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Efficient harvesting of fish stocks : the case of the Icelandic demersal fisheries Arnason, Ragnar 1984

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EFFICIENT HARVESTING OF FISH STOCKS: THE CASE OF THE ICELANDIC DEMERSAL FISHERIES By RAGNAR ARNASON M.Sc, London School of Economics, 1977 A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1984 © Ragnar Arnason, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of E c o n o m i c s The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 September 5, 1984 i i ABSTRACT A detailed model describing the economics of harvesting s e l f -renewable resources is constructed. The model i s developed in terms of the par t i c u l a r case of the Icelandic demersal f i s h e r i e s . Its basic structure i s nevertheless general enough to apply to a wide range of other self-renewable natural resources. The model involves a b i o l o g i c a l submodel, describing the internal dynamics of the resources, an economic submodel, describing the technology and dynamics of the harvesting process, and a special function providing the li n k between these two submodels. Ecological interactions are represented by a mixture of deterministic and stochastic r e l a t i o n s . The model is designed to cope with any f i n i t e number of self-renewable resources and harvesting technologies. In terms of f i s h e r i e s i t may thus be characterized as a multi-species, m u l t i - f l e e t model. The p a r t i c u l a r empirical case investigated, involves 3 species of f i s h and 2 types of fishing vessels. An objective function, mapping harvesting results into e f f i c i e n c y l e v e l s i s defined. B a s i c a l l y , this function s p e c i f i e s harvesting paths that maximize discounted economic rents as ef f ic ient. With the help of numerical search techniques, the model is employed to discover e f f i c i e n t harvesting paths for the Icelandic demersal f i s h e r i e s . The control variables in this maximization process are (i) investment in the fis h i n g f l e e t s and ( i i ) the al l o c a t i o n of the f l e e t s to f i s h e r i e s and/or idleness. For the case of the Icelandic demersal f i s h e r i e s , i t i s demonstrated that the current harvesting pattern i s very i n e f f i c i e n t and that an e f f i c i e n t harvesting path involves a s i g n i f i c a n t reduction in o v e r a l l f i s h i n g c a p i t a l as well as a reallocation of the f i s h i n g f l e e t s to the various f i s h e r i e s . The s e n s i t i v i t y of the e f f i c i e n t harvesting path to various model spe c i f i c a t i o n s i s b r i e f l y investigated. F i n a l l y , the model i s used to investigate the properties of e f f i c i e n t harvesting paths under a number of economic and ecological s p e c i f i c a t i o n s not necessarily related to the Icelandic demersal f i s h e r i e s . One notable result of t h i s investigation i s that the existence of s i g n i f i c a n t ecological interactions may invalidate certain economic relationships that are often taken for granted. iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv L i s t of Tables xi L i s t of Figures xv L i s t of Propositions and C o r o l l a r i e s x v i i i Preface xix Notation and Terminology xxi Part I. Introduction 1. Introduction 1 1.1 F i e l d of Study 1 1.2 Economics of Fisheries 2 1.3 Objectives of Study 4 1.4 Approach of Study 5 1.4 Limitations of Study 7 1.5 Structure of Presentation 8 Footnotes 10 2. Rent Maximization and Economic Welfare 11 Footnotes 16 V Part I I . The Empirical Model 3. The Empirical Model: An Outline 18 3.1 The Objective Function 19 3.2 The Empirical Model 20 3.2.1 The B i o l o g i c a l Submodel 20 3.2.2 The Economic Submodel 24 3.2.3 The Fishing Mortality Production Function 28 3.2.4 The Empirical Model: A Summary 29 3.3 The Control Variables 33 Footnotes 35 4. The B i o l o g i c a l Submodel 36 4.1 Some Fish Stock Growth Models 36 4.1.1 Aggregative population growth models 37 Appendix 4 . 1 . 1-A 42 Appendix 4.1.1-B , 48 4.1.2 Disaggregative growth models: The Beverton-Holt model 51 Appendix 4.1.2-A 57 Appendix 4.1.2-B 58 Appendix 4.1.2-C 60 Appendix 4.1.2-D 63 4.1.3 Ecological Considerations 65 4.1.4 An Ecological extention of the Beverton-Holt model 69 4.2 Icelandic Demersal Fish Stocks 72 vi 4.2.1 The Habitat 73 4.2.2 The Cod 76 4.2.2.1 B i o l o g i c a l Parameters 77 4.2.2.2 Recruitment Functions 78 4.2.2.3 Equilibrium Relationships 87 4.2.2.4 Simulations in the Data Period 90 4.2.3 The Haddock 93 4.2.3.1 B i o l o g i c a l Parameters 94 4.2.3.2 Recruitment Functions 95 4.2.3.3 Equilibrium Relationships 98 4.2.3.4 Simulations in the Data Period 101 4.2.4 The Saithe 104 4.2.4.1 B i o l o g i c a l Parameters 105 4.2.4.2 Recruitment Functions 106 4.2.4.3 Equilibrium Relationships 108 4.2.4.4 Simulations in the Data Period 110 4.2.5 Ecological Relationships 113 4.2.5.1 The Demersal Subsystem 115 4.2.5.2 Environmental Influences 134 Appendix' 4.2.5-A 139 Footnotes 140 5. The Economic Submodel 148 5.1 The Demersal Fishing Industry: A General Description 149 5.2 Processing Cost Functions 157 5.2.1 Processing Cost Functions: Theoretical Formulation 158 v i i 5.2.2 Processing Cost Functions: Empirical Estimation 163 Appendix 5.2-A 178 Appendix 5.2-B 180 Appendix 5.2-C 181 5.3 Harvesting Cost Functions 183 5.3.1 Harvesting Cost Functions: Theoretical Formulation 183 5.3.2 Harvesting Cost Functions: Empirical Estimation 188 Appendix 5.3-A 200 Appendix 5.3-B 201 5.4 The Revenue Function 203 5.4.1 The Quantity of F i n a l Products 204 5.4.2 The Process Allocation Parameters 212 5.4.3 Output Prices 219 Appendix 5.4-A 220 5.5 The Fishing Capital Function 222 5.5.1 Characterization of the Fishing Capital 222 5.5.2 Fishing Fleet Dynamics 223 5.5.3 Estimation of the Deterioration Function 226 5.5.4 Investment and Disinvestment in the Fishing Fleet 230 Appendix 5.5-A 233 5.6 The Rate of Discount 234 Footnotes 237 6. The Fishing Mortality Production Function 245 v i i i 6.1 A Brief Review of the Literature 246 6.2 Analysis 250 6.2.1 The Fishing Mortality Function 250 6.2.2 The Fishing Time Function 259 Appendix 6.2-A 270 6.3 Empirical Estimation 274 6.3.1 Data 274 Appendix 6.3.1-A 280 Appendix 6.3.1-B 281 6.3.2 Estimation of the Fishing Mortality Function 285 6.3.3 Estimation of the Fishing Time Function 291 6.3.4 A Joint Estimation of the Fishing Mortality and Fishing Time Functions 298 6.3.5 Estimation of the Fishing Mortality Function for Multipurpose Fishing Vessels .... 299 6.4 Theoretical Implications 310 Footnotes 314 Part I I I . Model Predictions 7. Simulations 319 7.1 The Structure of the Simulations 320 7.2 B i o l o g i c a l Simulations 322 7.3 Economic Simulations 328 7.4 Conclusions 333 Footnotes 335 ix 8. E f f i c i e n t Harvesting Programs 336 8.1 Solution Strategy 338 8.1.1 Control Variables 338 8.1.2 Numerical Techniques 344 8.2 Icelandic Demersal Fisheries 347 8.2.1 I n i t i a l Conditions 348 8.2.2 E f f i c i e n t Harvesting Programs 350 Appendix 8.2-A 368 Appendix 8.2-B 372 8.2.3 Variants of the Basic Case 373 8.2.3.1 Shorter Control Periods 373 8.2.3.2 Nonselective Fisheries 375 8.2.3.3 A Single Fishing Fleet 379 8.2.4 The Simplified Case 381 8.2.4.1 Different Degrees of M a l l e a b i l i t y 382 8.2.4.2 Ecological Weight Relationships 385 8.2.4.3 The Stock-effort Effect 389 8.2.5 Stochastic Recruitment 392 8.2.6 Icelandic Demersal Fishe r i e s : A Summary 398 8.3 More General Cases 401 8.3.1 E f f i c i e n t Adjustment Paths 401 8.3.2 Predator-prey Relationships 407 8.3.2.1 E f f i c i e n t Harvesting Paths 408 8.3.2.2 Relative Prices 412 Footnotes 415 X Part IV. Conclusions 9. Conclusions 421 L i s t of References 426 Appendix 1 : Data 436 Appendix 2: On Computer Programs 471 xi LIST OF TABLES The following l i s t s tables