l t i n g from the s imula t ion model with the h i s t o r i c a l r e s u l t s from the a c t u a l f i s h e r y suggests that escapement p o l i c y i n the l a t t e r has been somewhat l a r g e r than that which would r e s u l t from the optimized f i s h e r y . Table XI below presents the mean annual escapement and recruitment for the opt imal case I and case II s imulat ions and the a c t u a l f i s h e r y . The mean escapement and recrui tment f o r case I and case II r e s u l t s are very s i m i l a r . 147 Escapement observed on the Skeena River i s approximately 40% greater than f o r both of the s i m u l a t i o n s . Observed recruitment exceeds that f o r the case II s imula t ion by approximately 12%. Given that the models are a reasonable f a c s i m i l e of the Skeena River f i s h e r y , there i s c l e a r evidence TABLE XI COMBINED' MEAN.ANNUAL ESCAPEMENT AND RECRUITMENT CASE I , CASE I I MODELS AND ACTUAL FISHERY ESCAPEMENT (x 10 3 ) RECRUITMENT (x 10 3 ) CASE I* 954 2,611 CASE I I * 952 2,756 OBSERVATIONS FOR SKEENA RIVER** 1,637 3,139 * 50-year average * * Sockeye averaged over 35 years P ink averaged over 21 years suggest ing that the escapement p o l i c y has been too l i b e r a l over the past s e v e r a l decades. In conjunct ion wi th the opt imal escapement p o l i c y der ived from the case I and II models i s a f l e e t u t i l i z a t i o n p o l i c y which provides for opt imal harves t ing of the resource . In e a r l i e r d i s cuss ion of the s imula t ion r e s u l t s f o r case I and II models, the f l e e t u t i l i z a t i o n p a t t e r n assoc iated wi th the two f l e e t h i r i n g assumptions was presented s e p a r a t e l y . I t i s now appropriate to compare the case I and I I r e s u l t s both wi th each other and wi th the f l e e t u t i l i z a t i o n observed i n the Skeena R i v e r f i s h e r y . F igure 4-21 combines a t y p i c a l season's f l e e t u t i l i z a t i o n pat tern for both the opt imal case I and case II s i m u l a t i o n s . The f l e e t u t i l i z a t i o n p a t t e r n for each of Time In Weeks 149 the cases i s qu i t e d i s t i n c t as i s evident from Figure 4-21. The weekly v a r i a b i l i t y i n f l e e t u t i l i z a t i o n i s f a r greater f o r case I than for case I I . The t o t a l number of vesse l -days expended i n case I I , i . e . , 4,060, exceeded by 33% the number of v e s s e l days employed i n the case I season. While the t o t a l recruitment of 2.82 and 2.87 m i l l i o n f i s h for cases I and I I , r e s p e c t i v e l y , was approximately the same, the sockeye-pink composit ion was somewhat d i f f e r e n t f o r the two seasons. Thus, whi le the cost l e v e l of the case II season was approximately double that of case I , the harvest of case I I contained a l a r g e r percentage of h igher p r i c e d sockeye. The net r e s u l t was that t o t a l net returns f o r these two seasons were approximately e q u a l . However, i n other seasons the reverse s i t u a t i o n would occur and, f o r the two s imulat ions i n t o t a l , the f l e x i b l e case I f l e e t h i r i n g assumption r e s u l t e d i n a l a r g e r present va lue of net returns—$37.5 m i l l i o n versus $34.5 m i l l i o n . On a more g l o b a l s c a l e the comparison between case I and II f l e e t procurement assumptions has broader i m p l i c a t i o n s . The Case I I assumption i n e f f e c t ' B a l k a n i z e s ' the f i s h e r y f o r the durat ion of each season and allows no opportuni ty f o r i n t e r - f i s h e r y t r a d i n g o f vesse l s to achieve greater o v e r a l l e f f i c i e n c y i n the f i s h e r y . Thus, i n a d d i t i o n to the approximately $3 m i l l i o n d i f f e r e n c e i n the case I and I I present va lue of net p r o f i t s , one would add to the case I present va lue of net p r o f i t s the net returns of ve s se l s which f i shed i n a l t e r n a t i v e f i s h e r i e s under case I assumptions but could not have done so under case I I assumptions. A model of the e n t i r e salmon f i s h e r y would automat ica l ly incorporate these opportunity re turns i n a coast-wide comparison between case I and I I assumptions. The i n c l u s i o n of these opportuni ty returns i n t o the present r e s u l t s increases f u r t h e r the r e l a t i v e economic e f f i c i e n c y of case I over case II assumptions. 150 A v a i l a b l e estimates of e f f o r t l e v e l s on the Skeena R i v e r over the p e r i o d 1971 to 1975 i n d i c a t e a range between 7,700 vesse l -days annual ly and 12,800 vesse l -days a n n u a l l y . While these data are of uncer ta in accuracy2 they are corroborated by the data employed by Roberts i n h i s es t imat ion of the product ion func t ion for the Skeena R i v e r . ^ In a d d i t i o n , they prov ide the only bas i s f o r comparison of opt imal e f f o r t as determined by the s imula t ion model and that e f f o r t which i s present ly employed i n the a c t u a l f i s h e r y . Comparing e i t h e r the 4,060 vesse l -days from the case I I opt imal s i m u l a t i o n or the 2,715 vesse l -days from case I wi th the range of v e s s e l -days a c t u a l l y observed i n the Skeena R i v e r , one reaches the conc lus ion that the Skeena R i v e r f i s h e r y i s beset wi th excess capac i ty of a s i g n i f i c a n t magnitude. Quant i fy ing the extent o f excess c a p a c i t y , however, i s a d i f f i c u l t task f o r s e v e r a l reasons. F i r s t , a r e l i a b l e estimate of the a c t u a l number of ve s se l s f i s h i n g at any p a r t i c u l a r time i s d i f f i c u l t to o b t a i n . Second, i t i s c l e a r both from observations o f the a c t u a l f i s h e r y and of the model that e f f o r t must change i n response to recrui tment f l u c t u a t i o n s , not to mention p r i c e and cost changes. Thus i t i s , s t r i c t l y speaking , c o r r e c t to speak of opt imal e f f o r t only i n r e l a t i o n to opt imal escapement and recru i tment . The opt imal c o n t r o l model, i t w i l l be r e c a l l e d , so lved t h i s problem by assuming constant escapement and recru i tment . T h i r d , i t i s c l e a r from knowledge of the e x i s t i n g salmon f i s h e r y i n B r i t i s h Columbia that the number of vesse ls f i s h i n g i s an inadequate index of e f f o r t . C u r r e n t l y , the number of vesse l s i n the g i l l n e t f i s h e r y i s d e c l i n i n g but the aggregate a b i l i t y of the remaining g i l l n e t f l e e t to harvest the resource i s improving due to the i n c o r p o r a t i o n of t e c h n o l o g i c a l developments i n t o the remaining v e s s e l s . Thus, s i g n i f i c a n t 151 excess- capac i ty remains i n the f i s h e r y . In s p i t e of these reservat ions a r a t i o n a l e can be developed f o r q u a n t i f y i n g a suggested reduct ion i n the number of vesse l -days p r e s e n t l y expended on the Skeena R i v e r . The maximum number of vesse l -days employed dur ing any season of the opt imal case I and case II s imulat ions was 3,700 and 6,620 r e s p e c t i v e l y . Given that the range of h i s t o r i c a l l y observed recruitment was encompassed by both the models, i t i s p o s s i b l e to conclude t h a t , f o r weekly f l e e t h i r i n g , the number of vesse l -days expended i n a season should not exceed approximately 6,600. To disaggregate t h i s r u l e f u r t h e r , the maximum number of vesse l -days employed dur ing the week i n which the maximum season vesse l -days were employed was 690 and 760 for cases I and I I , r e s p e c t i v e l y . Thus opt imiza t ion of the f i s h e r y would suggest that i n the peak week of a season with very large r e c r u i t m e n t , no more than approximately 700 vesse l -days should be used under weekly f l e e t h i r i n g ; under annual f l e e t h i r i n g , no more than approximately 760 vesse l -days should be employed. Given that an extension of the number of days of f i s h i n g permit ted each week beyond that observed i n the Skeena R i v e r (approximately 3 per week) would increase the rate of capac i ty u t i l i z a t i o n , the 690 and 760 vesse l -days convert to 138 and 152 v e s s e l s , r e s p e c t i v e l y , assuming a 5-day f i s h i n g week. In most seasons, f l e e t capac i ty would be l ess than the above due to the fac t that recrui tment would be l e s s . Fo l lowing t h i s r u l e would r e s u l t i n at l eas t 14% l e s s vesse l -days expended i n the Skeena R i v e r for case II assumptions as compared x^ith the minimum observed i n the a c t u a l f i s h e r y ; a l t e r n a t i v e l y , f o r case I assumptions, at l eas t 52% l e s s vesse l -days would be employed. 152 FOOTNOTES TO CHAPTER 4 1. T h i s i s not to suggest that recrui tment i s expected to increase i n d e f i n i t e l y to c o n t i n u a l increases i n escapement. Indeed, i t i s s ta ted elsewhere i n the text t h a t , w i th c o n t i n u a l l y increased escapement, recrui tment would eventua l ly begin to d e c l i n e i n abso lute terms. No s imulat ions were run with large enough des i red escapements to produce t h i s r e s u l t . 2. The data problems r e f e r r e d to here are dea l t with i n greater d e t a i l i n Appendices A and B. 3. R. F . A . Roberts , "A Commercial F i s h e r i e s Product ion F u n c t i o n : The Skeena River Sockeye Salmon G i l l n e t F i s h e r y , " Unpublished Master 's T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1971, pp. 57-58. 153 CHAPTER 5 SUMMARY OF FINDINGS, POLICY IMPLICATIONS AND SUGGESTIONS FOR FURTHER RESEARCH 5.0 In troduct ion The purpose of t h i s chapter i s to review and h i g h l i g h t the major f ind ings of t h i s study and to p lace these f ind ings i n a p o l i c y and f i s h e r i e s management context . The chapter begins wi th a very b r i e f review of the models developed i n the s tudy. Thi s w i l l serve as background for a d i s c u s s i o n of the f ind ings and p o l i c y i m p l i c a t i o n s of the f i n d i n g s . The chapter c loses w i th a b r i e f d i s c u s s i o n of important areas for fur ther research which suggested themselves dur ing the course of the r e s e a r c h . 5.1 Review of the Study The study began by showing that the major i ty of research to date i n f i s h e r i e s economics has viewed f i s h e r i e s as e q u i l i b r i u m systems which could be analyzed w i th the a i d of long- term, steady s ta te models. I t was shown that few researchers had concerned themselves wi th shor t - t erm h a r v e s t i n g and only one author concerned h imse l f concurrent ly w i th both s h o r t - and long-term harves t ing economics. F u r t h e r , i t was shown, both through statements of f i s h e r y economists who are f a m i l i a r wi th P a c i f i c salmon f i s h e r i e s and through a c t u a l development of the models h e r e i n , that anadromous, P a c i f i c salmon f i s h e r i e s have a d i s t i n c t i v e b i o l o g i c a l and t e c h n o l o g i c a l nature which r e a l l y requ ires the simultaneous modeling of wi th in-season and in ter tempora l h a r v e s t i n g . Th i s e s tab l i shed the r a t i o n a l e for the s e r i e s of models developed f o r the r e s e a r c h . 