appearing in the main text of the essay. Tables contained in appendices are not included in the l i s t . Page 3.1 Main Categories of Variables in Empirical Model 30 4.1 Estimates of B i o l o g i c a l Parameters for Cod 78 4.2 Estimation of Recruitment Functions for Cod 85 4.3 Comparison of Simulated to "Actual" Values for Cod 91 4.4 Estimates of B i o l o g i c a l Parameters for Haddock 94 4.5 Estimation of Recruitment Functions for Haddock 96 4.6 Comparison of Simulated to "Actual" Values for Haddock 102 4.7 Estimates of B i o l o g i c a l Parameters for Saithe 105 4.8 Estimation of Recruitment Functions for Saithe 107 4.9 Comparison of Simulated to "Actual" Values for Saithe 111 4.10 Maximum Joint Sustainable Y i e l d E f f o r t Levels 121 4.11 Ecological Weight Functions: Estimation Results 127 4.12 Ecological Weight Functions: S t a t i s t i c s and Tests 127 4.13 Ecological Weight Functions: E l a s t i c i t i e s at Sample Means 129 4.14 Estimated Recruitment Functions: Correlation of Residuals 134 4.15 Structure of Recruitment Residuals 137 xi i 5.1 Processing Cost Functions: Tests of Functional Forms 166 5.2 Fish Meal and O i l Processing Cost Function: Estimation Results 169 5.3 Specialized Freezing Processing Cost Function: Estimation Results 171 5.4 S a l t f i s h and Stockfish Cost Function: Estimation Results 173 5.5 Multi-process Plants: Estimation Results 174 5.6 Aggregative Processing Cost Function: Estimation Results 176 5.7 Harvesting Cost Functions: Tests of Simplifying Restrictions 193 5.8 Harvesting Cost Functions: Aggregation Tests 194 5.9 Multi-Purpose Fishing Vessels: Harvesting Cost Functions 196 5.10 Deep-Sea Trawlers: Harvesting Cost Functions 198 5.11 Harvesting Cost Functions: Estimates of Total E l a s t i c i t i e s 199 5.12 Transformation Coefficients 209 5.13 Technical Transformation Coefficients 211 5.14 Estimation of Processing Allocation Equations 218 5.15 Estimation of the Fleet Remainder Function 228 5.16 Fishing Capital Investment Prices 231 5.17 Iceland's Real Rate of Interest 236 6.1 Testing Restrictions on the x i i i Fishing Mortality Function 286 6.2 Estimates of the Fishing Mortality Function 290 6.3 Testing the Specification of the Fishing Time Function 293 6.4 Estimates of the Fishing Time Function 295 6.5 Testing the Specification of Equation (24) 303 6.6 Estimation of A Fishing Mortality Function For the Multi-purpose Fleet 305 6.7 E l a s t i c i t y of Fishing Mortality w.r.t. Vessel Specific Stock 307 7.1 B i o l o g i c a l Simulations: Distinguishing C h a r a c t e r i s t i c s 323 7.2 B i o l o g i c a l Simulations: F i t with Observed Catch Levels. 326 7.3 Simulations of Harvesting P r o f i t a b i l i t y : F i t with Observed Levels 331 8.1 Standardized Fishing Vessels 339 8.2 I n i t i a l B i o l o g i c a l Stocks and Base Fishing M o r t a l i t i e s 349 8.3 I n i t i a l Economic Stocks 349 8.4 E f f i c i e n t Harvesting Program: Some Key Results 352 8.5 E f f i c i e n t Harvesting Program: Sectoral P r o f i t a b i l i t y 354 8.6 E f f i c i e n t Harvesting Programs: Different Control Periods 374 8.7 E f f i c i e n t Harvesting Programs: x i v Selective and Nonselective Fisheries 376 8.8 E f f i c i e n t harvesting Programs: Single Fleet 380 8.9 Ecological Weight Relationships and E f f i c i e n t Harvesting Programs 388 8.10 Stock-effort Effect and E f f i c i e n t Harvesting Programs 391 8.11 Effects of Stochastic Recruitment: A Summary 394 8.12 E f f i c i e n t Harvesting Paths: I n i t i a l Conditions 402 8.13 S p e c i f i c a t i o n of Control Regions 402 8.14 E f f i c i e n t Paths: Different I n i t i a l Conditions 403 XV LIST OF FIGURES Page 3.1 The Empirical Model: Basic Structure 32 4.1 Equilibrium Cohort Biomasses 44 4.2 Alternative Estimates of the Aggregative Growth Function 46 4.3 The System of Currents around Iceland 74 4.4 Spawning Grounds and Migration of the Cod 79 4.5 Icelandic Cod: Recruitment Functions 86 4.6 Icelandic Cod: Sustainable Yi e l d Functions 88 4.7 Icelandic Cod: Biomass per Recruit 89 4.8 Icelandic Cod: Catch Simulations 92 4.9 Icelandic Haddock: Recruitment Functions 98 4.10 Icelandic Haddock: Sustainable Yield Functions 100 4.11 Icelandic Haddock: Biomass per Recruit 101 4.12 Icelandic haddock: Catch Simulations 103 4.13 Icelandic Saithe: Recruitment Functions 108 4.14 Icelandic Saithe: Sustainable Yield Functions 109 4.15 Icelandic Saithe: Biomass per Recruit 110 4.16 Icelandic Saithe: Catch Simulations 112 4.17 Predation Relationship: Joint Sustainable Yie l d Contours 120 5.1 Fishing Ports and Fishing Grounds 1975-6 151 5.2 The Estimated Fleet Remainder Function and Deterioration rates 229 xvi 6.1 The S(y) Function.. 267 6.2 The Fishing Time Function 269 6.3 The Estimated Fishing Time Stock Size Relationship 297 6.4 The Estimated Fishing Mortality Stock Size Relationship 307 7.1 Annual Catch of Cod: Simulation Results 324 7.2 Annual Catch of Haddock: Simulation Results 324 7.3 Annual Catch of Saithe: Simulation Results 325 7.4 Recruitment of Cod: Actual and Predicted 327 7.5 P r o f i t a b i l i t y in the Harvesting Sector: Simulation Results 330 7.6 P r o f i t a b i l i t y in the Processing Sector: Simulation Results 330 7.7 P r o f i t a b i l i t y of the Harvesting A c t i v i t y : Simulation Results 331 7.8 Processing P r o f i t s : Different Process Alloca t i o n Specifications 333 8.1 Trawlers: E f f i c i e n t Investment, Fleet Size and Idle Vessels 356 8.2 Multipurpose Vessels: E f f i c i e n t Investment Fleet Size and Idle Vessels 357 8.3 Trawlers: E f f i c i e n t Allocation to Fisheries 360 8.4 Multipurpose Vessels: E f f i c i e n t A l l o c a t i o n to Fisheries 361 8.5 E f f i c i e n t Harvesting Program: Catch Levels 363 8.6 E f f i c i e n t Harvesting Program: Biomass Levels 364 xvi i 8.7 E f f i c i e n t Harvesting Program: State Space 366 8.8 E f f i c i e n t Harvesting Paths: Different Control Periods 374 8.9 E f f i c i e n t Harvesting Programs: Selective and Nonselective Fisheries 378 8.10 Different Degrees of M a l l e a b i l i t y : Total Capital 384 8.11 Different Degrees of M a l l e a b i l i t y : Active C a p i t a l . . . . 384 8.12 E f f i c i e n t Harvesting Programs: Total Capital 386 8.13 E f f i c i e n t Harvesting Programs: Active Capital 386 8.14 Different Stock-effort Assumptions: Total C a p i t a l . . . . 390 8.15 Different Stock-effort Assumptions: Active C a p i t a l . . . 390 8.16 E f f i c i e n t Investment: Deterministic and Stochastic Recruitment 395 8.17 E f f i c i e n t Total Fleet: Deterministic and Stochastic Recruitment 395 8.18 E f f i c i e n t Idle Vessels: Deterministic and Stochastic Recruitment 396 8.19 E f f i c i e n t Harvesting Paths: Aggregate State-space Diagram 406 8.20 Different Predation Relations 409 8.21 E f f i c i e n t Fishing Capital Levels 410 8.22 E f f i c i e n t Biomass Levels 411 8.23 E f f i c i e n t Vessel Allocation to Fisheries under Predation.... 413 xvi i i LIST OF PROPOSITIONS AND COROLLARIES Page A. Propositions 2.1 On economic rents and Pareto improvement 13 2.2 On economic rents and economic well-being 14 4.1 On aggregative growth functions 40 6.1 On s p e c i f i c a t i o n of fis h i n g time functions 236 6.2 On fi s h i n g time and resource stocks 268 B. C o r o l l a r i e s . 4.1 On nonexistence of aggregative growth functions 41 6.1 On f i s h i n g e f f o r t and f i s h stocks 263 6.2 On s p e c i f i c a t i o n of aggregative harvesting functions ..265 x i x PREFACE This thesis has been in progress for a number of years. The i n i t i a l research took place at the London School of Economics and P o l i t i c a l Science as a part of my studies for a M.Sc degree in mathematical economics and econometrics in 1977. Most of the work, however, was carried out at the University of B r i t i s h Columbia during my residence there as a member of the Programme in Natural Resource Economics 1977-1980. Subsequently, the work was continued at the University of Iceland where I have held a post in economics since 1980. As b e f i t s such a long gestation period, the l i s t of individuals that have contributed to the completion of this thesis is long. F i r s t and foremost I wish to thank my disser t a t i o n committee and, in pa r t i c u l a r , i t s chairman, professor P. A. Neher, for valuable advice and support. I would also l i k e to express special gratitude to professor G. R. Munro, who guided my work during my stay at the U.B.C. and to Professors A. D. Scott and R. S. Uhler at the Department of Economics, C. W. Clark at the Department of Mathematics and C. J. Walters at the Institute of Animal Resource Ecology for numerous helpful suggestions. I am also indebted to Professor W. M. Gorman currently at Nuf f i e l d College Oxford and P. Dasgupta at the London School of Economics who supervised my early research in f i e l d of natural resource economics and to the directors and staff of Icelandic Fisheries Association, National Enterprise Development Institute, the Marine Research Institute and National Economic XX Institute for having made a great deal of unpublished data available to me. Financial support from the Programme in Natural Resource Economics at the U.B.C., the National Enterprise Development Institute and the Icelandic Science Fund i s gr a t e f u l l y ac knowledged. xx i ORGANIZATION, NOTATION AND TERMINOLOGY. The main text of t h i s thesis i s arranged into parts and chapters. The parts separate l o g i c a l l y d i s t i n c t segments of the work. The chapters, which are ordered in a h i e r a r c h i a l manner into sections, subsections and sub-subsections, deal with the corresponding topics and subtopics. Footnotes are l i s t e d at the end of each chapter. Appendices are generally located at the end of the respective chapters, sections and subsections and are correspondingly l a b e l l e d . Thus appendix 4.2.5-A i s the f i r s t appendix of subsection 5 of section 2 in chapter 4. Most of the unpublished empirical data used in this essay are, however, l i s t e d in a separate appendix at the end of the thesis. Tables and figures are referred to by means of a binary number, the f i r s t of which indicates the respective chapter and the second the order of appearance. Thus, for instance, table 5.13 refers to the 13th table in chapter 5. Equations are referred to by running numbers, restarted at 1 in each new chapter. Equations appearing in appendices are a d d i t i o n a l l y related to the p a r t i c u l a r appendix in question by means of the corresponding l e t t e r . Thus equation (A.2) i s equation 2 in appendix A of some chapter or section. Due to the lim i t a t i o n s of the word processor employed to generate the text, the mathematical notation in t h i s essay is rather awkward. Although there are several exceptions to t h i s convention, the f i r s t part of the roman alphabet is generally used to represent constants and parameters, the middle part indices and xx i i the last part variables. Functions are generally indicated by c a p i t a l l e t t e r s . Thus, y is a function of x i s usually written as y=Y(x). Matrices are indicated by upper case and vectors lower case l e t t e r s with the symbol "~ " underneath. Thus U denotes the matrix U. The following table l i s t s some of the less standard mathematical notation employed in the thesis: Symbol Meaning The n-dimensional space of nonnegative real numbers. * Mult i p i i c a t ion. ** Exponentat ion. ABS(x) Absolute value of x. INT(x) The integer nearest to x,including x i t s e l f . SIN(x) The sine function of x. X 3x/3t, where t indicates time. Y' (x) 3Y(x)/3x. Y, (x) 3Y(x)/3x. Yx (x) 3Y(x)/3x. x~NIID x i s normally, i d e n t i c a l l y and independently d i s t r i b u t e d X' Transpose of the matrix X. The term " f i s h i n g industry" i s used to cover both the harvesting and the processing sectors of the f i s h e r i e s process. Values in Icelandic currency are in old Icelandic crowns (Ikr.) unless otherwise s p e c i f i e d . In 1974, 1 Ikr. = 0.0084 USD (US d o l l a r s ) . xxi i i PART I INTRODUCTION 1 1 . Introduction. In t h i s chapter we attempt to define the position of t h i s study within the f i e l d of natural resource economics in general and fishery economics in p a r t i c u l a r . We w i l l , moreover, describe the main objectives of the work and explain i t s approach, limi t a t i o n s and organization. 1 .1 F i e l d of Study. This i s a study in the f i e l d of natural resource economics. More precisely, i t belongs to the area frequently referred to as the economics of replenishable or renewable natural resources. An exact d e f i n i t i o n of the concept of natural resources i s not c r u c i a l in the context of t h i s study. For our purposes, a general wide d e f i n i t i o n of natural resources such as the one proposed by Kerry Smith and K r u t i l l a <1> i s p e r f e c t l y adequate. Within t h i s d e f i n i t i o n , however, the boundary between replenishable and nonreplenishable resources i s somewhat arbi t r a r y . Given long enough time frame, most, perhaps a l l , natural resources are renewable. Thus, for instance, o i l resources, which are generally regarded as nonrenewable, are continuously being renewed via natural processes a l b e i t at a very slow rate. This essay deals exclusively with a well defined subset of replenishable natural resources that I have chosen to c a l l s e l f -renewable. The defining c h a r a c t e r i s t i c of these resources is that their growth rate depends upon their own stock s i z e . The 2 essence of t h i s d e f i n i t i o n i s contained in the following characterization of a self-renewable natural resource. (1) x(t)=dx(t)/dt=G(x(t), ), where x(t) represents a measure of the resource stock at time t. The function, G( ), which depends on x(t) and possibly other variables, is referred to as the resource growth function. Obvious examples of self-renewable natural resources are b i o l o g i c a l resources including, of course, f i s h stocks. Several nonbiological resources, e.g. gases and water resources, may also exhibit the property described by (1) <2>. The s p e c i f i c subject of t h i s essay i s the economics of the Icelandic demersal f i s h stocks. This means that we w i l l be primarily interested in the properties of e f f i c i e n t harvesting programs for these f i s h stocks. However, since the Icelandic demersal f i s h stocks constitute an example of self-renewable natural resources, i t seems l i k e l y that some of the results to be derived may also apply to the u t i l i z a t i o n of other s e l f -renewable natural resources, in pa r t i c u l a r other f i s h stocks. 1.2 Economics of F i s h e r i e s . The branch of natural resource economics that deals with the economics of harvesting f i s h stocks i s c a l l e d economics of f i s h e r i e s . Economics of f i s h e r i e s is a f a i r l y recent addition to 3 economic theory. Its beginnings, as a formal academic subject, is marked by the papers by Gordon in 1954 and Scott in 1955 <3>. As early as 1911, however, Warming, writing in Danish, had already i d e n t i f i e d many of the pa r t i c u l a r e f f i c i e n c y problems of fi s h e r i e s <4>. His ideas, however, do not seem to have had any impact on the mainstream of economic thought. Thus, Marshall, in his P r i n c i p l e s of Economics, makes several references to fi s h e r i e s to i l l u s t r a t e general p r i n c i p l e s of commodity supply <5>, without an apparent r e a l i z a t i o n of their peculiar c h a r a c t e r i s t i c s from an economic point of view. Since 1954 s i g n i f i c a n t progress has been made in the economics of f i s h e r i e s . This holds especially in the area of fishery dynamics <6> and, more recently, the areas of stochastic <7> and multispecies f i s h e r i e s <8>. Fishery economics are primarily concerned with e f f i c i e n t harvesting of f i s h stocks. This l o g i c a l l y involves the following three sub-areas. (1) Demonstration of the i n e f f i c i e n c y of competitive harvesting of f i s h stocks <9>. (2) I d e n t i f i c a t i o n and description of the properties of e f f i c i e n t harvesting paths <10>. (3) The s p e c i f i c a t i o n of e f f i c i e n t i n s t i t u t i o n a l arrangements capable of generating an e f f i c i e n t harvesting pattern <11>. The present work contributes to the f i r s t and second 4 of these areas. More precisely, i t provides an empirical example of the i n e f f i c i e n c y of competitive harvesting of f i s h stocks and describes the attributes of the corresponding e f f i c i e n t harvesting programs. 