154 The s imples t model was developed f i r s t . This model focused on the wi th in-season harves t ing process for a salmon f i s h e r y and i n v o l v e d the use of a product ion func t ion based on the technology of h a r v e s t i n g of salmon wi th g i l l n e t v e s s e l s . The s tock d i s t r i b u t i o n f u n c t i o n , o r , t ime-o f - en try curve was introduced for purposes of t h i s model. This func t ion i s p e c u l i a r to anadromous (more broadly gauntlet) f i s h e r i e s and i s the fundamental reason behind the importance of wi th in-season a n a l y s i s i n t h i s type of f i s h e r y . Given t o t a l recruitment f o r the season, estimates of the parameters of the product ion funct ion and, t ime-of -entry r e l a t i o n s h i p for the Skeena R i v e r g i l l n e t salmon f i s h e r y and the customary p r i c e and cost assumptions, i t was p o s s i b l e to so lve t h i s model a n a l y t i c a l l y and to prov ide a numerica l s o l u t i o n to the 'optimal* number of vesse l -days f o r the season. The emphasis of the second model developed i n chapter 2 changed markedly from wi th in-season to i n t e r t e m p o r a l . The long-term model was set up as a problem i n opt imal c o n t r o l theory , f o l l owing the approach of much of the recent l i t e r a t u r e of f i s h e r i e s economics. C e n t r a l to the cons truc t ion of t h i s model i s the r e l a t i o n s h i p which l i n k s current escape-ment and future recru i tment , i . e . , the spawner-recrui t or R i c k e r r e l a t i o n s h i p . For g iven values of the parameters of t h i s r e l a t i o n s h i p and p r i c e s and costs based on the customary assumptions i t was p o s s i b l e to so lve t h i s model for the opt imal escapement, i . e . , that which maximized the present va lue of net p r o f i t s f o r a l l t ime. Obta in ing t h i s s o l u t i o n was enabled by recent work of C. W. C l a r k who has shown the general form of s o l u t i o n to d i s c r e t e models of the type constructed here . Development of the two models d iscussed above was i n s t r u c t i v e f o r , i n the case of the wi th in-season model, i t al lowed ana lys i s of the wi th in-season 155 harves t ing problem i n a gauntlet f i s h e r y which inc luded the important t ime-o f - en try r e l a t i o n s h i p . The in ter tempora l model provided a s o l u t i o n to the opt imal escapement i n a p a r t i c u l a r salmon f i s h e r y . Both models contain cons iderable h e u r i s t i c value and provide a good bas i s f o r amalgamation of the wi th in-season and in ter seasona l problems i n t o one model—an accomplishment achieved wi th the a i d of the computer s imulat ion models. Notwithstanding the usefulness of these models, however, they were found to have shortcomings which j u s t i f i e d f u r t h e r e f f o r t s to develop a more comprehensive model. The b a s i c shortcoming of the wi th in-season model i s i t s shor t - t erm nature , i . e . , i t does not r e f l e c t the renewable, c a p i t a l t h e o r e t i c aspect of a f i s h e r y . While the opt imal c o n t r o l model c losed t h i s p a r t i c u l a r gap, i t i s subject to other shortcomings. One of the main shortcomings was the requ ired suppression of wi th in-season ana lys i s v i a the constant e f f o r t assumption. In a d d i t i o n , the m u l t i - s p e c i e s , age - s tructure c h a r a c t e r i s t i c of a l l salmon f i s h e r i e s could not be handled i n the opt imal c o n t r o l model. The d e t e r m i n i s t i c nature of the model and i t s s t eady-s ta te behavior i s sLmply not what one f inds i n a salmon f i s h e r y i n which the biomass i s subject to s t o c h a s t i c m o r t a l i t y i n f l u e n c e s . In sum, a model combining these f ea tures , i . e . , i n t e g r a t i o n of wi th in-season and i n t e r s e a s o n a l a n a l y s i s , a more comprehensive b i o l o g i c a l model i n c o r p o r a t i n g m u l t i p l e - s p e c i e s and age s t ruc tures was c a l l e d f o r . The computer s imula t ion model was designed to incorporate the main u s e f u l features of both the within-season and in ter tempora l models; i . e . , the f i s h e r y product ion funct ion and the capac i ty for wi th in-season ana lys i s as w e l l as the capac i ty f o r i n t e r s e a s o n a l op t imiza t ion v i a determination of the opt imal escapement p o l i c y . In a d d i t i o n to the preserva t ion of these 156 fea tures , the s imula t ion model incorporates a more complete model of the b io logy of P a c i f i c salmon which appl i e s the various types of m o r t a l i t y phenomena experienced by a salmon stock at var ious stages i n i t s l i f e c y c l e . Th i s model a l s o inc ludes a s t o c h a s t i c process i n the m o r t a l i t y sequence to s imulate the e f f ec t s of environmental f a c t o r s on the s u r v i v a l of salmon. Two vers ions of the model were developed: one incorporates the assumption of weekly f l e e t h i r i n g by the so le owner and the other incorporates the assumption of annual f l e e t h i r i n g . The main purpose f o r developing these two vers ions of the model i s to i n v e s t i g a t e the extent of a d d i t i o n a l e f f i c i e n c i e s which can be der ived by i n t e r - f i s h e r y u t i l i z a t i o n of the f l e e t as opposed to commitment of a p o r t i o n of the f l e e t to a p a r t i c u l a r f i s h e r y for the durat ion of the season. 5.2 Findings and P o l i c y Impl i ca t ions . An opt imal escapement p o l i c y emerged from the tr ia l -^and^error process of repeated s imula t ion experiments us ing d i f f e r e n t l e v e l s of minimum annual des i red escapement f o r both the case I and case II formulat ions . The escapement p o l i c y which maximized the present value of net re turns us ing a discount rate of 10% per annum turned out to be the same for both cases , i . e . , 300,000 sockeye and 400,000 p i n k . However, we found t h a t , whi le the d e c i s i o n process of the case II so le owner had exhausted a l l the oppor tun i t i e s f o r opt imal a l l o c a t i o n of a v a i l a b l e e f f o r t throughout the season, the e f f i c i e n c y of the case I regime was increased by the a p p l i c a t i o n of a c o n s t r a i n t on the weekly escapement. Thus, the escapement p o l i c y which maximized the present value of net p r o f i t s f o r the case I model was a 157 minimum desired escapement of 300,000 sockeye and 400,000 pink i n addition to a minimum weekly escapement of 30% of the stock available for that week. While minimum desired escapement was the same for both cases, actual escapement exceeded t h i s minimum by 266,000 f i s h for case I I and 274,000 f i s h for case I (the excess of actual over desired escapement i s based on the mean actual escapement f o r the two models). Based on comparisons of actual escapement for the two simulations with r e s u l t s from the actual fishery, i n addition to other evidence from the simulation r e s u l t s , i t was concluded that escapement p o l i c y i n the actual f i s h e r y has probably been more l i b e r a l than economic optima would dictate. The implications of t h i s finding for management of the actual fishery can be interpreted f a i r l y d i r e c t l y . Given that mean h i s t o r i c a l escapement of sockeye and pink i s approximately 639,000 and 682,000 f i s h , respectively, and mean simulated escapement i s 396,000 and 578,000 sockeye and pink, respectively, both sockeye and pink harvest could be increased. Using the average f i s h prices and weights employed i n the simulation model, the gross annual value of reducing mean escapement 05y increasing the harvest) i n the actual fishery to the mean simulated levels would be approximately $1,035 m i l l i o n . I t i s f a i r to state that the Fisheries and Marine Services knows more about and pays more attention to the b i o l o g i c a l implications of the p o l i c i e s which they adopt. This i s perhaps natural, given the s p l i t j u r i s d i c t i o n , i . e . , the industry being responsible for p r o f i t a b l e harvesting and the Fisheries and Marine Service being responsible for stock management, and the professional background and interests of the majority of Fisheries and Marine Service personnel. However, our resu l t s indicate that t h i s b i o l o g i c a l 1 5 ? emphasis may be r e l a t i v e l y c o s t l y . A c o m a p r i s o n of the r e s u l t s o f the o p t i m a l s i m u l a t i o n r e s u l t s f o r c a s e I and I I assumptions p o i n t e d to s e v e r a l c o n t r a s t s between t h o s e r e s u l t s w h i c h have i m p o r t a n t p o l i c y and/or management i m p l i c a t i o n s . The marked c o n t r a s t between w i t h i n - s e a s o n f l e e t u t i l i z a t i o n p a t t e r n s under the two c a s e s i s one of t h e most o b v i o u s comparisons. The d i f f e r e n t i a l f l e e t u t i l i z a t i o n p a t t e r n s r e s u l t i n d i f f e r e n t i a l w i t h i n - s e a s o n h a r v e s t i n g p a t t e r n s w h i c h a f f e c t the i n t e n s i t y o f o v e r a l l e x p l o i t a t i o n o f t h e s t o c k as w e l l as the e x p l o i t a t i o n of t h e r a c e s w h i c h make up the s t o c k . The p o l i c y and management i m p l i c a t i o n s of t h e se f i n d i n g s a r e s i g n i f i c a n t . I f one v i e w s e s t a b l i s h m e n t of c l e a r l y d e f i n e d p r o p e r t y r i g h t s to f i s h i n a r i v e r system as a means of i m p r o v i n g the performance o f t h e h a r v e s t i n g i n d u s t r y , i t i s c l e a r t h a t the method of e s t a b l i s h i n g s o l e ownership and a t t e n d a n t i n s t i t u t i o n a l r u l e s w i l l r e q u i r e c a r e f u l a t t e n t i o n . E s t a b l i s h m e n t of s o l e ownership of the f i s h e r y by a p r i v a t e f i r m r e q u i r e d to r e n t i t s c a p i t a l equipment on a y e a r l y b a s i s would r e s u l t i n an u n d e r u t i l i z a t i o n of t h e Skeena R i v e r f i s h e r y i n t h e sense t h a t under d i f f e r e n t o r g a n i z a t i o n a l r u l e s a l a r g e r f l e e t c o u l d be u t i l i z e d a t c e r t a i n p e r i o d s d u r i n g the s e a s o n . C o n v e r s e l y , a t o t h e r times d u r i n g the s e a s o n t h e f l e e t s i z e would be l a r g e r than t h a t d i c t a t e d by a s e t o f more f l e x i b l e i n s t i t u t i o n a l r u l e s . The r e s u l t of t h e se e x c e s s e s and d e f i c i e n c i e s of h a r v e s t i n g c a p a c i t y would be l e s s e r p r e s e n t v a l u e of n e t p r o f i t s under the c a s e I I regime than under the e a s e l regime as shown by the s i m u l a t i o n r e s u l t s . Were s o c i e t y t o a u c t i o n r i g h t s to the f i s h e r y , the d i s c o u n t e d sum of n e t p r o f i t s i s an a p p r o x i m a t i o n to the p r i c e a p r i v a t e owner would be w i l l i n g t o pay f o r the r i g h t of s o l e a c c e s s . Thus, i t i s i n s o c i e t y ' s i n t e r e s t t o e s t a b l i s h i n s t i t u t i o n a l r u l e s w h i c h maximize the p r e s e n t v a l u e of n e t p r o f i t s of the f i s h e r y . A c c o r d i n g l y , i t a p pears t h a t a p r i v a t e s o l e 159 owner who would rent vesse l s on a weekly b a s i s could opt imize the f i s h e r y by adjus t ing the f l e e t capac i ty as d i c t a t e d by the stock a v a i l a b l e f o r h a r v e s t . Moreover, i n a d d i t i o n to the reduced present value of net p r o f i t s which r e s u l t s from use of case II assumptions, s o c i e t y must reckon the a d d i t i o n a l net re turns which the vesse ls could generate were they not s i t t i n g i d l e during the e a r l y and l a t e weeks o f the season. On the other hand, there i s no obvious reason why the r i g h t s of so le access should be s o l d to a p r i v a t e i n d i v i d u a l or o r g a n i z a t i o n . The F i s h e r i e s and Marine Service could act as sole owner and could contract with v e s s e l owners and skippers to e x p l o i t the f i s h e r y . Given t h i s arrange-ment, part of the economic rent o f the resource could be returned to the owners of the resource who could at l e a s t p a r t l y o f f s e t the costs of management with the funds thereby d e r i v e d . T h i s would be a d e s i r a b l e r e s u l t i n s e v e r a l r e s p e c t s . The economic performance of the f i s h e r y could be improved provided that the F i s h e r i e s and Marine S e r v i c e , as sole owner, employed economic c r i t e r i a i n determining the f l e e t s i z e to l e a s e . Fishermen as vessel-owners could remain as p r i v a t e entrepreneurs ra ther than employees as they would be under the case II s t r u c t u r e . While the opt imal o r g a n i z a t i o n f o r the f i s h e r y i n terms of the maximization of net re turns appears to be a v a r i a n t of the case I so l e ownership r u l e s t h i s a l so impl ies a f l e e t s i z e which f l u c t u a t e s markedly from week-to-week w i t h i n the season and from season to season. I f par t of the management s t ra tegy i s to maintain r e l a t i v e l y s tab le and continuous employment i n the f i s h e r y the case I so l e ownership regime as descr ibed here appears to do l i t t l e to achieve t h i s r e s u l t . However, as descr ibed below, coast-wide a p p l i c a t i o n of case I f l e e t h i r i n g r u l e s mi t iga te the apparent d i f f i c u l t y which r e s u l t s from the Case I regime to a s ing l e salmon f i s h e r y . 