1.3 Objectives of the Study. A primary objective of t h i s study i s to discover e f f i c i e n t harvesting paths for the Icelandic demersal f i s h e r i e s <12>. This aim necessarily involves the construction of a f a i r l y detailed empirical model of the economics and population dynamics of the Icelandic demersal f i s h e r i e s as well as the s p e c i f i c a t i o n of a relevant s o c i a l objective function. This part of the study is e s s e n t i a l l y normative. It may be regarded as a s o c i a l cost benefit analysis with respect to these f i s h e r i e s . This study i s also concerned with two important positive objectives. F i r s t l y , i t i s hoped that the very construction of an empirical model applicable to the Icelandic demersal f i s h e r i e s w i l l provide useful suggestions as to the appropriate s p e c i f i c a t i o n of f i s h e r i e s models in general; both from an empirical as well as an a n a l y t i c a l point of view. Secondly, the intention i s to build enough f l e x i b i l i t y into the model to allow investigation of the effects of some pa r t i c u l a r fishery situations on the nature of 5 e f f i c i e n t harvesting paths, independently of the of the empirical r e a l i t i e s of the Icelandic demersal f i s h e r i e s . Among such features are stochastic recruitment, f i s h i n g c a p i t a l m a l l e a b i l i t y , d i f f e r e n t ecological conditions etc. In t h i s way, some l i g h t may hopefully be thrown upon some issues of general interest in fishery economics. 1.4 Approach of the Study. In order to at t a i n the objectives described in the previous section, our approach is to develop a f a i r l y general model capable of imitating a wide range of empirical fishery situations <13>. Although the model i s es s e n t i a l l y a representation of the Icelandic demersal f i s h e r i e s , t h i s representation i s organized in such a way as to make the model much more generally applicable. To appreciate t h i s , i t is convenient to distinguish between the basic structure of the model, on the one hand, and i t s quantitative content, on the other. The basic structure of the model consists of a c o l l e c t i o n of variables and relationships devoid of any quantitative empirical content. As w i l l become apparent in part II below, th i s structure is mostly derived from the relevant b i o l o g i c a l and economic theory with added features suggested by the observed attributes of the Icelandic demersal f i s h e r i e s . Hence, at least to the extent that the broad c h a r a c t e r i s t i c s of the Icelandic demersal f i s h e r i e s 6 are representative of other harvesting processes, t h i s structure should have a general a p p l i c a b i l i t y . The quantitative content of the model i s e n t i r e l y independent of the basic structure. In th i s particular study, the quantitative data are mostly derived from the Icelandic demersal f i s h e r i e s . In p r i n c i p l e , however, the data may be derived from any source. Once the data have been supplied, however, the model becomes operational in the sense of being able to generate r e s u l t s . The nature of the results depends on the data supplied to the model. General results can be obtained by simply feeding data of a general nature to the model. Conversely, s p e c i f i c results may be derived by supplying s p e c i f i c data to the model. This p a r t i c u l a r approach to model building, which may be referred to as the empirical or numerical approach, has important advantages compared with the a n a l y t i c a l approach. Most importantly, a n a l y t i c a l models, due to the limi t a t i o n s of a n a l y t i c a l techniques, must be extremely simple. Hence their relevance to the real situations i s always subject to question. The numerical approach is designed to overcome thi s d i f f i c u l t y . Given s u f f i c i e n t computational power, numerical models can, in p r i n c i p l e , provide any desired approximation to actual situations. On the other hand, there are also important disadvantages with the numerical approach. Numerical models are comparatively cumbersome to describe and d i f f i c u l t to manipulate. Hence their use generally requires the assistance of high capacity computers <14>. A more serious 7 problem i s that there are fundamental obstacles to establishing the generality of results derived on the basis of numerical models. These considerations suggest the d e s i r a b i l i t y of an i t e r a t i v e process between the a n a l y t i c a l and numerical approaches to modelling. In fact, the current work should be regarded as an attempt towards one numerical i t e r a t i o n in accordance with t h i s methodological view. 1.5 Limitations of the Study. The scope of t h i s inquiry i s r e s t r i c t e d in several important respects most of which w i l l become clearer below. Here, however, i t i s helpful to draw attention to some fundamental r e s t r i c t i o n s . F i r s t , the model to be developed constitutes only a p a r t i a l view of the relevant r e a l i t y . The model concentrates on the fi s h i n g industry and i t s constituent b i o l o g i c a l and economic subsets. The economics and biology of the harvesting process are, on the other hand, embedded in much more extensive economic "and b i o l o g i c a l systems. These overall systems are regarded as exogenous in t h i s study. The depth of the study i s also limited. This refers i . a . to the l e v e l of disaggregation. Many di f f e r e n t types of f i s h i n g vessels, fi s h i n g grounds, f i s h products, fishing seasons etc. are bunched together in the model. To a large 8 extent t h i s i s prescribed by the a v a i l a b i l i t y of data. Partly t h i s aggregation constitutes a si m p l i f i c a t i o n , to render the model more manageable <15>. F i n a l l y , the extent of model manipulations undertaken in t h i s work i s very l i m i t e d . The impact of r e l a t i v e l y few po t e n t i a l control v ariables is considered. S i m i l a r l y few s p e c i f i c a t i o n s of exogenous conditions are investigated. Consequently, in t h i s area, there i s ample room for further work. 1.6 Structure of Presentation. The contents of t h i s work are broadly arranged as follows: Part I, i . e . the current part, provides an introduction to the to the main subject of the th e s i s . Part II i s devoted to the construction of an empirical model of the Icelandic demersal f i s h e r i e s . F i r s t , in chapter 3, an out l i n e of the model i s provided. In chapters 4 and 5, the b i o l o g i c a l and economic submodels are developed. Chapter 6 deals with the l i n k between these two submodels, in the form of the so-called f i s h i n g mortality production function. In part I I I , the properties of the empirical model are investigated. F i r s t , in order to assess i t s a p p l i c a b i l i t y to Icelandic demersal f i s h e r i e s , the model i s employed to simulate these f i s h e r i e s during 1960-80. In chapter 8, 9 e f f i c i e n t harvesting programs are considered. F i r s t , e f f i c i e n t programs for the Icelandic demersal f i s h e r i e s are discovered and examined. Secondly, the s e n s i t i v i t y of these programs to model sp e c i f i c a t i o n s are checked. Thirdly, the model i s used to investigate e f f i c i e n t harvesting paths under a few other conditions of interest. F i n a l l y , in part IV, the main results of the study are summarized. This study u t i l i z e s a great deal of empirical data. As a rule, only those subsets of the data that have not been published elsewhere are l i s t e d in the thesis. Most of these can be found in Appendix 1 at the end of the thesis. 1 0 Footnotes. 1. This b a s i c a l l y defines natural resources as those endowments that are o r i g i n a l to earth as well as the services therefrom. See Kerry Smith and K r u t i l l a , 1979, pp. 4-9. 2. Consider e.g. natural underground water res e r v o i r s . The momentary rate of inflow into such reservoirs i s t y p i c a l l y a decreasing function of the volume i t contains at that instance. 3. See Gordon, 1954 and Scott, 1955. 4. See Warming, 1911 and 1931. The resemblance of the arguments in Warming's 1911 paper with those in Gordon's 1954 paper without any evidence of the l a t t e r being f a m i l i a r with the former i s remarkable. 5. See Marshall, 1930, e.g. pp 369-71. 6. See e.g. C r u t c h f i e l d and Zellner 1962, Smith 1968, Clark and Munro 1975 and Clark, Clarke and Munro, 1979. 7. Examples are given by Reed, 1974, Charles, 1981b, Ludwig and Walters, 1982 and McKelvey, 1983. 8. See e.g. Quirk and Smith, 1970, S i l v e r t and Smith, 1977, Mendelssohn, 1980 and Hannesson, 1983. 9. The phrase "competitive harvesting" refers here simply to two or more agents harvesting the same resource. For further d e t a i l s see appendix 5.3-A. 10. For references see e.g. those quoted in footnotes 6, 7 and 8 above. For empirical references see e.g. Hannesson, 1974, Wilen, 1976, Arnason, 1977, M i t c h e l l , 1979 and Charles, 1981a. 11. See e.g. Sigurdsson, 1979 and Clark 1980. 12. E f f i c i e n t harvesting paths are, broadly speaking, those that maximize the economic contribution of the demersal f i s h e r i e s to the Icelandic people. A more precise d e f i n i t i o n of the concept is given in section 3.1 below. 13. And perhaps some other self-renewing natural resources as well. 14. For use in t h i s study, a c o l l e c t i o n of computer programs were developed. A b r i e f description of these i s provided in appendix 2 on computer programs at the end of the t h e s i s . 15. Notice i . a . that the aggregation over vessels excludes the p o s s i b i l i t y of e x p l o i t i n g differences in the e f f i c i e n c y of i n d i -vidual vessels and t h e i r crews in devising harvesting programs. 11 2. Rent Maximization and Economic Welfare. One of the main objectives of this study i s to discover harvesting paths that maximize the contribution of the demersal f i s h e r i e s to the economic well-being of the Icelandic people. The purpose of t h i s chapter i s to translate t h i s rather vague objective into a more tangible economic measure. Since t h i s , while fundamental in a wider sense, is somewhat peripheral to the main topics of this essay, the discussion w i l l proceed in rather simple terms. Nevertheless, within t h i s simple framework, i t turns out that the conditions under which th i s translation i s possible are somewhat r e s t r i c t i v e <1>. Consider an economy composed of individuals, firms and a government. Let there be a f i n i t e number of dated commodities, m, say, in t h i s economy <2>. The firms, which are owned by the consumers, produce a l l the commodities except labour which i s supplied by the consumers. The role of the government i s limited to levying taxes and paying subsidies to the firms. What i s l e f t of the output is consumed by the consumers. Represent the net production of commodities by the (mxl) vector y with negative elements denoting net inputs and positive elements net outputs of the respective commodity. The vector y belongs, of course, to a production p o s s i b i l i t y set c a l l e d Y. Moreover, l e t p represent a (1xm) vector of nonnegative prices corresponding to y. Now define the value of the production a c t i v i t y as the value of the net production vector: (1) r=py, peR?, yeY. 1 2 The variable r, thus, represents the value of the production a c t i v i t y after a l l inputs have been paid. It corresponds exactly to production p r o f i t s before taxes and subsidies. In what follows, r w i l l be frequently referred to as economic rents. The net production vector can be decomposed into the production of the fis h i n g industry, y(1), and the production of the rest of the economy, y(2). In other words: (2) y=y(1)+y(2), y(1),y(2),yeY. S i m i l a r l y , economic rents may be decomposed as follows: (3) r=r(1)+r(2)=p(y(1)+y(2)), where r(l)=py(l) denotes the economic rents in the fishery sector. Now, assuming a balanced government budget, the consumers are bound by the following constraint: (4) pc=r, where c represents the (mxl) net consumption vector including labour supply <3>. Equation (4) b a s i c a l l y defines the upper boundary of the consumption p o s s i b i l i t y set. More precisely, assuming p is fixed, (4) implies that the attainable consumption le v e l increases with r. Moreover, since r=r(l)+r(2), i t increases, c e t e r i s paribus, with r ( l ) . Thus, we have arrived at the following proposition: 1 3 Proposition 2.1 Under the conditions s p e c i f i e d , an increase in the l e v e l of economic rents generated in the fis h i n g industry creates an unambiguous potential for a Pareto improvement <4>. This conclusion can be strengthened, at some cost in generality, with the help of the following argument. Let the consumption behaviour of the consumer be described as a solution to the problem: Max U(c) c s.t. pc=r, where U(c) may be regarded as a well-being function for the consumers as a whole <5>. The solution to th i s problem, assuming i t exi s t s , allows us to define the following indi r e c t economic well-being function <6>: (5) v=V(p,r). Given nonsatiation and certain regularity conditions on U(c) <7>, i t follows that V(p,r) i s increasing in r. 1 4 Now, assume that the economy is in a state of equilibrium characterized by y*,c* and p* and the condition c*=y*. Assume, also, that y ( l ) depends on a vector of decisions or controls, z, say, so that y ( l ) = Y l ( z ) . Consider the following maximization problem: (6) Max r=p*y=p*y(1)+p*y(2), Z s . t. (i) yeY, ( i i ) y(1)=Y1(z). The solution to (6) gives r i s e to a certain level of economic rents, r**, say. Since r** i s the highest l e v e l of economic rents attainable, the solution to (6) also maximizes the indirect economic well-being function V(p,r) with respect to z. Moreover, since by assumption y(2) is independent of z, the following proposition has been established: Proposition 2.2 Under the' conditions described, the controls, z, that maximize the l e v e l of economic rents in the f i s h i n g industry also maximize the indirect economic well-being function defined by (5). 