160 To t h i s juncture the management of the Skeena River has been considered l a r g e l y i n i s o l a t i o n from other c o a s t a l salmon f i s h e r i e s . Consider f o r a moment the establishment of so le ownership i n a l l such f i s h e r i e s with the F i s h e r i e s and Marine Serv ice a c t i n g as a s o l e owner who l i c e n s e s ves se l s to e x p l o i t the var ious f i s h e r i e s . I f the t iming of the d i s c r e t e f i s h e r i e s i s such that some peak whi le others wane i t i s c l e a r that the case I regime need not r e s u l t i n wide f l u c t u a t i o n s of employment of ve s se l s coastwide. This adds new scope to the determinat ion of opt imal f l e e t c a p a c i t y . I t i s easy to see how the model developed i n t h i s research could be repeated for a l l f i s h e r i e s coastwide and l i n k e d through response funct ions which would a l l o c a t e ve s se l s to var ious f i s h e r i e s according to t h e i r comparative margina l net re turns i n the d i f f e r e n t f i s h e r i e s . Coastwise a set of so le owners has w i t h i n i t a g iven f l e e t of vesse ls which they d e s i r e to d i spatch to the appropr ia te f i s h e r y much as the owner of a f l e e t of t a x i s d e s i re s to p lace them opt imal ly with respect to p o t e n t i a l f a r e s , w i t h i n the season the f l e e t s i z e i s g iven but vesse l s may be r e t i r e d or constructed from season-to-season. The t iming and s i z e of the v a r i o u s runs w i l l then determine both the s i z e of the f l e e t and the degree to which i t i s f u l l y u t i l i z e d at any po in t i n the season. I t was concluded from a l l the s i m u l a t i o n r e s u l t s that the f l e e t s i z e which opt imized the e x p l o i t a t i o n of the f i s h e r y would have to f l u c t u a t e both w i t h i n the season and from season-to-season, i r r e s p e c t i v e of the f l e e t procurement r u l e s which p r e v a i l e d . Given t h i s conc lus ion some d i f f i c u l t y was experienced i n s p e c i f y i n g the reduc t ion i n present f l e e t s i z e r e q u i r e d to achieve o p t i m a l i t y . A r a t i o n a l e was developed xvrhich r e s u l t e d i n an > order of magnitude estimate of the extent of present excess h a r v e s t i n g c a p a c i t y . Depending upon the f l e e t procurement r u l e s p r e v a i l i n g we concluded 161 that the maximum number of vesse l -days which would be opt imal ly employed i n the Skeena R i v e r dur ing any season would be 14% l e s s f o r case II and 52% less for case I than the minimum number o f vesse l -days a c t u a l l y employed i n any one year over the p e r i o d 1971-1975. T h i s i s i n d i c a t i v e of s i g n i f i c a n t excess h a r v e s t i n g c a p a c i t y . The f i n a l observat ion which can be supported by the s imula t ion r e s u l t s i s t h a t , given the s t r u c t u r e of the model and the p r i c e and cost estimates u t i l i z e d i n the var ious s i m u l a t i o n s , i t does not appear that a r a t i o n a l so l e owner f o l l o w i n g the p r o f i t motive would e x p l o i t the s tock to a p o i n t approaching e x c t i n c t i o n i n a v i r t u a l l y unconstrained f i s h e r y . From t h i s i t would appear that o f f i c i a l management a c t i v i t y i n the f i s h e r y , given the establ ishment of so l e ownership, could not be j u s t i f i e d on stock maintenance grounds. T h i s c l e a r l y strengthens the case for measuring o p t i m a l i t y by the economist ' s , not the b i o l o g i s t ' s , measure. 5.3 Suggestions for Further Research  and Extensions of the Model A number of areas for fur ther research have suggested themselves dur ing the course of the process of development, cons truc t ion and c a l i b r a t i o n of t h i s model. Data are almost always i n s u f f i c i e n t , of poor q u a l i t y or not c o l l e c t e d i n the appropriate form for d i r e c t use . The author's experience has been no exception to th i s general r u l e . However, a general and concerted e f f o r t to expand or improve the data base i s not recommended. Rather, some very goal d i r e c t e d primary research and data c o l l e c t i o n programs would pay d iv idends . For example, fur ther examination of the product ion func t ion i s important for s e v e r a l reasons and the i n v e s t i g a t i o n should be conducted on 162 s e v e r a l l e v e l s . F i r s t , the assumption that vesse ls are homogeneous should be subjected to hypothesis t e s t i n g . I t may be that a s h o r t - r u n model might j u s t i f i a b l y employ the assumptions that un i t s are homogeneous and that there i s l i t t l e or no opportuni ty for input s u b s t i t u t i o n . However, i n a long-run model one should not ignore the v a r i a t i o n i n earnings among skippers which suggests that labor and/or c a p i t a l are not homogeneous. Second, the s u b s t a n t i a l c a p i t a l i z a t i o n of vesse ls i n recent years suggests that t e c h n o l o g i c a l change has markedly a l t e r e d the product ion p r o c e s s . An i n v e s t i g a t i o n o f the v e s s e l product ion func t ion should prov ide answers to these ques t ions . T h i r d , aggregating to the indus try l e v e l , i n v e s t i g a t i o n s of the degree to which gear c o n f l i c t s cause product ion e x t e r n a l i t i e s i s necessary for f l e e t o p t i m i z a t i o n . F i n a l l y , s e v e r a l d i r e c t i o n s i n which t h i s model could be extended are apparent . The most obvious extension i s to model a number o f major salmon f i s h e r i e s along the B r i t i s h Columbia coast , l i n k i n g the i n d i v i d u a l f i s h e r y models by some form of adaptive expectat ions func t ion def ined upon r e l a t i v e f i s h i n g success i n each f i s h e r y p r o x i e d , f o r example, by the s i z e of net r e t u r n s . In such a model adjustment costs would n e c e s s a r i l y be i n c l u d e d . This would permit an i n v e s t i g a t i o n of the e f f e c t of the t iming of runs i n each system on the s i z e of the opt imal f l e e t . In p a r t i c u l a r , whi l e i t was necessary to conclude on the b a s i s of the model developed here ( for case I assumptions) that the opt imal f l e e t s i z e f l u c t u a t e d markedly, i t i s l i k e l y t h a t , cons ider ing the e n t i r e c o a s t a l salmon f i s h e r y , f l e e t u t i l i z a t i o n would be s i g n i f i c a n t l y more constant . One would then expect that the r e s u l t s f o r case I and case II would begin to merge and that the f l e e t u t i l i z a t i o n patterns would become more s i m i l a r . 163 That i s , the case I I so l e owner, having committed h imse l f to a f l e e t for the e n t i r e season, may now u t i l i z e what would otherwise be i d l e ves se l s i n some other f i s h e r y . In many respects the present research has paved the way f o r t h i s coast-wide extension which i s c l e a r l y a very ambitious undertaking . For the cons tra ined s imulat ions reported i n the previous chapter the so le owner a p p l i e d the same l e v e l of escapement c o n t r o l year a f t e r year without i n c o r p o r a t i n g any knowledge r e s u l t i n g from experience. An improve-ment i n t h i s r u l e i s appropr ia te so that the escapement c o n t r o l l e v e l could be adjusted from y e a r - t o - y e a r based on the s o l e owner's accumulated knowledge of how the prev ious l e v e l s of the c o n t r o l were a f f e c t i n g the l e v e l of net r e t u r n s . T h i s would i n v o l v e the establishment of a more complicated feedback loop . In a d d i t i o n to the coast-wide extension of the o p t i m i z a t i o n model m o d i f i c a t i o n s should be made so as to inc lude a l l species captured i n each f i s h e r y as w e l l as a l l gear types — g i l l n e t , se ine and t r o l l . Th i s w i l l r e q u i r e that the research address the quest ion of whether, under the so le ownership circumstances es tabished i n t h i s model, the se ine f i s h e r y could continue to pay by the share system. Payment by the share system would have to be r e c o n c i l e d wi th the assumption that the opportuni ty cost of product ive resources i s def ined by what they could earn i n an a l t e r n a t i v e f i s h e r y . 164 APPENDIX A ESTIMATION OF COEFFICIENTS AND PARAMETERS 41.0 In troduct ion Thi s Appendix contains the es t imates , es t imat ion methods and data sources for the parameters and c o e f f i c i e n t s employed i n the s i m u l a t i o n model descr ibed i n chapter 3. C e r t a i n of these parameters and c o e f f i c i e n t s were a l so employed i n the a n a l y t i c a l models developed i n chapter 2. A.1.1 I n i t i a l Condi t ions The biomass of sockeye and pink salmon and the propor t ions of sockeye maturing at var ious ages are i n i t i a l c o n d i t i o n s . The sockeye biomass for two races—Babine and non-Babine—estimated by L a r k i n and McDonald were employed i n t h i s model."'" L a r k i n and McDonald estimated the e q u i l i b r i u m sockeye biomass by averaging the 1912-22 catch data of Babine and non-Babine sockeye assuming that t h i s data represented i n i t i a l r e l a t i v e abundance of the two races . The i n i t i a l biomass i s 980,000 pieces f o r the Babine race and 320,000 pieces for the non-Babine races . For the odd-year and even-year pink races , 1955-73 estimates of b i - y e a r l y biomass supp l i ed by the 2 F i s h e r i e s and Marine Service were averaged. These averages were taken as represent ing i n i t i a l r e l a t i v e abundance of the two p ink races i n e q u i l i b r i u m . The even-year pink biomass was ca l cu la ted at 1.587 m i l l i o n salmon while the odd-year estimate was se t at 1.935 m i l l i o n 165 A . 1 . 1 . 1 . Formation o f the Run The values of the proport ions of the sockeye biomass by race which mature at var ious ages are shown i n Table XII below. These proport ions are used as i n i t i a l condi t ions u n t i l the model has generated s u f f i c i e n t data for endogenous determinat ion of these porport ions . TABLE XII PROPORTIONS OF SOCKEYE MATURING AT VARIOUS AGES PROPORTION SOCKEYE MATURING — — AT AGE BABINE NON-BABINE 3 0.05 0.05 4 0.49 0.30 5 0.46 0.54 6 0.0 0.11 These same propor t ions were employed by L a r k i n and McDonald i n t h e i r model 3 o f the Skeena R i v e r sockeye. A l . 2 " Time of Entry C o e f f i c i e n t s The time of entry c o e f f i c i e n t s employed to determine the weekly stock from the annual recrui tment are shown i n Table XIII below. 166 TABLE XIII TIME OF ENTRY COEFFICIENTS Sockeye P ink Week Babine Non-Babine Odd Even 1 0.0022 0.0022 0.0 0.0 2 0.0092 0.0092 0.0 0.0 3 0.0246 0.0246 0.0 0.0002 4 0.0430 0.0430 0.0 0.0012 5 0.0702 0.0702 0.0010 0.0034 6 0.117 6 0.1176 0.0082 0.0074 7 0.2050 0.2050 0.0550 0.0306 .8 0.2040 0.2040 0.1414 0.1064 9 0.1792 0.1792 0.2492 0.2324 10 0.1006 0.1006 0.2686 0.2654 11 0.0288 0.0288 0.1914 0.2166 12 0.0110 0.0110 0.0740 0.0972 13 0.0030 0.0030 0.0096 0.0326 14 0.0014 0.0014 0.0016 0.0060 15 0.0 0.0 0.0002 0.0010 Source: Northern Operations Branch, F i s h e r i e s and Region, Vancouver, B r i t i s h Columbia. Marine S e r v i c e , P a c i f i c These c o e f f i c i e n t values are averages of t iming d i s t r i b u t i o n c o e f f i c i e n t s a c t u a l l y observed i n the Skeena River f i s h e r y over the p e r i o d 1955-72. The source of t h i s data i s the Northern Operations Branch of the F i s h e r i e s and Marine S e r v i c e , P a c i f i c Region, Vancouver, B r i t i s h Columbia. 167 A1.3 Es t imat ing the Parameters of the  Product ion Funct ion A l . 3 . 1 Introduct ion The theory under ly ing the product ion funct ion employed i n the s imula t ion model was developed i n sec t ion 2.2 of Chapter 2. The object of the d i s c u s s i o n developed below i s to descr ibe and i n t e r p r e t the r e s u l t s of the e m p i r i c a l es t imat ion of the parameters of t h i s func t ion based on data s er i e s generated.by the a c t u a l g i l l n e t f i shery of the \ Skeena R i v e r . A 1.3.2 The Time P e r i o d for Es t imat ion The b a s i c time p e r i o d of the s imulat ion model descr ibed above i s the week. There fore , the appropriate parameter estimates must be based upon a weekly es t imat ion p e r i o d . This poses some d i f f i c u l t y due mainly to the manner i n which data i s accumulated. The fo l lowing w i l l b r i e f l y describe the current management of the f i s h e r y and the means f o r ad jus t ing for the d i f f i c u l t i e s r e f e r r e d to above. The des i red escapement ( in t o t a l ) and d i s t r i b u t i o n of escapement (based on b i o l o g i c a l objec t ives ) i s achieved by opening and c l o s i n g the f i shery to commercial vesse ls at the i n d i c a t e d t ime. Normally , f i s h i n g commences each week on a Sunday evening. An i n i t i a l p r e d i c t i o n i s made of the number of vesse ls which w i l l f i s h and the t o t a l f i s h i n g e f f o r t which the stock i s capable of wi ths tanding . This informat ion i s t r a n s l a t e d to a p r e d i c t i o n of the number of days the f i shery w i l l remain open to commercial 168 h a r v e s t i n g . In recent years the modal number of days has been approximately three per week. By monitor ing the catch and combining th i s informat ion with pre-season p r e d i c t i o n s as to s tock s i z e , f i shery managers can determine whether to c lose the f i s h e r y e a r l y , leave i t open a day longer or stand by t h e i r o r i g i n a l es t imates . During t h i s p e r i o d any v e s s e l can and w i l l make one or more d e l i v e r i e s e i t h e r to i t s home p o r t or to a packing v e s s e l i n the f i s h i n g a r e a . The number of d e l i v e r i e s made dur ing the 'open' p e r i o d w i l l depend upon a v a r i e t y of circumstances among them, s i z e of catch , s i z e of v e s s e l h o l d , weather, d i s tance to home p o r t , and so f o r t h . The d i f f i c u l t y th i s presents f o r est imat ion of a product ion funct ion i s that for any week, f i s h i n g ' i n t e n s i t y ' may vary with the number of days the f i s h e r y i s open and/or the number of vesse ls on the f i s h i n g grounds. One thus requires data on the number of days the season i s open each week dur ing each year and data on the number of vesse ls f i s h i n g each day. To r e t u r n to the time p e r i o d quest ion there are three a l t e r n a t i v e s based on the a v a i l a b l e d a t a — d a i l y , weekly and seasonal . A d a i l y es t imat ion i s not e n t i r e l y appropriate f o r t h i s model s ince the fundamental time p e r i o d i s the week. This Is fortunate s ince d a i l y catch data i s not r e a d i l y a v a i l a b l e f o r the Skeena R i v e r f i s h e r y . Since the s imula t ion model i s a long p e r i o d model an es t imat ion of the product ion funct ion over a long span of time would be h e l p f u l to determine parameter values i n l i g h t of l o n g r - p e r i o d changes. In th i s respect a season product ion funct ion es t imat ion would be warranted. However, i n a long-per iod es t imat ion of t h i s funct ion one would be required to account for t e c h n o l o g i c a l change, a d i f f i c u l t problem both t h e o r e t i c a l l y and e m p i r i c a l l y . In a d d i t i o n , the range of v a r i a t i o n i n number of vesse l s f i s h i n g during any one season can 169 be s u b s t a n t i a l thus render ing a seasonal est imate of vesse ls f i s h i n g some-what suspect . A g a i n , the seasonal product ion funct ion est imat ion would q u i c k l y encounter data l i m i t a t i o n s f o r , whi le catch and recruitment s e r i e s beginning i n 1945 are a v a i l a b l e , s er i e s on the number of days the f i s h e r y was open each year and the number of vesse l s f i s h i n g each day are not a v a i l a b l e p r i o r to 1969. The conc lus ion therefore i s that data a v a i l a b i l i t y and the requirements of the model mesh i n t h i s case and the parameters of a weekly product ion funct ion have been est imated. Al . 3 . 3 Data Requirements The product ion funct ion contains three v a r i a b l e s for which data are r e q u i r e d . These are the stock of f i s h i n the area during a given week and season, the number of vesse ls f i s h i n g on any given day dur ing which the f i s h e r y i s open and the weekly ca tch . These data are a v a i l a b l e from the records of the F i s h e r i e s and Marine S e r v i c e . Catch s t a t i s t i c s for s t a t i s t i c a l area 4 (the main area of the Skeena g i l l n e t f i s h e r y ) by week and species were obtained from the 'D' sheets of the p u b l i c a t i o n B r i t i s h Columbia Catch S t a t i s t i c s . These data were aggregated to form a weekly t o t a l catch s e r i e s for both spec ie s . Data for the t o t a l run of each species i s a v a i l a b l e on an annual bas i s on ly . Th i s data was obtained from b i o l o g i s t s of the Northern Operations Branch of the F i s h e r i e s and Marine Service r e g i o n a l o f f i c e i n Vancouver. T o t a l run s i z e i s determined by summing escapement and catch on a seasonal b a s i s . Escapement data are obtained by counting the spawning f i s h which pass through the Babine counting fence and by spawning ground surveys and counts for races which do not spawn i n the Babine system or above the Babine fence. Catch data are obtained by sales r e c e i p t s which document sa les by f ishermen, packers and processors . 170 The annual t o t a l run sizes were d i s t r i b u t e d to weeks during the season on the basis of the same c o e f f i c i e n t s employed i n the simulation model (see page 166, Table XITT).' The c o e f f i c i e n t s used are,.averages of the percentage of the t o t a l run for the season observed i n the fishery during each week of the season. The l i m i t a t i o n on the number of years which could be included i n the estimation was caused by paucity of data on the number of vessels f i s h i n g each week. The source of these data i s the monthly b u l l e t i n of the Skeena River Management Committee which i s submitted to the Regional Director of F i s h e r i e s . This b u l l e t i n describes the a c t i v i t y i n the Skeena fishery during the month. Included i n t h i s descriptive information are estimates of the number of vessels i n s t a t i s t i c a l area 4 during each week. This information i s accompanied by a report on the number of days the fishery was open during each week. The b u l l e t i n covers the period 1969-1974 and yield s 48 observations on these two variables. An unsuccessful attempt to extend the period of estimation as far back as 1963 was made i n the following fashion. In 1963 the Fisheries Service began publishing a series on the number of deli v e r i e s made per week by g i l l n e t , seine and t r o l l vessels. I f a consistent and stable relationship could be hypothesized to exist between the number of vessels f i s h i n g as reported i n the Skeena River Management Committee (SRMC) b u l l e t i n and the number of d e l i v e r i e s made by g i l l n e t vessels per week, t h i s l a t t e r variable could be adjusted as required and used as a proxy for vessels i n the estimation. A correl a t i o n was run between the number of vessels f i s h i n g and the number of d e l i v e r i e s during the period 1969 to 1974. The resu l t s were disappointing. The simple correlation c o e f f i c i e n t i s 0.60. In addition, 171 a regress ion of d e l i v e r i e s on vesse l s f i s h i n g was run s imultaneously wi th equa l ly d i s a p p o i n t i n g r e s u l t s . In the equation without a constant term 2 the R turned out to be 0.24 whi le i n the equation w i th a constant term 2 the R was 0.36. In s p i t e of the fac t that the c o e f f i c i e n t of d e l i v e r i e s was s i g n i f i c a n t at the 99% l e v e l i n both regress ions the r e l a t i o n s h i p between vesse l s f i s h i n g and d e l i v e r i e s i s not s u f f i c i e n t l y strong to employ the l a t t e r as a proxy f o r the former. I t was thus necessary to use the SRMC data and a smal ler sample p e r i o d . This data i s arrayed i n Table XV. A l . 3 . 4 Funct iona l Form The product ion func t ion was estimated us ing the l o g a r i t h m i c t r a n s -formation of the exponent ia l form on the U n i v e r s i t y of B r i t i s h Columbia implementation of TSP. Repeating equation (2.15) for convenience v/e have 6 + 1 6 + 1 (A .1) C = AV R The l o g a r i t h m i c form of ( A . l ) i s the s p e c i f i c a t i o n used for es t imat ion purposes. Taking logs of ( A . l ) y i e l d s (A.2) l n C = l n A + (6^1) l n V + (6 2 +l) In R A l . 3 . 5 Regression Results The i n i t i a l regress ions were run with the constant term and r e s u l t e d i n the fo l l owing parameter va lues . 172 (A.3) In C = -2.8251 + l n 0.7371 l n V + 0.72 l n R (-1.48) (2.93) (3.42) A l l regress ions contained 42 observat ions of the endogenous and exogenous v a r i a b l e s . I t was necessary to exclude s i x sets of observat ions due to u n a v a i l a b i l i t y of data f o r one or the other of the three v a r i a b l e s . The t - s t a t i s t i c s are shown i n brackets below the c o e f f i c i e n t est imates . The 2 - 2 adjusted R , R f o r the above r e g r e s s i o n i s 0.62. The Durbin-Watson s t a t i s t i c of 2.6036 f a l l s i n s i d e the i n c o n c l u s i v e reg ion i n d i c a t i n g that t h i s t es t for the absence of a u t o c o r r e l a t i o n f a i l s . Both c o e f f i c i e n t s i n the above r e g r e s s i o n have the c o r r e c t s ign and are s i g n i f i c a n t at the 99% l e v e l of conf idence . The constant term i n t h i s equation i s j u s t s i g n i f i c a n t at the 90% l e v e l of conf idence . A second r e g r e s s i o n was run wi th no constant term. F o r c i n g the regres -s ion l i n e through the o r i g i n i s t h e o r e t i c a l l y j u s t i f i e d by the f a c t that without ves se l s f i s h i n g and/or without f i s h a v a i l a b l e f o r h a r v e s t i n g , no harvest can r e s u l t . The r e s u l t s of t h i s r egres s ion are d i sp layed i n equation ( A . 4 ) . (A.4) l n C = 0.754 In V + 0.486 l n R . (2.96) (3.46) The e l i m i n a t i o n of the constant term has had only a smal l e f f ec t on the v e s s e l c o e f f i c i e n t , changing i t from 0.74 to 0.75. The e f f e c t on the c o e f f i c i e n t of the stock i s more marked. The va lue drops from 0.72 to 0.486. Both c o e f f i c i e n t s i n the l a t t e r r e g r e s s i o n have the c o r r e c t s ign and are s i g n i f i c a n t at the 99% l e v e l . The Durbin-Watson s t a t i s t i c again f a l l s i n the i n c o n c l u s i v e reg ion—the hypothesis of no a u t o c o r r e l a t i o n cannot be 2 r e j e c t e d . The R for t h i s equation drops s l i g h t l y to 0.59. A rough and ready method for determining the extent of t e c h n o l o g i c a l 1 7 3 change, i f any, i s to inc lude a time trend among the explanatory v a r i a b l e s i n the r e g r e s s i o n . This was done w i th s u r p r i s i n g r e s u l t s . The c o e f f i c i e n t of the time trend v a r i a b l e c a r r i e d a negat ive s ign i n d i c a t i n g t h a t , wi th the in f luence of vesse l s and stock removed, catch i s d e c l i n i n g over the es t imat ion p e r i o d . I f t e c h n o l o g i c a l change i s a f a c t o r , other in f luences have swamped i t s e f f e c t on ca tch . The regress ion wi th the i n c l u s i o n of the time trend and exc luding the constant term i s (A..5) In C = 0.7798 In V + 0.55 In R - 0.344 In T (3.15) (3.94) (-1.94) -2 The R for th i s equation i s 0.61 and the Durbin-Watson s t a t i s t i c of 2.30 f a l l s j u s t i n s i d e the reg ion for which the no a u t o c o r r e l a t i o n hypothesis i s r e j e c t e d . One fur ther mod i f i ca t ion to the b a s i c s p e c i f i c a t i o n was made i n order to complete the es t imat ion of the product ion f u n c t i o n . During c e r t a i n days or weeks throughout the es t imat ion p e r i o d fishermen belonging to the United Fishermen and A l l i e d Workers Union (UFAWU) were on s t r i k e . During the same per iods members of the Pr ince Rupert Fishermen's Co-op may or may not have honored the UFAWU s t r i k e . A b inary dummy v a r i a b l e t a k i n g the value 1 during s t r i k e days and zero otherwise was created to model the e f f e c t of s t r i k e s . Inc lus ion of t h i s v a r i a b l e re su l t ed i n the f o l l o w i n g regress ion r e s u l t s (A.6) In C = 0.822 In V + 0.44 In R + 0.74 D (3.185) (3.10) (1.32) -2 The adjusted R for t h i s equation i s 0.59. The Durbin-Watson s t a t i s t i c of 3.02 f a l l s i n the i n c o n c l u s i v e region i n which the no a u t o c o r r e l a t i o n 174 hypothesis cannot be r e j e c t e d . The dummy v a r i a b l e i s not s t a t i s t i c a l l y s i g n i f i c a n t at normally accepted l e v e l s of confidence. Since the 90% confidence i n t e r v a l for t h i s c o e f f i c i e n t estimate inc ludes zero , no s i g n i f i c a n c e can be attached to the s ign of the c o e f f i c i e n t es t imate . A f i n a l equation with a l l the above in f luences inc luded was run to determine the explanatory power of the f u l l set of v a r i a b l e s . Resu l t s of t h i s r e g r e s s i o n were as fo l lows: (A.7) l n C = 0.846 l n V + 0.51 l n R -0.342 l n T + 0.729 l n D (3.39) (3.58) (-1.94) (1.35) The Durbin-Watson s t a t i s t i c for t h i s r e g r e s s i o n a l so f a l l s i n the 2 i n c o n c l u s i v e r e g i o n . The adjusted R has a va lue of 0.59. In view of the regres s ion r e s u l t s reported here i t i s concluded that the c o e f f i c i e n t estimates obtained i n equation (A.4) w i l l be employed i n i n i t i a l runs of the s imula t ion model. The i n c l u s i o n of the time trend and the dummy v a r i a b l e to account for s t r i k e s d id not have s u f f i c i e n t margina l explanatory power to warrant t h e i r i n c l u s i o n i n the s imula t ion model s ince t h i s would have necess i ta tes forecas t s for these v a r i a b l e va lues throughout the s imulat ion p e r i o d . Appendix B report s the r e s u l t s of a s e r i e s of s imulat ions which were run wi th the product ion f u n c t i o n parameters as estimated here (equation A.4) as compared to the estimates obtained by Roberts .