1 5 The results expressed in propositions 2.1 and 2.2 are based on a number of r e s t r i c t i v e assumptions that are important to keep in mind. F i r s t , the deduction of proposition 2.2 i s c r i t i c a l l y dependent upon the existence of a c o l l e c t i v e well-being function with the required properties. This is questionable as previously noted. Secondly, both propositions depend on the assumption that prices remain constant. If a new harvesting pattern were, in fact, adopted, the price vector would, in general, be altered. Therefore, as suggested by the arguments of the indirect well-being function, so would the le v e l of economic well-being. It follows that a harvesting program e f f i c i e n t under the old prices would generally not be e f f i c i e n t under the new ones. In fact, i f the price vector changes as a result of a new harvesting program, the opportunity set of the consumers may not have been increased. Thirdly, as pointed out at the outset, the a n a l y t i c a l framework employed above is highly s i m p l i f i e d . Hence, there is no assurance that the same results would apply in a more r e a l i s t i c setting. 16 Footnotes. 1. This, of course, is the usual situation in cost benefit analyses. 2. The concept of dated commodities is a standard t r i c k to account for time within an otherwise s t a t i c framework. A f i n i t e number of such commodities, however, implies added r e s t r i c t i o n s ; f i r s t l y discrete time and secondly a f i n i t e time horizon. 3. It i s informative to notice that equations (1) and (4) imply Walras law. 4. In fact, i t constitutes a d e f i n i t e s o c i a l improvement on the scales of the Hicks-Kaldor welfare measure. See e.g. Ng, 1980, pp. 60-62. 5. There are some fundamental problems with these kinds of functions, e.g. concerning their existence. For d e t a i l s see e.g. the discussion in Ng, 1980, pp. 111-18. 6. This function is pe r f e c t l y analogous to standard indirect u t i l i t y functions. 7. It i s s u f f i c i e n t that U(c) be l o c a l l y concave in the neighbourhood of maximum. ~ 1 7 PART II THE EMPIRICAL MODEL 18 3. The Empirical Model: An Outline. In t h i s part of the thesis an empirical model of the bioeconomics of the Icelandic demersal f i s h e r i e s w i l l be developed. As the model i s somewhat complex i t may be helpful to present here a summary of i t s main structure. As discussed in the introduction, one of the basic aims of this study is to identify and investigate the properties of so-c a l l e d e f f i c i e n t harvesting programs for the Icelandic demersal f i s h e r i e s . A model designed for this purpose must c l e a r l y include, at least, the following components: (i) A set of controls to define d i f f e r e n t harvesting programs. ( i i ) An empirical model describing the b i o l o g i c a l and economic relationships that map given values of the controls into harvesting r e s u l t s . ( i i i ) An objective function that associates given harvesting results with certain e f f i c i e n c y l e v e l s . Beginning with the objective function, we w i l l , in the following three sections b r i e f l y describe each of these components of the model. 19 3.1 The Objective Function. The management of a resource may, in p r i n c i p l e , be concerned with a great number of objectives. In fact, the empirical model to be developed below can accomodate many such objectives. However, as stated in section 1.3 above, we w i l l , in this study, mainly confine our attention to the discovery of harvesting paths that maximize the contribution of the demersal f i s h e r i e s to the economic well-being of the Icelandic people. On the basis of arguments forwarded in chapter 2, we assume that a sati s f a c t o r y measure of t h i s contribution is provided by the value of harvesting programs. The value of a harvesting program is defined to be the discounted sum of the value of the harvesting a c t i v i t y in each period <1>. More precisely, the value of a p a r t i c u l a r harvesting program, A, i s defined by the expression: (1) V(A)= I V(t)d(t)= L ( R ( t ) - C ( t ) ) d ( t ) . Where V(A) denotes the value of harvesting program A. V(t)=R(t)~ C(t) is the value of the harvesting a c t i v i t y during [ t - 1 , t ] , where R(t) and C(t) denote the respective revenue and cost functions. d(t) represents the time discount factor during [ t -1,t], T stands for the t o t a l number of time periods considered. In standard economic terminology, V(A) is simply the discounted p r o f i t function of program A over the period [0,T]. A l t e r n a t i v e l y , V(A) may be referred to as the economic rents of program A. T, of course, may not be f i n i t e . Harvesting paths having the property of maximizing V(A), or 20 other objective functions that may be defined, w i l l be c a l l e d e f f i c i e n t . 3.2 The Empirical Model. The empirical model consists of b i o l o g i c a l and economic submodels and the link between them. In what follows, we outline, in very general terms, the structure of each of these components of the model. 3.2.1 The B i o l o g i c a l Submodel. The b i o l o g i c a l submodel comprises 3 demersal species; cod, haddock and saithe <2>. It describes the relevant biology of these species, in p a r t i c u l a r their population dynamics. The central descriptive variables of the population at a given date are the number and average weight of the individuals belonging to the population at that date. The multiple of these variables constitutes the biomass of the species. The number of individuals changes over time due to mortality on the one hand and so-called recruitment or procreation on the other. Mortality i s due to both natural causes and f i s h i n g . Recruitment is generally a function of the state of the population, especially the size of i t s spawning stock. The structure of the b i o l o g i c a l model permits some f a i r l y general population interactions between the three species. Most 21 importantly, the model contains empirical estimates of the effects of food competition on the average weight of the individuals belonging to the species. Mortality interactions in terms of predator-prey relationships etc. are also represented in the model. Since, however, empirical knowledge about the nature and magnitude of these relationships i s very limited, this p a r t i c u l a r c a p a b i l i t y i s empirically rather vacuous. Nevertheless, from a formal point of view, the description of the 3 demersal species contains many of the key elements of a small ecosystem. For lack of a better label t h i s system w i l l be referred to as the demersal subsystem. The demersal subsystem i s of course but a small part of a much larger ecosystem. It i s axiomatic that the state of the ecosystem as a whole aff e c t s the demersal subsystem in various ways. Some channels of this nature are included in the bi o l o g i c a l submodel. As very l i t t l e empirical data on these influences are available, however, th i s c a p a b i l i t y of the model is again e s s e n t i a l l y formal. Feed-back p o s s i b i l i t i e s , i . e . the reverse e f f e c t s from the demersal subsystem to the remainder of the ecosystem, are not included in the model. The mathematical structure of the b i o l o g i c a l submodel is based upon a formulation that has become known as the Beverton-Holt f i s h e r i e s model <3>. According to the Beverton-Holt model, a general version of the stock size and catch equations for each species may be written as: (2) x(t)=X({f},{m},{w},{r}) /w <s* 22 (3) y(t)=Y({f},{m},{w},{r}), a l l t. *V **** Where x(t) and y(t) denote the aggregate biomass and catch respectively at time t for the species in question, i, m and w are vectors of fis h i n g mortality rates, natural mortality rates and individual weights. The dimensionality of these vectors equals the number of exploited cohorts in the fishery, J, say. r represents recruitment, i . e . the size of new cohorts recruited to the fishery. The curly brackets indicate that s u f f i c i e n t l y long time paths of the respective variables, in th i s case covering J periods, are involved. Now, l e t u stand for the lead time between spawning and the subsequent recruitment and l e t s(t-u) represent the size