^ The comparative s imula t ion r e s u l t s reported there are the b a s i s for a d e c i s i o n to use the parameters estimated by Roberts i n fur ther s imula t ions . The parameter estimates are d i sp layed i n Table XIV. 175 TABLE XIV PRODUCTION FUNCTION PARAMETERS LOOSE ROBERTS 0.541 0.754 0.3335 0.486 0.7788 A1.4 Values of Economic Parameters and V a r i a b l e s A l . 4 . 1 P r i c e Est imates In Table XVI below a twenty-four year s e r i e s of wholesale p r i c e s for net caught sockeye and pink i s d i s p l a y e d . This Table was der ived from the p u b l i c a t i o n B r i t i s h Columbia Catch S t a t i s t i c s for the p e r i o d 1951 to 1974 . 5 Constant of P r o p o r t i o n a l i t y , A E l a s t i c i t y of output WRT Vesse l -days , S-j+l E l a s t i c i t y of output WRT Stock S i z e , 6 9 + l - - \ : - ""176" " I TABLE" XV " ~ ' — DATA EMPLOYED IN THE PRODUCTION FUNCTION ESTIMATION 1 1969-19-74 Observation Total Total Day's Total Year Number Catch 1969 1 ' 1 24,404 . 2 _ 61,029 - 3 79,865 4 177,560 ' * 5 '- 127,503 6 -_ _129.,135 7 113,676 ' 8 " 53,566 1970 Z "32,828 J V 2 33,521 3 55,484 . 4 - 204,840 ^.^-^ =^^^^S - - - = = - - - = = = . = ^ - 6 v-275 6 93,698 " " " "7 412,497 . 8 207,683 1971 1 1,714 2 7,458 3 . 28,820 4 146,071 - 5 274,640 6 223,232 7 577,973 8 126,742 1972 1 15,944 2 40,372 - 3 50,462 4 107,001 _ 5 -1827640 - " 6 182,191 - - - 7 263,963 — — - T . _ _ l — : 8"- ""- •3l8- r72l 1973 1 29,203 - - - 2-- 33;881 -.— - 3 - -:r -—-—100 r5 8 2 A 572,105 5 388,193 6 290,073 7 - 253,429 8 97,924 1974 1 364 2 25,215-; " 3 145,057 4 129,883 5 390,362 6 467,166 7 186,702 8 645 Vessels Fishing Stock 300 2.0 45,540 . 317 . .. 2.0„_ 75,355 - -423 _.. 2.0 - 173,871 549 2.0 268,862 665 3.5 409,697 . 55.4 ... 1.0 .. 452,918 525 1.5 526,196 ' 349 " 1.5 " "353,930 " 250 " .1.5 ... . 86,796 : 433 - 1.-5 _ -165,687 - -382 . 1.5 274,415 375 2.0 322,151 460 - 2^0 ---- -261,505 -n.a. 0.0 939,368 _._Ln.a. .1.5 . 7 676,080 524 . 3.0.. 331,131 35 2.0 27,637 40 2.0 42,376 387 " 1.0 156,602 623 2.0 471,939 563 : 3.0 545,401 639 2.0 758,472 670 4.0 1,056,241 574 2.0 465,816 256 2.0 33,182 312 2.5 60,114 --r—249 " 2 5 • 69,884 282 1.0 502,969 500 3.0 378,960 664 2.0 983,790 n.a. 4.0 815,162 ---485 . : -. . 4 . 6 749,321 245 2.0 62,847 "349 ~ 2.0 — 83,796 - 588 6.0 —619,153 521 5.0 815,114 542 3.0 517,964 433 3.0 718,182 281 2.0 " 689,677 n.a. n.a. 340,867 409 2.0 65,321 -—422 - 3.0— —--89,102 778 3.0 648,190 501 6.0 810,254 862 5.0 411,229 623 3.0 487,883 n.a. n.a. 250,227 90 2.0 203,644 177 TABLE XVI PRICES OF NET CAUGHT SOCKEYE AND PINK SALMON, 1951-1974 C / l b Most recent ten-year average Most recent f i v e - y e a r average 43.9 50.2 Year Sockeye Pink 1951 25.0 9.5 1952 25.0 8.0 1953 22.3 7.3 1954 22.1 7.8 1955 24.0 8.9 1956 27.5 9.0 1957 28.1 9.4 1958 28.1 9.2 1959 31.2 10.9 1960 35.3 11.9 1961 33.3 11.3 1962 33.5 11.6 1963 34.0 10.0 1964 36.0 11.0 1965 37.1 11.6 1966 37.1 11.6 1967 37.5 12.0 1968 37.8 12.4 1969 38.8 14.3 1970 39.4 14.7 1971 41.5 15.8 1972 43.0 17.0 1973 65.0 27.0 1974 62.0 23.0 Sockeye Pink Twenty-four year average 35.19 12.3 15.94 19.50 The trend p r i c e of salmon has c l e a r l y been i n c r e a s i n g over the past decade and more. Convert ing the above s e r i e s to an index wi th a base year (1961-100) we f i n d that the index p r i c e of sockeye i n 1962 i s 100.6 and that of p ink i n the same year i s 102.7. The 1974 index p r i c e s f o r sockeye and pink r e s p e c t i v e l y are 186.2 and 203.5. For comparison, the Vancouver Consumer P r i c e Index (1961=100) had a J u l y 1962 value of 102.3. 178 The July 1974 value of t h i s index i s 176.6. Thus, while the trend price of salmon has d e f i n i t e l y increased, r e l a t i v e to other consumer products the price of salmon has remained remarkably stable over the past twenty years. While the prices for net caught sockeye and pink salmon displayed i n Table XVI are wholesale prices and therefore not d i r e c t l y comparable to the r e t a i l p r i c e index quoted, the alter n a t i v e comparison to a wholesale price index would not be e n t i r e l y v a l i d since no such indices are reported on a regional basis. The conclusion we seek to draw i s whether the r e l a t i v e price of salmon has changed i n the recent past. On the basis of the information presented, one could tentatively conclude that there has been no marked change i n the r e l a t i v e price of salmon, p a r t i c u l a r l y i f wholesale and r e t a i l prices can be assumed to move together. For purposes of the computer model i t i s assumed that r e l a t i v e prices of salmon do not change throughout the simulation period. The 1974 prices w i l l be employed i n the model. The prices are expressed i n cents per pound of f i s h while the output of the harvesting routine i s given i n terms of the number of f i s h . Estimates of the average weights of sockeye and pink salmon are therefore required. Such estimates were obtained from b i o l o g i s t s at the Fisheries and Marine Service.*' The estimated average weight per sockeye salmon i s 6.2 pounds and that of pink salmon i s 3.9 pounds. Al.4.2 Cost Estimates The ca l c u l a t i o n of t o t a l cost of f i s h i n g for the sole owner contains several important components. I t w i l l be recalled from the discussion of chapter 2 that the sole owner i s assumed to have the option of renting vessels by the week (case I) and by the year (case I I ) . In both these 179 cases the so le owner must pay the v a r i a b l e costs of operat ing a v e s s e l each day. Since fishermen (vesse l owners) are assumed to have the opt ion of f i s h i n g i n some f i s h e r y other than the Skeena R i v e r , the so le owner of the Skeena R i v e r must pay fishermen the p o t e n t i a l net returns of t h e i r f i s h i n g i n o ther r i v e r s . This opportunity cost i s assumed to be the d a i l y net r e t u r n that v e s s e l owners could rece ive by f i s h i n g elsewhere. There fore , i n order to r e l a t e our r e s u l t s to the condi t ions a c t u a l l y o c c u r i n g , estimates of gross returns and costs of f i s h i n g are r e q u i r e d . While in format ion o f t h i s s o r t i s l i m i t e d , estimates have been constructed from a v a r i e t y of sources . The b a s i c source used to develop the fo l lowing c a l c u l a t i o n s i s found i n a 1969 p u b l i c a t i o n by the F i s h e r i e s and Marine Service e n t i t l e d , Returns from F i s h i n g Vesse ls i n B r i t i s h Columbia.^ Thi s p u b l i c a t i o n reports the r e s u l t s of a three-year (1966-1968) study i n t o the economics of the i n d i v i d u a l f i s h i n g e n t e r p r i s e i n B r i t i s h Columbia. Gross returns for the season and costs of f i s h i n g are estimated f o r d i f f e r e n t v e s s e l types—salmon g i l l n e t vesse ls among them. These data can be employed to der ive estimates of net returns for g i l l n e t v e s s e l s . The f i r s t step i n developing estimates of net returns to g i l l n e t vesse l s was to determine the s i z e and replacement cost of the (assumed) homogeneous vesse l s f i s h i n g i n the Skeena R i v e r f i s h e r y . Data on the s i z e s and cons truc t ion costs of a l l g i l l n e t vesse ls b u i l t i n 1974 was obtained 8 from the v e s s e l insurance o f f i c e r of the F i s h e r i e s and Marine S e r v i c e . From t h i s data a weighted average ves se l s i z e was c a l c u l a t e d (see Table XVII for the data ) . Th i s weighted average i s 34.4 f ee t . The weighted average cost of t h i s v e s s e l i s $29,524. To s i m p l i f y c a l c u l a t i o n s i t i s assumed that the replacement cost i s $30,000 and that i t s length i s 35 f e e t . F i n i s h i n g touches on the v e s s e l and i n s t a l l a t i o n of e l e c t r o n i c 180 9 gear are estimated to add another 10% to the replacement cost. This brings the t o t a l replacement cost to $33,000. The net returns for t h i s vessel are displayed i n Table XVIII with the 1966-1968 returns f o r the same size vessel as reported i n the study c i t e d above.^ With the information contained i n Table XVIII i t i s now possible to specify the minimum amount which the sole owner must offer to pay a vessel owner to f i s h i n the Skeena River. From the point of view of the vessel owner the variable cost of operating a vessel would be incurred whether he fished i n the Skeena River or i n a free access f i s h e r y . However, i n the l a t t e r case the catch value belongs to the vessel owner who, assuming r a t i o n a l i t y and perfect foresight, would not f i s h unless the catch value equalled or exceeded variable costs. I f the vessel owner fishes under contract to the sole owner, the catch value obtained i s the property of the sole owner. Hence, variable costs must be covered by the sole owner's contract price to h i r e a vessel. The same i s also true of fixed costs since any gross returns i n excess of variable costs make a contribution to payment of fixed costs when the vessel owner fishes for his own account. The opportunity cost of c a p i t a l i s an opportunity cost of f i s h i n g under contract to the Skeena sole owner assuming that vessel owners could earn t h i s return by exercising the option to f i s h elsewhere. Any gross receipts the vessel owner could earn i n excess of variable costs, fixed costs and opportunity costs of c a p i t a l are attributed to the vessel owner as his personal opportunity cost. Thus i t i s concluded that the net return estimated above i s the average and marginal opportunity cost of a vessel with skipper and gear. This i s the minimum payment which the Skeena River sole owner must pay to contract the services of vessel owners. 181 TABLE XVII CALCULATION OF WEIGHTED AVERAGE 1974 REPLACEMENT COST** OF THE WEIGHTED AVERAGE LENGTH OF SALMON GILLNET VESSEL* CONSTRUCTED IN 1974 Average Cost T o t a l Costs Length (feet) Number of Vesse ls per Vesse l (Do l lars ) for Length (Dol lars ) 30 7 $20,286 $142,002 31 - - • 32 1 30,000 30,000 33 35 22,385 783,475 34 1 35,000 35,000 35 2 35,000 70,000 36 17 33,575 570,775 37 5 45,720 228,600 38 - - -39 1 54,000 54,000 40 1 60,000 60,000 41 5 45,000 225,000 42 1 45,000 45,000 43 - - -T o t a l s 76 $2,243,852 Weighted Average Replacement Cost = 2,243,852/76 = $29,524 $29,524 + 10% for f i n i s h i n g , e tc : = $33,000 * G i l l n e t vesse l s on ly ; excludes combination vesse l s * * Source: F i s h e r i e s and Marine S e r v i c e , P a c i f i c Region, Vancouver, B . C . 182 TABLE XVIII ESTIMATED DAILY NET RETURNS OF A VESSEL WHICH GILLNETTED FOR SALMON ONLY 1974 14 1966 Average V e s s e l P a r t i c u l a r s Length V a l u e 1 1 Gross Rece ipts 12 Salmon sa les ' V a r i a b l e Costs Repairs and Maintenance—Hul l , engine and e l e c t r o n i c equipment Repair and Maintenance on gear!3 F u e l and l u b r i c a t i o n Foodl5 Deprec ia t ion M i s c e l l a n e o u s ^ 13 T o t a l V a r i a b l e Costs F ixed Costs Marine Insurance 13 13 Wharfage and S l i p Charges T o t a l F i x e d Costs 17 Return on Investment Net Returns to V e s s e l Owner 35 feet $33,000 $338 22 21 14 11 32 18 118 6 3 9 44 167 35 feet $14,300 $200 13 12 8 7 14 10 64 3 2 5 13 118 183 TABLE XIX GILLNET VESSEL VARIABLE OPERATING COSTS PER DAY Cost Category 75 Day Season Repair and Maintenance^ ( H u l l , Engine and E l e c t r o n i c Equipment) $ 22.00 Repair and Maintenance on Gear 21.00 3 F u e l and L u b r i c a t i o n 14.00 F o o d 2 10.00 . . 