of the spawning stock at t-u. This spawning stock thus gives r i s e to a recruitment to the fishery u periods l a t e r , i . e . r ( t ) . Moreover, l e t Z(t) denote the vector of relevant environmental conditions and X(t) the vector of cohort stock sizes of the three species in the demersal subsystem. A f a i r l y general form of the recruitment function can now be spec i f i e d as: (4) r(t)=R(s(t-u),{X},{Z}), a l l t, where the curly brackets here suggest that u periods of the respective variables are involved. S i m i l a r l y we may specify natural mortality and weight functions as: (5) m(t)=M({X},{Z}), a l l t, 23 (6) w(t)=W({X},{Z}), a l l t. Equations (2)-(6) describe the essential structure of the b i o l o g i c a l submodel. The model b a s i c a l l y explains the stock size and catch in terms of current and previous f i s h i n g and natural mo r t a l i t i e s , individual weights, recruitment and environmental conditions. Of these variables only f i s h i n g mortality and environmental conditions are exogenous. Hence, substituting equations (4)-(6) into (2) and (3) yi e l d s the stock and catch dynamics for each species in terms of the exogenous variables and the i n i t i a l cohort stock sizes of a l l the species. I.e. (7) x(t)=X({f},{Z},X(0)), a l l t, (8) y(t)=Y({f},{Z},X(0)), a l l t. Where X(0) denotes the vector of stock sizes of the species at /NS t = 0 . Equations (2) to (6) are f i n a l form equations that summarize the dynamics of the b i o l o g i c a l submodel. According to these equations, the development of the fishery i s completely defined by i t s i n i t i a l b i o l o g i c a l stock lev e l s and the time path of fishing m o r t a l i t i e s and environmental conditions. Clearly equations (7) and (8) constitute, in general, a dynamic system of nonlinear equations. Due to the nature of the b i o l o g i c a l renewal processes, i . e . the annual recruitment process, the dynamics are formulated in terms of difference equations with a basic time period of one year <4>. As indicated 24 above, the order of these difference equations may be high. Hence the b i o l o g i c a l submodel may exhibit complex dynamics. Although not e x p l i c i t in the above equations, stochastic effects can ea s i l y be included in the b i o l o g i c a l submodel, e.g. as a part of the Z(t) vector. 3.2.2 The Economic Submodel. The economic submodel describes the structure and operations of the fis h i n g industry. It distinguishes between two sectors of the fis h i n g industry; the harvesting and f i s h processing sectors. The harvesting sector involves several types of fishing vessels, distinguished according to location, size, age and other c h a r a c t e r i s t i c s . The harvesting sector employs these vessels and other economic f i s h i n g inputs to generate fi s h i n g m o r t a l i t i e s and subsequently catch. The central stock variable of the harvesting sector i s the f i s h i n g f l e e t . At any point of time the available f l e e t imposes an upper l i m i t on the amount of fish i n g e f f o r t , so to speak, that can be exerted. The fis h i n g f l e e t changes over time due to depreciation and investment. The f i s h processing sector comprises several f i s h processing plants that operate one or more of four basic production processes av a i l a b l e . With the help of these plants and other economic inputs the processing sector transforms landed catch into f i n a l f i s h products. The v e r t i c a l flow of f i s h from the harvesting sector through the various stages of processing i s determined by certain empirically estimated input-25 output c o e f f i c i e n t s . As is the case with the b i o l o g i c a l submodel the economic submodel is only a subset of a larger system. The surrounding economy supplies i t with inputs of labour, c a p i t a l and materials and serves as an outlet for i t s outputs. The value of these input-output flows is determined with the help of prices which are generated by the complete economic system. The values of the inputs and outputs constitute the costs and revenues of the harvesting a c t i v i t y , i . e . i t s value. In t h i s way the economic system, or, more precisely, the prices i t generates, af f e c t s the economic submodel. As in the case of the b i o l o g i c a l submodel, we assume that there are no feed-backs; i . e . whatever happens in the economic submodel does not aff e c t the remainder of the economic system. In p a r t i c u l a r , input and output prices are taken to be exogenous to the economic submodel. The functional structure of the economic submodel may be summarized as follows: The revenue function i s defined by: (9) R(t)=p(t)q(t), where p(t) i s the row vector of output prices and q(t) is the column vector of f i n a l i z e d outputs. The output of f i s h products at time t is given by: (10) q(t)=T(p,t)y(t), Where q(t) represents the vector of f i n a l f i s h products during 26 [ t - 1 , t ] , y(t) is the catch vector and T(p,t) i s a matrix of input-output c o e f f i c i e n t s that transform a given catch vector into a vector of f i n a l outputs. Notice that some of the elements of T are taken to be variable depending on a vector of relevant prices, p , as well as time. The t o t a l cost of the harvesting a c t i v i t y during a given time i n t e r v a l , [ t - 1 , t ] , say, i s the sum of harvesting, processing and c a p i t a l adjustment costs, as defined by the cost functions C1(t), C2(t) and C3(t), respectively. (11) C(t)=C1(t)+C2(t)+C3(t). Total costs of the harvesting a c t i v i t y during a period of time are the sum of the harvesting costs for each individual vessel. Those, in turn, depend on the vessels' use of economic inputs, here represented by the vector e(t) and the corresponding prices, w(t). Included in the vector of economic inputs are, of course, both stock and flow variables. Among the stock variables are the size, type and other c h a r a c t e r i s t i c s of the vessels. Also included in £,(t) are t y p i c a l harvesting decisions or control variables such as f i s h i n g time, type of gear, fishery pursued etc. More formally: N (12) C1(t) = Z C 1 ( e ( i , t ) , w ( t ) , i ) , where N i s the t o t a l number of vessels. Total processing costs are the sum of the processing costs of the various processing plants. Each plant's processing costs 27 are assumed to depend on the volume of catch processed during [ t - l , t ] , i . e . y ( t ) , as well as the input prices, w(t). So: M (13) C2(t)= Z C 2 ( y ( i , t ) , w ( t ) , i ) , where M i s the t o t a l number of processing plants. The c a p i t a l in the economic submodel is mostly in the form of i n d i v i s i b l e units, namely fis h i n g vessels and processing plants of various types. The c a p i t a l dynamics are consequently concerned with changes in the number of these units, i . e . N and M. For convenience of presentation refer to those c a p i t a l units by the common label k. Given t h i s , the c a p i t a l dynamics are determined by a function of the following type: (14) k(t)=k(t-1)-D(k(t-1))+I(t), where k(t) denotes c a p i t a l at time t. The function D(.) represents depreciation of c a p i t a l and l ( t ) net investment or disinvestment in c a p i t a l during [ t - 1 , t ] . Associated with c a p i t a l adjustments there are, in general, certain costs. Let these be represented by: (15) C3(t)=C3(I(t)). Equations (9)-(1 5) describe the essential structure of the economic submodel. Of the main variables involved, economic fishing inputs, c a p i t a l , catch, prices, investment and some of 28 the elements of the input-output matrix, T, are exogenous. The other variables are endogenous. Notice, that according to equation (14), the economic submodel adds some dynamics of i t s own to the empirical model. In conformity with the dynamic