1 Deprecxatxon 32.15 3 4 Misce l laneous ' 18.00 T o t a l V a r i a b l e Costs per Day 115.00 Assuming a 13 year l i f e , 5% salvage value and s t r a i g h t - l i n e economic d e p r e c i a t i o n ( i . e . , same number of days of operat ion per y e a r ) . $33,000 x 0.95 = $31,350 7 13 = 2411.59 per year; assuming 75 days operat ion per year = $32.15 per day. See Table XX. V a n c o u v e r CPI (1960 = 100); J u l y 1967 = 11.4; J u l y 1974 = 158.5; $500 x 158.5 / 111.4 = 711.40 / 30 = $23.71; $711.40 / 75 = 10.00. 3 Based on wholesale p r i c e index for Canada. Source - Canada Yearbook, 1974. 1935-39 = 100; 1967 = 264.1; 1974 = 460.6. Includes wages of he lper . 184 TABLE XX CALCULATION OF NET RETURNS PER VESSEL-DAY 1. Roberts assumed a 25 week season for a 35 foot v e s s e l such as we are es t imat ing costs of here . The average number of f i sh ing-days per week i n 1966-68 was 2.98 - 3. Hence, i n c a l c u l a t i n g costs per day, e t c . , we assume a 75 day season. 2. C a l c u l a t i o n of Gross Returns: Assumed a 75 day f i s h i n g season and took Roberts f i gure of $15,000 gross f o r 35 foot g i l l n e t t e r . This was d i v i d e d by 75 to get $22 per day. This d a i l y f igure was i n f l a t e d by the WPI to get $348 (200 x 460.6 / 264.1) . Gross Returns $348.00 Gross Costs V a r i a b l e costs from Table XIX 115.00 F ixed costs Marine Insurance Wharfage and s l i p charges 6.00 3.00 Return to c a p i t a l (assumed 10% opportunity cost) 40.00 T o t a l Costs $164.00 Net Economic Returns $184.00 I n f l a t e d by WIP 1935-39 = 100; 1967 = 264.1; 1974 = 460.4 185 A1.5 Parameters of the B i o l o g i c a l Model A l . 5 . 1 Estimates of Egg Product ion Factors The sockeye egg product ion fac tors employed i n t h i s model are taken 18 from the L a r k i n and McDonald study and are d i sp layed i n Table XXI below. TABLE XXI EGG PRODUCTION FACTORS SOCKEYE Race Age ' • Babine Non-Babine 3 0.0 0.0 4 0.951 0.951 5 1.152 1.152 6 1.152 1.152 Three year o l d sockeye happen a l l to be males, hence the zero valued egg product ion f a c t o r s . Pink salmon do not e x h i b i t the v a r i a b l e age of r e t u r n . While there i s d e f i n i t e l y a s i z e d i s t r i b u t i o n of r e t u r n i n g pinks and while the number of eggs produced w i l l vary p o s i t i v e l y with the s i ze of the female, for p r a c t i c a l purposes i t w i l l be assumed that each female deposits the same quant i ty of eggs of average. Data on the parameters of the populat ion b io logy of p ink salmon i s l i m i t e d compared to that a v a i l a b l e for sockeye. Several researchers have publ i shed s tudies on p ink salmon which inc luded data on egg product ion but 186 only one source per ta ined to the Skeena R i v e r . Data for Sashin Creek, 19 Alaska cover ing the years 1934-1966 publ i shed by Skud i n d i c a t e that the average fecundi ty per female l i e s i n the range 1732 to 2299 eggs. Skud employed an average of 2000 eggs for h i s work. 20 The 1969 Annual Report of the Skeena River Management Committee reports the r e s u l t s of a l ength- f ecundi ty i n v e s t i g a t i o n conducted on Skeena R i v e r p i n k females f o r the years 1960, 1962, 1965, 1967, 1968 and 1969. A s i n g l e regress ion between number of eggs and t a i l fork length was est imated each y e a r . The p r e d i c t e d value of the number of eggs per spawning female—the dependent v a r i a b l e — t o g e t h e r w i th other estimates 21 generated by the i n v e s t i g a t i o n i s given i n the fo l lowing T a b l e . TABLE XXII LENGTH-•FEDUNDITY RELATIONSHIP FOR SKEENA RIVER PINK SALMON . Year Number of Eggs (Y) Intercept Slope Fork , Length (mm) Number of Observations 1960 1807.3 -2045.2 7.18 536.5 92 1962 1755.3 -3138.3 9.02 542.4 16 1965 1553.54 -1686,7 6.25 517.9 77 1967 1842.3 -1030.2 5.40 532.0 25 1968 1408.5 -2064.0 7.83 476.7 96 1969 1821.2 -1312.5 5.99 523.5 98 P a r k e r , Z Z r e p o r t i n g data c o l l e c t e d for Hooknose Creek, B . C . , found that 89,207 adul t spawners had a p o t e n t i a l egg depos i t ion of 75,896,000 eggs f o r an average of 850.8 eggs per adult p ink (both sexes cons idered) . Based on an assumed sex r a t i o of 50 percent females and an egg d e p o s i t i o n 187 of 1,600 eggs per spawning female, the egg product ion f a c t o r employed i n t h i s model f o r p ink salmon w i l l be 800 eggs per spawning i n d i v i d u a l . A l . 5 . 2 Est imates of Compensatory M o r t a l i t y C o e f f i c i e n t s The parameters employed i n the compensatory m o r t a l i t y r o u t i n e are d i sp layed i n Table XXIII . TABLE XXIII COMPENSATORY MORTALITY COEFFICIENTS Sockeye P ink  C o e f f i c i e n t Babine Non-Babine Odd Even a l a 2 D 1.75 1.25 1.25 1.25 1.75 1.25 1.25 1.25 0.75 0.75 0.75 0.75 The c o e f f i c i e n t s f o r the sockeye races are those employed by L a r k i n 23 and McDonald i n a Skeena R i v e r sockeye salmon s i m u l a t i o n . No d i r e c t estimates of compensation c o e f f i c i e n t s and compensation asymptotes f o r the p ink races were a v a i l a b l e . U n a v a i l a b i l i t y of estimates of these parameters for the Skeena R i v e r p ink races i s i n d i c a t i v e of a general p a u c i t y of data for other than sockeye for most r i v e r systems, the Skeena R i v e r i n c l u d e d . Contacts were made with knowledgeable I n d i v i d u a l s at the I n s t i t u t e of Animal Resource Ecology at the U n i v e r s i t y of B r i t i s h Columbia and the Federa l F i s h e r i e s and Marine S e r v i c e , Northern Operations Branch i n an attempt to obta in data on p ink e g g - t o - f r y m o r t a l i t y rates which cou ld be used as a bas i s for cons truc t ion of parameters of the compensation and 188 d e p e n s a t i o n r o u t i n e s . From t h e s e d i s c u s s i o n s i t became e v i d e n t t h a t no such d a t a were a v a i l a b l e . A s e c o n d approach w h i c h a l s o p r o v e d u n s u c c e s s f u l was t o o b t a i n d a t a f o r e g g - t o - f r y m o r t a l i t y r a t e s f o r a n o t h e r r i v e r system. S i n c e the F r a s e r R i v e r has r e c e i v e d c o n s i d e r a b l y more r e s e a r c h a t t e n t i o n than t h e Skeena R i v e r i t was presumed t h a t s uch d a t a would be a v a i l a b l e . Indeed, t h e 1973 A n n u a l Re p o r t o f th e I n t e r n a t i o n a l P a c i f i c Salmon F i s h e r i e s Commission (IPSFC) c o n t a i n e d o b s e r v a t i o n s f o r f r e s h w a t e r and m a rine s u r v i v a l r a t e s f o r p i n k salmon f o r th e odd y e a r s between 1961 and 1971, i n c l u s i v e . I f i t can be h y p o t h e s i z e d t h a t t h e major f a c t o r s a f f e c t i n g e g g - t o - f r y s u r v i v a l r a t e s a r e t h e same i n b o t h r i v e r s , t h e n the F r a s e r R i v e r p i n k salmon e g g - t o - f r y s u r v i v a l d a t a c a n be used t o c o n s t r u c t e s t i m a t e s o f the c o e f f i c i e n t s o f the Skeena R i v e r compensatory m o r t a l i t y f u n c t i o n f o r p i n k salmon. I n d i s c u s s i o n w i t h b i o l o g i s t s knowledgeable of the e c o l o g y o f b o t h r i v e r systems i t became c l e a r t h a t w h i l e b o t h systems were d i f f e r e n t i n t h e i r h y d r o l o g y and e c o l o g y , the major f a c t o r a f f e c t i n g e g g - t o - f r y s u r v i v a l i n b o t h systems i s li-the magnitude and t i m i n g of t h e maximum d a i l y d i s c h a r g e from the r i v e r . Data was o b t a i n e d f o r the maximum d a i l y d i s c h a r g e s f r o m each of the r i v e r s u s i n g t h e gauges a t M i s s i o n on the F r a s e r R i v e r and a t Usk on the Skeena R i v e r . The d a t a f o r the two r i v e r s i s d i s p l a y e d below i n T a b l e XXIV. A comparison o f the mean and v a r i a b i l i t y of the d i s c h a r g e s from the two r i v e r s i n a d d i t i o n to t h e t i m i n g o f o c c u r r e n c e of the maximum d i s c h a r g e i n each y e a r i n d i c a t e s t h a t the major c a u s a l f a c t o r b e h i n d o b s e r v e d e g g - t o - f r y s u r v i v a l i n the two r i v e r s i s s u f f i c i e n t l y d i s s i m i l a r t h a t t h e F r a s e r R i v e r s u r v i v a l r a t e s cannot r e a s o n a b l y be a p p l i e d to the Skeena R i v e r . I n a d d i t i o n , the Skeena R i v e r c o n t a i n s b o t h an odd-year and even-year p i n k salmon run w h i l e the p i n k salmon r u n o f t h e F r a s e r i s odd-year o n l y . 189 TABLE XXIV MAXIMUM DAILY DISCHARGE FROM SKEENA AND FRASER RIVERS, 1965-1973 (X i o 3 ) Skeena River F r a s e r River Year Discharge* Date Discharge* Date -1965 161 June 29 332 June 7 1966 211 June 8 330 June 22 1967 197 June 7 477 June 22 1968 195 May 22 359 J u l y 10 1969 159 May 25 341 June 10 1970 181 June 4 338 June 9 1971 173 June 24 338 June 8 .1972 275 June 12 507 June 17 1973 164 May 17 317 June 28 Mean Maximum D a i l y Discharge 190.67 371.0 Standard D e v i a t i o n 36.37 69.89 * c fs = cubic feet per second Source: H y d r o l o g i c a l Records, Water Resources Serv ice , V i c t o r i a , B . C . In view of the apparent i m p o s s i b i l i t y of obta in ing observed data for es t imat ion of p ink e g g - t o - f r y m o r t a l i t y ra tes we adopt the a l t e r n a t i v e s trategy of employing the sockeye e g g - t o - f r y s u r v i v a l rates to pink salmon as w e l l . To make some assessment of the appropriateness of t h i s assumption a s e n s i t i v i t y a n a l y s i s of por t ions of the m o r t a l i t y rout ine was conducted. The main conc lus ion r e s u l t i n g from t h i s a n a l y s i s i s that the egg - to - f ry 190 sur v i v a l rates have substantially less e f f e c t on the outcome of the mortality routine than the egg production factors, the most important parameter i n the routine. This conclusion resulted from the analysis displayed i n Table XXV. TABLE XXV SENSITIVITY ANALYSIS ON PERCENTAGE MORTALITY RATES WITH VARYING PARAMETER VALUES (AFTER APPLICATION OF COMPENSATORY MORTALITY) Egg Deposit i n Stock Units  1.0 1.2 1.6 Compensation Coefficient 1.25 0% 22% 47% Compensation Asymptote 0.75 Compensation Coefficient 1.25 Compensation Asymptote 0.9 Compensation Coefficient 1.5 227 47% Compensation Asymptote 0.75 The percentage reduction i n the size of the brood unit after application of compensatory mortality for given values of the compensation co e f f i c i e n t and asymptote can be read across a row for various values of the s i z e of the spawn. At 1.0 brood units the spawning population j u s t replaces i t s e l f . At larger brood u n i t s , the percentage reductions become larger. Increasing the compensation c o e f f i c i e n t from 1.25 to 1.5 has very l i t t l e e f f e c t (compare f i r s t and t h i r d set of rows). Likewise, increasing the compensation asymptote from 0.75 to 0.9 has very l i t t l e e f f e c t on the outcomes (compare f i r s t and second set of rows). Thus i t i s apparent that applying the observed sockeye egg-to-fry s u r v i v a l rates to pink should not unduly bias r e s u l t s . 191 FOOTNOTES TO APPENDIX A 1. P. A. Larkin and J. G. McDonald, "Factors i n the Population Biology of the Sockeye Salmon of the Skeena River," Journal of Animal Ecology, XXXVII, (1968), p. 250. 2. E. R. Zyblut, Northern Operations Branch, Fisheries and Marine Service. 3. Larkin and McDonald, "Factors," p. 250. 4. R. F. A. Roberts, "A Commercial Fisheries Production Function: The Skeena River Sockeye Salmon G i l l n e t Fishery," Unpublished Master's Thesis, University of B r i t i s h Columbia, 1971. 5. B r i t i s h Columbia Catch S t a t i s t i c s , 1951-1974. 6. E. R. Zyblut, B i o l o g i s t , Northern Operations Branch, Fisheries and Marine Service. 7. Canada Department of Environment, Fisheries and Marine Service, P a c i f i c Region, "Returns from Fishing Vessels i n B r i t i s h Columbia," by Blake Campbell, Vancouver, 1969. 8. Ian P e r c i v a l , Vessel Insurance O f f i c e r , Fisheries and Marine Service, P a c i f i c Region, Vancouver, B.C. 9. Ian P e r c i v a l , personal communication through W. Massie. 10. Campbell, "Returns from Fishing," p. 55. 11. For 1974, replacement cost as described i n text. 12. Fish sales i n 1974 for t h i s vessel were estimated by assuming the same physical catch as for the 1966-68 study. Catch value was determined by assuming a 75%/25% s p l i t i n the vessel's catch value between sockeye and pink, respectively. Thus, an average price per pound of f i s h i n 1967 could be estimated and used as a base for an index (1967 was the midpoint of the study period). The same average p r i c e could be calculated for 1974 and indexed. The 1974 index value was then applied to the 1966-68 catch of 48,130 pounds reported i n the study. The 192 c a l c u l a t i o n i s as fo l lows: .3754 x 75% = .2816 _ . . . . . . . 12 x 25% = 0 3 -3116 (average 1967 prxce) This r e s u l t i s supported by an a l t e r n a t e c a l c u l a t i o n reported i n Table XX. .6239 x 75% = .4679 , . . .2344 x 25% = .0586 = * 5 2 6 5 < a v e r a S e 1 9 7 4 P r i c e > 48,140 pounds x .5265 = $25,345/75 = $338. 13. I n f l a t e d by WIP 1935-39 = 100; 1967 = 264.1; 1974 = 460.6. 14. The study c i t e d i n footnote ;7 assumed a twenty-f ive week season. The average number of f i s h i n g days per week i n 1966-68 was 2.98. Hence, i n c a l c u l a t i n g d a i l y costs and r e t u r n s , a seventy - f ive day season has been assumed. 15. I n f l a t e d by Vancouver CPI , 1961 = 100; J u l y 1967 = 111.4; . J u l y 1974 = 158.5. 16. Assuming a t h i r t e e n - y e a r l i f e , f i v e percent salvage value and s t r a i g h t l i n e economic d e p r e c i a t i o n , i . e . , same number of days of operat ion per year . $33,000 x 0.95 = $31,350/13 = $2411.54 per year / 75 days = $32.15 per day. See Table XX f o r replacement cost c a l c u l a t i o n . 17. Ten percent opportuni ty cost of c a p i t a l assumed; 1966 study used 7%. 18. L a r k i n and McDonald, "Factors ," p . 251. 19. Bernard Skud, "Factors Regulat ing the Product ion of Pink Salmon," F i s h Stocks and Recruitment , Proceedings of a Symposium h e l d i n  A l a s k a , ed. by B. B. P a r r i s h , C o n s e i l I n t e r n a t i o n a l Pour L ' E x p l o r a t i o n de l a Mer. (Char lo t ten lund Slot—Danemark), p . 108. 20. Canada Department of Environment, Skeena R i v e r Management Committee  Annual Report , 1969; T e c h n i c a l Report P A C / T — 7 5 , Vancouver, B . C . 193 21. I b i d . , p . 285. 22. Robert R. Parker , "Est imation of Sea M o r t a l i t y Rates for the 1960 Brood-Year Pink Salmon on Hooknose Creek, B r i t i s h Columbia ," J o u r n a l of the F i s h e r i e s Research Board of Canada, 21 (No. 5) pp. 1019-34. 23. L a r k i n and McDonald, " F a c t o r s , " p . 251. 24. E . R. Z y b l u t , Northern Operations Branch, F i s h e r i e s and Marine S e r v i c e ; J im Wit tee , B i o l o g i s t , I n t e r n a t i o n a l P a c i f i c Salmon F i s h e r i e s Commission. 194 APPENDIX B. PILOT TESTS OF PRODUCTION FUNCTION PARAMETER ESTIMATES Given the q u a l i t y of the data employed i n the es t imat ion of the parameters of the product ion funct ion reported i n Appendix A , the low 2 R 's obtained on v i r t u a l l y a l l the equations est imated, and the opportuni ty provided by the ex is tence of the estimates obtained by Roberts to tes t the r e l a t i v e performance of the two sets of parameters, p i l o t t e s t s of the case I and I I models were r u n . Based on the r e s u l t s of these tes ts i t was concluded that the estimates obtained by Roberts were super ior performers i n respect of t h e i r a b i l i t y to generate r e s u l t s of a reasonable f a c s i m i l e to the observed f i s h e r y . One s imulat ion experiment was run with each of the case I and case I I formulat ions of the model. The major c r i t e r i a used for comparison of the r e s u l t s of these s imulat ions wi th the r e s u l t s observed i n the a c t u a l f i s h e r y i s the mean and range of the endogenous v a r i a b l e s h a r v e s t , escapement and recrui tment . Table XXVI below por trays the comparison of the s i m u l a t i o n r e s u l t s for case I with the r e s u l t s from the h i s t o r i c a l f i s h e r y taken from Table V of chapter 3. Table XXVI revea l s very c l e a r l y that the r e s u l t s of the s imulat ion are not a very accurate representa t ion of the f i s h e r y under study. The average catch for the case I s imulat ion i s 16% of the mean h i s t o r i c a l l y observed harves t—a s i g n i f i c a n t departure . The minimum and maximum t o t a l harvest for the case I s i m u l a t i o n a l so diverges s i g n i f i c a n t l y from the h i s t o r i c a l r e s u l t s as does the sockeye-pink breakdown of the harves t for a l l three s t a t i s t i c s presented. TABLE XXVI COMPARISON OF ACTUAL AND SIMULATED CATCH, ESCAPEMENT AND STOCK FOR THE SKEENA RIVER FISHERY CASE I MODEL* (X 10 3) Case I H i s t o r i c a l - S e a s onal** Simulat ion-Seasonal Minimum Maximum Average Minimum Maximum Average Sockeye harvest 142 1,499 786 11 747 185 Pink harvest 281 2,410 1,000 16 224 98 T o t a l Harvest 423 3,909 1,786 27 971 283 Sockeye Escapement 110 1,147 639 285 2,010 1,390 Pink Escapement 261 1,753 998 599 3,173 1,422 T o t a l Escapement 371 2,900 1,637 884 5,183 2,812 Sockeye Recruitment 285 2,599 1,426 397 2,758 1,390 Pink Recruitment 841 3,380 1,979 647 3,576 1,705 T o t a l Recruitment 1,126 5,979 3,405 1,044 6,334 3,095 * employing parameters as est imated i n Appendix A. * * Taken from Table V . 196 In contras t to the harvest s t a t i s t i c s of the case I s i m u l a t i o n , the escapement measures l i e s i g n i f i c a n t l y above the escapement recorded i n the observed f i s h e r y . T o t a l combined mean annual escapement i s approximately 32% l a r g e r for the case I s i m u l a t i o n than for the observed f i s h e r y . The minimum combined annual escapement of the case I s imulat ion i s more than double that for the observed f i s h e r y whi le the maximum combined annual escapement of the case I s i m u l a t i o n i s 78% l a r g e r than that for the observed f i s h e r y . In contras t to both the harves t and escapement s t a t i s t i c s presented i n Table XXVI, s t a t i s t i c s for the recruitment v a r i a b l e . d i s p l a y s i m i l a r i t y to the recrui tment s t a t i s t i c s of the a c t u a l f i s h e r y . Mean annual t o t a l recruitment for both species d i f f e r s by 300 thousand f i s h , or 10%, f o r the a c t u a l and s imulated r e s u l t s . The minima and maxima show a s i m i l a r r e l a t i o n s h i p . That the recruitment v a r i a b l e f o r the case I s imula t ion and the a c t u a l f i s h e r y are so s i m i l a r i n s p i t e of the divergent escapement and recruitment v a r i a b l e values reported above i s not e n t i r e l y s u r p r i s i n g i n l i g h t of the r e s u l t s reported i n chapter 4. There i t was shown that greater long-term escapement d i d not n e c e s s a r i l y lead to greater l o n g -term recruitment because, i n t h i s model as i n the salmon f i s h e r y , f i s h i n g m o r t a l i t y and n a t u r a l m o r t a l i t y act as subs t i tu te s to a c e r t a i n extent . Based on the r e s u l t s reported here as compared with the r e s u l t s presented i n chapter 3 us ing the parameter estimates obtained by Roberts , i t i s concluded that f u r t h e r s imulat ion experiments us ing the case I and II models w i l l employ the parameters estimed by Roberts . 197 BIBLIOGRAPHY ARTICLES A g n e l l o , R ichard J . and Lawrence P . Donnel ley. "Prices and Property Rights i n the F i s h e r i e s . " Southern Economic J o u r n a l . XXXXII, No. 2 (1975), 253-262. A g n e l l o , R ichard J . and Lawrence P. Donnel ley . " E x t e r n a l i t i e s and Property Rights i n the F i s h e r i e s . " Land Economics. L I I . No. 4 (19 76), 518-529. A l l e n , K . Radway. "The Influence of Random F l u c t u a t i o n s i n the Stock-Recruitment Re la t ionsh ip on the Economic Return from Salmon F i s h e r i e s . " mimeo. Anderson, Lee G . . "Analys is of Open-Access Commercial E x p l o i t a t i o n and Maximum Economic Y i e l d i n B i o l o g i c a l l y and T e c h n o l o g i c a l l y Interdependent F i s h e r i e s . " J o u r n a l of F i s h e r i e s Research Board of Canada. XXXII , No. 10 (1975), 1825-1842. Anderson, Lee G. "The Re la t ionsh ip Between Firm and Fishery i n Common Property F i s h e r i e s . " Land Economics, L I I , No. 2 (1976), 179-191. A r c h i b a l d , G. C . "The Q u a l i t a t i v e Content of Maximizing Models ." J o u r n a l of P o l i t i c a l Economy. LXXIII , (1965), 27-36. B e l l , F r e d e r i c k W. "Technolog ica l E x t e r n a l i t i e s and Common Property Resources: An E m p i r i c a l Study of the U . S . Northern Lobster F i s h e r y . " J o u r n a l o f P o l i t i c a l Economy. LXXX (1972), 148-158, B r a d l e y , P a u l G . , "Some Seasonal Models of the F i s h i n g Indus try ." Economics  of F i s h e r i e s Management: A Symposium. E d i t e d by A . D. Scot t , H . R. MacMil lan Lectures on F i s h e r i e s , Vancouver, The U n i v e r s i t y of B r i t i s h Columbia (1970), 33-44. Brown, Gardner, J r . "An Optimal Program for Managing Common Property Resources wi th Congestion E x t e r n a l i t i e s , " Journa l of P o l i t i c a l Economy. LXXXII (1974), 163-174. B u t l i n , J . A . , G. R. Munro, C. W. C l a r k and J . M. Tomkins. " E m p i r i c a l Es t imat ion and F i s h e r i e s Dynamics: The Manx H e r r i n g F i s h e r y . " mimeo. C l a r k , C o l i n W. "The Economics of O v e r e x p l o i t a t i o n . " Sc ience . CIXC (1973). 630-634. C l a r k , C o l i n W. " P r o f i t Maximization and the E x t i n c t i o n of Annual Spec ies ." J o u r n a l of P o l i t i c a l Economy. LXXXI (1973), 950-961. C l a r k , C o l i n W. "Contro l Theory i n F i s h e r i e s Economics: F r i l l or Fundamental." Economic Impacts of Extended F i s h e r i e s J u r i s d i c t i o n . E d i t e d by L . G. Anderson. Ann A r b o r : Ann Arbor Science P u b l i s h e r s , 1977, pp , 317-330. 198 Clark, Colin W. and Gordon R. Munro. "The Economics of Fishing and Modern Capital Theory: A Si m p l i f i e d Approach," Journal of Environmental  Economics and Management. 11 (1975), 92-106. Crutchf i e l d , James A. "Some Economic Aspects of the Halibut Program," B i o l o g i c a l and Economic Aspects of Fisheries Management. Edited by James A. Crutchfield. Proceedings of a conference held under the auspices of the College of Fisheries and the Department of Economics of the University of Washington at Seattle. February 17-19, 1959. Crut c h f i e l d , James A. "The Marine Fisheries: A Problem i n International Cooperation." American Economic Review. Papers and Proceedings (1964) 287-283. Cru t c h f i e l d , James A. "C o l l e c t i v e Bargaining i n the P a c i f i c Coast Fish e r i e s : The Economic Issues." International Labor Relations Review. V I I I , (1955) 541-556. ~ ~ ~ Foskett, D. R. "The Rivers Inlet Sockeye Salmon." Journal of the Fisheries Research Board of Canada. XV, No. 5 (1958), 867-869. Fullenbaum, Richard F. Ernest W. Carlsen and Frederick W. B e l l , "On Models of Commercial Fishing: A Defense of the Traditional L i t e r a t u r e . " Journal of P o l i t i c a l Economy. LXXX, No. 4 (1972), 761-768. Godfrey, H. "Comparison of the Index of Return for Several Stocks of B r i t i s h Columbia Salmon to Study Variations i n Survi v a l . " Journal of  the Fisheries Research Board of Canada. XV, No. 5 (1958), 891-908. Gordon, H. Scott, "The Economic Theory of a Common Property Resources: The Fishery." Journal of P o l i t i c a l Economy. LXII, No. 2 (1959), 124-142. Gould, J . R. " E x t e r n a l i t i e s Factor Proportions and the Level of Free Access Resources." Economica. XXXIX (1972), 383-401. Gould, J . R. "A Reply to Newberry on Free Access Resources." Economica. XXXXIII (1976), 299-303. Hannesson, Rognvaldur "Fishery Dynamics: A North A t l a n t i c Cod Fishery." Canadian Journal of Economics. V I I I , No. 2 (1975), 151-173. Huang, David S. and Chae W. Lee. "Toward a General Model of Fishery Production." Southern Economic Journal. XXXXIII, No. 1 (1976), 846-854. Larki n , P. A., R. F. Raleigh and N. J . Wilimovsky. "Some Alternative Premises for Constructing Theoretical Reproduction Curves," Journal  of the Fisheries Research Board of Canada. XXI, No. 3 (1964), 477-484. Larkin, P. A. and A. S. Hourston. "A Model for Simulation of the Population Biology of P a c i f i c Salmon." Journal of the Fisheries Research Board  of Canada. XXI, No. 5 (1964), 1245-1265. Larkin, P. A. and J. G. McDonald. "Factors i n the Population Biology of the Sockeye Salmon of the Skeena River." Journal of Animal Ecology. XXXVII (1968), 229-258. 199 Loose, Verne f . and G, C Robinson. " E x t e r n a l i t i e s and Property Rights i n the F i s h e r i e s : A R e p l y . " Land Economics, ( forthcoming). M i l n e , D. J . "The Skeena River Salmon F i s h e r y , With S p e c i a l Reference to Sockeye Salmon." J o u r n a l of the F i s h e r i e s Research Board of Canada. X I I , No. 3 (1955), 451-485. Munro, Gordon R. "Canada and F i s h e r i e s Management wi th Extended J u r i s d i c t i o n : A P r e l i m i n a r y View." Economic Impacts of Extended J u r i s d i c t i o n . E d i t e d by L . G. Anderson, Ann Arbor : Ann Arbor Science P u b l i s h e r s . 1977, 29-49. Neher, P h i l i p A . "Notes on the V o l t e r r a - Q u a d r a t i c F i s h e r y , " J o u r n a l of  Economic Theory . VIII (19 74), 39-49. Newberry, David M. G . . "Congestion and Overexp lo i ta t i on of Free Access Resources ." Economica. XXXXII (1975), 243-259. Parker , Robert , "A Concept of the Dynamics o f Pink Salmon P o p u l a t i o n s , " Symposium on Pink Salmon. E d i t e d by N. J . Wilimonsky. H . R. MacMil lan Lectures i n F i s h e r i e s , Vancouver: The U n i v e r s i t y of B r i t i s h Columbia, (1962), 203-211. P a u l i k , G. J . and J . W. Greenough, J r . "Management A n a l y s i s f o r a Salmon Resource System." Systems A n a l y s i s i n Eco logy . E d i t e d by K. E . F . Watt, New York: Academic Pres s , 1966. P a u l i k , G. J . , A . S. Hourston and P . A . L a r k i n . " E x p l o i t a t i o n of M u l t i p l e Stocks by a Common F i s h e r y . " J o u r n a l of the F i s h e r i e s Research Board  o f Canada. XXIV, No. 12 (1967), 2527-2537. Peterman, Randa l l M. "New Techniques f o r P o l i c y E v a l u a t i o n i n E c o l o g i c a l Systems: Methodology for a Case Study of P a c i f i c Salmon F i s h e r i e s , " J o u r n a l of the F i s h e r i e s Research Board of Canada. XXXII, No. 11 (1975), 2179 P lourde , C. G . "A Simple Model of Replenishable N a t u r a l Resource E x p l o i t a t i o n . " American Economic Review. LX (1970), 518-522. P lourde , C . G. " E x p l o i t a t i o n of Common Property Replenishable N a t u r a l Resources." Western Economic J o u r n a l . IX (1971), 256-266. Q u i r k , James P . and Vernon L . Smith. "Dynamic Economic Models of F i s h i n g . " Economics of F i s h e r i e s Management: A Symposium. E d i t e d by A . D. S c o t t , H. R. MacMil lan:Lectures i n F i s h e r i e s , Vancouver, The U n i v e r s i t y of B r i t i s h Columbia, 1970. R i c k e r , W. E . "Maximum Sustained Y i e l d s from F l u c t u a t i n g Environments and Mixed S tocks ," J o u r n a l of the F i s h e r i e s Research Board of Canada. XV, No. 2 (1958), 991-1006. R i c k e r , W. E . "Stock and Recruitment ." J o u r n a l of the F i s h e r i e s Research  Board of Canada. X I , No. 5 (1964), 559-623. Samuelson, P a u l A . "Economics of Fores try i n an Evo lv ing S o c i e t y . " Economic I n q u i r y . XIV, No. 4 (1976), 466-492. 200 S c o t t , A . D. "The F i s h e r y : The Object ives of Sole Ownership." J o u r n a l  of P o l i t i c a l Economy. L X I I I , No. 2 (1955), 116-24. Sco t t , A . D. and C l i v e Southey. "The Problem of Achiev ing E f f i c i e n t Regulat ion of a F i s h e r y . " Economics of F i s h e r i e s Management: A  Symposium. E d i t e d by A . D. Sco t t . H . R. MacMil lan Lectures i n F i s h e r i e s , Vancouver, The U n i v e r s i t y of B r i t i s h Columbia (1970), 47-59. Schaefer, M i l n e r B . . "Methods of Es t imat ing E f f e c t s of F i s h i n g on F i s h P o p u l a t i o n s . " Transact ions of the American F i s h e r i e s S o c i e t y . XCVII ( J u l y , 1968), 231-241. Schaefer , M i l n e r B . "Some Cons iderat ions of Populat ion Dynamics and Economics i n R e l a t i o n to the Management of Commercial Marine F i s h e r i e s . " J o u r n a l of the F i s h e r i e s Research Board of Canada. XIV, No. 5 (1957), 669-81. Schaefer , M. B . and R. J . Beverton. I n t e r p r e t a t i o n . " The Sea, V o l . In tersc i ence P u b l i s h e r s , 1963. "Fishery Dynamics - T h e i r A n a l y s i s and I I , E d i t e d by M. N. H i l l , New York: Shephard, M. P . and F . C. W i t h l e r , "Spawning Stock Size and Resultant Product ion f o r Skeena Sockeye," J o u r n a l of the F i s h e r i e s Research  Board of Canada. XV, No. 5 (1958), 1007-1025. Skud, Bernard . "Factors Regulat ing the Product ion of Pink Salmon." F i s h  Stocks and Recruitment. Proceedings of a Symposium held i n Aahrus. E d i t e d by B. B. P a r r i s h . C o n s e i l I n t e r n a t i o n a l Pour L ' E x p l o r a t i o n de l a Mer. Char lo t ten lund Slot-Danemark, (1970), 106-112. Smith, Vernon L . "Economics of Product ion from Natura l Resources ." American Economic Review. L V I I I , No. 2 (1968), 409-31. -Smith, Vernon L . . "On Models of Commercial F i s h i n g , " Journa l of P o l i t i c a l  Economy, LXXVII , No. 2 (1969), 181-98. Smith , Vernon L . "On Models of Commercial F i s h i n g : The T r a d i t i o n a l L i t e r a t u r e Needs No Defenders." J o u r n a l of P o l i t i c a l Economy. LXXX, No. 4 (1972), 776-779. Southey, C l i v e . " P o l i c y P r e s c r i p t i o n s i n Bionomic Models: The Case of the F i s h e r y . " J o u r n a l of P o l i t i c a l Economy. LXXX, No. 4 (1972), 769-775. Spence, M i c h a e l , "Blue Whales and A p p l i e d C o n t r o l Theory." I n s t i t u t e of Mathematical Studies i n the S o c i a l Sc iences . T e c h n i c a l Report No. 108, Palo A l t o : Stanford U n i v e r s i t y (1973). S trand , I . E . and D. L . Heuth. "A Management Model f or a M u l t i - S p e c i e s F i s h e r y , " Economic Impacts of Extended J u r i s d i c t i o n . Ed i t ed by L , G. Andersen, Ann Arbor : Ann Arbor Science P u b l i s h e r s , 1977, pp. 331-346. Turvey, Ralph . "Optimizat ion i n F i shery Regu la t ion ." American Economic Review. LIV, No. 2, Part I (1964), 64-76. 201 Z e l l n e r , A . . "Management of Marine Resources: Some Key Problems Requir ing A d d i t i o n a l A n a l y s i s , " Economics of F i s h e r i e s Management: A Symposium, E d i t e d by A. D. Sco t t , H. R. Macmillan Lectures i n F i s h e r i e s , Vancouver: The U n i v e r s i t y of B r i t i s h Columbia, 1970. BOOKS Bel lman, Richard E . and Stuart E . Dreyfus , A p p l i e d Dynamic Programming, P r i n c e t o n : Pr ince ton U n i v e r s i t y Pres s , 1962. Beverton, R. J . H . and S. J . H o l t . On The Dynamics of E x p l o i t e d F i s h  P o p u l a t i o n s . London: Her Majesty 's S ta t ionary O f f i c e , 1957. Chiang, Alpha C. . Fundamental Methods of Mathematical Economics. New York: M c G r a w - H i l l Book Company, I n c . , 1967. C h r i s t y , F . T . and A . D . Scott . , The Common Wealth i n Ocean F i s h e r i e s . Resources for the F u t u r e , I n c . . Ba l t imore , Johns Hopkins Pres s , 1965. C l a r k , C o l i n W. Mathematical Bioleconomics: The Optimal Management of  Renewable Resources, New York: John Wiley and Sons, 1976. C r u t c h f i e l d , James A . . "Economic Object ives of F i shery Management," The  F i s h e r i e s : Problems i n Resource Management. E d i t e d by James A. C r u t c h f i e l d , S e a t t l e : U n i v e r s i t y of Washington Pres s , 1965. C r u t c h f i e l d , James A . and G i u l i o Pontecorvo. The P a c i f i c Salmon F i s h e r i e s : A Study of I r r a t i o n a l Conservat ion , Resources for the Future , I n c . , B a l t i m o r e : Johns Hopkins Pres s , 1969. F o r r e s t e r , Jay W. I n d u s t r i a l Dynamics. Cambridge: M . I . T . P r e s s , 1961. Hadley , G. Nonl inear and Dynamic Programming. Palo A l t o : Addison-Wesley P u b l i s h i n g Company, 1964. Henderson, James and Richard E . Quandt. Microeconomic Theory. New York: McGraw H i l l Book Company, I n c . , 1958. Kmenta, J a n , Elements of Econometrics . New York: C o l l i e r - M a c M i l l a n L i m i t e d , 1971. L o t k a , A . J . Elements of Mathematical B i o l o g y . New York: Dover P u b l i c a t i o n s , 1956. M a r s h a l l , A l f r e d . P r i n c i p l e s of Economics, 8th e d . , London: Macmil lan and Company, 1962. Nay lor , Thomas H . Computer S imulat ion Experiments with Models of Economic  Systems. E d i t e d by T . H . Nay lor , New York: John Wiley and Sons, I n c . , 1971. 202 Roberts , R. F . A . . "A Commercial F i s h e r i e s Product ion Funct ions: The Skeena R i v e r Sockeye Salmon G i l l n e t F i s h e r y . " Unpublished Master 's T h e s i s . U n i v e r s i t y of B r i t i s h Columbia, 1971. R i c k e r , W. E . . Computation and I n t e r p r e t a t i o n of B i o l o g i c a l S t a t i s t i c s of  F i s h P o p u l a t i o n s . Canada. Department of Environment, F i s h e r i e s and Marine S e r v i c e , B u l l e t i n of the F i s h e r i e s Research Board of Canada, Ottawa, 1975. Royce, W i l l i a m F . , Donald E . Bevan, James A . C r u t c h f i e l d , Gerald J . P a u l i k and Robert L . F l e t c h e r . Salmon Gear L i m i t a t i o n i n Northern Washington  Waters . P u b l i c a t i o n i n F i s h e r i e s , New S e r i e s , V o l . I I , No. 1. S e a t t l e : U n i v e r s i t y of Washington, 1963. Southey, C l i v e , "Studies i n F i s h e r i e s Economics," Unpublished Ph .D . D i s s e r t a t i o n . U n i v e r s i t y of B r i t i s h Columbia, 1969. GOVERNMENT DOCUMENTS AND REPORTS Canada, Department of Environment. F i s h e r i e s S e r v i c e . P a c i f i c Region. An A n a l y s i s of Gross Returns from F i s h i n g Vessels i n B r i t i s h Columbia, by Michae l Hunter. Vancouver, B r i t i s h Columbia, 1971. Canada, Department of Environment, F i s h e r i e s S e r v i c e , P a c i f i c Region. B r i t i s h Columbia Catch S t a t i s t i c s . S t a t i s t i c s Canada. Ottawa Ontar io : Queen's P r i n t e r , 1945-1974. Canada, Department of Environment, F i s h e r i e s and Marine S e r v i c e , P a c i f i c Region, Returns from F i s h i n g Vesse ls i n B r i t i s h Columbia, by Blake A . Campbel l , Vancouver, B r i t i s h Columbia, 1969. Canada. Department of F i s h e r i e s and F o r e s t r y , F i s h e r i e s S e r v i c e , P a c i f i c Region. Rivers I n l e t Sockeye, by F . E . A . Wood. Vancouver, B . C . , 1970. Canada, Department of Environment. F i s h e r i e s S e r v i c e . P a c i f i c Region, Skeena R i v e r Management Committee, Annual Report , 1969, T e c h n i c a l Report PAC/T-75 -17 , Vancouver, B r i t i s h Columbia, 1969. Canada, Department of Environment. F i s h e r i e s . S e r v i c e . P a c i f i c Region. Some Economic Aspects of Commercial F i s h i n g i n B r i t i s h Columbia. Vancouver, B r i t i s h Columbia, 1971. Canada. Department of Environment, F i s h e r i e s S e r v i c e , P a c i f i c Region. The Importance of the Commercial F i s h i n g Industry to Se lected Remote C o a s t a l Communities of B r i t i s h Columbia, by W. F . S i n c l a i r , Vancouver B r i t i s h Columbia, 1971. 203 Canada. Department of Environment, Fisheries Service, P a c i f i c Region, The Socio-Economic Background of Commercial Fishing i n B r i t i s h Columbia, by W. Alan Wilson, D e t a i l Report No. 1, Vancouver, B r i t i s h Columbia, 1970. Interamerican Tropical Tuna Commission, Some Aspects of the Dynamics of Populations Important to the Management of Commercial Marine Fisheries, by Milner B. Schaefer, B u l l e t i n I, No. 1, La J o l l a , C a l i f o r n i a , 195A. Organization for Economic Cooperation and Development, Directorate of Agriculture, Fisheries D i v i s i o n . "Simulation Programmes for Selected F i s h e r i e s . " by James A. Crutchfield, Economic Aspects of Fish Production, International Symposium on Fisheries Economics. P a r i s , 1972. Organization for Economic Cooperation and Development, Directorate of Agriculture. Fisheries D i v i s i o n . "Rationalization of Canada's West Coast Salmon Fishery." by P. H. Pearse. Economic Aspects of Fish  Production. International Symposium on Fisheries Economics, P a r i s , 1972. United Nations. Food and Agriculture Organization, Economic Effects of  Fishery Regulation: Report of an Expert Meeting at Ottawa. Edited by R. Hamlisch, Rome, 1962. United Nations. Food and Agriculture Organization. The Economics of Fisheries: Proceedings of a Round Table Discussion Organized by the  International Economics Association, Edited by Ralph Turvey and Jack Wiseman, Rome, 1957. U.S. Department of the I n t e r i o r , Fish and W i l d l i f e Service, Bureau of Commercial Fish e r i e s . Economic Aspects of the P a c i f i c Halibut Fishery, by James Crutch f i e l d and Arnold Zellner. Fishery I n d u s t r i a l Research Volume 1. Number 1. Washington, D.C.: Government P r i n t i n g Office, 1963. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0100244/manifest

Comment

Related Items