structure of the b i o l o g i c a l submodel, the economic dynamics are also formulated in discrete time with a basic time period of one year. 3.2.3 The Fishing Mortality Production Function. The b i o l o g i c a l and economic submodels are linked by a special relationship c a l l e d the fis h i n g mortality production function. This function b a s i c a l l y maps the economic f i s h i n g inputs of the economic submodel into the fis h i n g mortality rates in the b i o l o g i c a l submodel. Regarding f i s h i n g m o r t a l i t i e s as outputs, this function i s , in important respects, analogous to a standard production function in economic theory. In general the state of the demersal species as well as that of the ecological environment w i l l a f f e c t the production of fi s h i n g m o r t a l i t i e s . Thus a formal s p e c i f i c a t i o n of the fis h i n g mortality production function i s : (16) f ( t ) = F ( e ( t ) , X ( t ) , Z ( t ) ) , a l l t, where |,(t) represents the vector of fis h i n g mortalities at time t. 29 3.2.4 The Empirical Model: A Summary. The essential structure of the empirical model has now been outlined and formally expressed in equations (2)—(16) above. Here we attempt to provide a somewhat more compact picture of the model and how i t works. F i r s t , i t is convenient to summarize the main categories of variables involved in the model. This i s provided in table 3.1. This table also serves another purpose. In addition to defining the included variables, the empirical model also specifies which of the included variables are endogenous and which are exogenous. Since, by defination, control variables must be exogenous the empirical model also defines the set of l o g i c a l l y permissable controls. The empirical content of the model, in turn, indicates which of these variables constitute reasonable controls from an empirical point of view. This p a r t i c u l a r c l a s s i f i c a t i o n is also indicated in table 3.1. 30 Table 3.1 The Main Categories of Variables in the Empirical Model. Variables Endogenous Exogenous Controllable Biology: Environm. effects Z Biomass X + Catch y + Recruitment r + Nat. m o r t a l i t i e s m + Fish, m o r t a l i t i e s f + Indiv. weight w + Economy: Prices p Tot. f i s h , costs C + Harvest, costs C1 + Process, costs C2 + Cap. adjm. costs C3 + Fish, c a p i t a l k + Depreciation D + F i n a l products q + Fish, inputs e - + + Invm. f i s h . cap. I - + . + Input-output cof's T* +* +* + Discount factor d - + Total number 19 14 Note: 1) + suggests that the respective variable has the i n d i -cated property. 2) - means that the variable does not have the indicated property. * Some of the elements of T are endogenous. Others are exogen-ous and c o n t r o l l a b l e . According to table 3.1 there are 19 main categories of variables in the empirical model. Of these 14 are endogenous. Out of the 6 exogenous variables, 3 are regarded c o n t r o l l a b l e . Prices, p, environmental e f f e c t s , Z, and the rate of discount, 31 d, are according to the structure of the model, not controllable. More formally, the essential structure of the model may be summarized in the following f i n a l form equation representing the value of harvesting program A: (17) V(A)=F(X(0),K(0);{Z},{p},{d};{e(A)},{T(A)},{l(A)}). *SJ SNS ^ y>W 4S*t Where X (0) and K (0) represent the i n i t i a l state of the biology and economy ( c a p i t a l ) , respectively. {Z}, {p} and {d} represent the time paths of the exogneous variables, namely environmental influences, prices and discount factor. {e(A)}, {T(A)} and {1(A)}, f i n a l l y , represent the time path of the control variables according to harvesting program A. The i n t e r r e l a t i o n s between the variables as well as the general structure of the empirical model are further summarized in the diagram in figure 3.1. 32 Figure 3.1 The Empirical Model: Basic Structure. E c o L o & y ENVl 10NMEN1 AL FECTS OCX EFFOIl EFFECT S H H I 6 M O R l A L I T i £5 f i S H i N & MoRTAL i r y P R O D U C T I O N Ft/WCTioN E COUOHi£ F I SHMf r INPU7.S C A T C H INPUTS E C O N O M y PRICES F I S H I N G - I N D U S T R Y Purs With the help of figure 3.1 the effects of a change in the exogenous variables may be traced out. Assume, for instance, that the system is i n i t i a l l y in a steady state and consider a change in the use of economic f i s h i n g inputs. This sets into motion a process of dynamic adjustments. Immediately affected are (i) harvesting costs and ( i i ) fishing m o r t a l i t i e s (through the f i s h i n g mortality production function). The change in fishi n g m o r t a l i t i e s , in turn, changes the catch rate and, subsequently, the population dynamics of the demersal subsystem. The l a t t e r , which incidently may a l t e r the f i s h i n g mortalities via the f i s h i n g mortality production function, may require many periods to reach equilibrium, assuming i t is stable. The i n i t i a l and subsequent changes in the volume of catch a f f e c t , in turn, (i) the quantity of f i n a l products and ( i i ) the harvesting and processing costs in the economic submodel. Thus the value of the 3 3 objective function i s altered throughout the adjustment process due to varying revenues and costs. 3 . 3 The Control Variables. According to table 3 . 1 above, the l o g i c a l l y and empirically permissable control variables of the model are investment in fish i n g c a p i t a l , £(t), the use of economic inputs, e,(t), and some of the elements of the input-output matrix, T_ < 5 > . Any well defined path of the control variables w i l l be c a l l e d a harvesting program. Investment in fi s h i n g c a p i t a l involves both investment in new c a p i t a l and disinvestment in the old. The economic fishing inputs cover variables such as fis h i n g time, the choice of fish i n g gear and fishery, cohort selection etc. The controllable elements of the matrix T are primarily the allocations of catch to the d i f f e r e n t processes. It may be noted that these are precisely the variables that a sole owner of the fishery would generally adjust in order to maximize his objective function under standard market conditions. Although, in this essay, the f i s h e r i e s problem is generally viewed from the standpoint of a sole owner, th i s may be an appropriate place to point out that the empirical model can, with l i t t l e modification, describe the behaviour of a decentralized fishery. By that we mean a fishery consisting of fishing and f i s h processing firms that are motivated by a well defined objective such as p r o f i t maximization. Within that framework i t would then be possible to examine to what extent a 34 central authority would be able to at t a i n a s o c i a l objective by manipulating the economic environment of the private agents, e.g. by imposing taxes on inputs or outputs, issuing f i s h i n g licences etc. 35 Footnotes. 1. As explained in section 3.2 below, the model i s formulated in discrete time. 2. These three species have, in recent years, accounted for over 80% of the t o t a l demersal catch. Redfish that accounts for the bulk of the remaining demersal catch was not included in the study because of lack of appropriate data. 3. See Beverton and Holt, 1957. 4. In addition, perhaps partly as a result of this feature of the biology, much of the available empirical data i s in a yearly discrete form. 5. Notice, however, that these are only the main categories of control variables defined by the empirical model. As w i l l become apparent later on, the model includes numerous other potential control variables of i n t e r e s t . 36 4. The B i o l o g i c a l Submodel. The purpose of this chapter i s to develop the b i o l o g i c a l part of the empirical model. While our main concern is to model the population dynamics of the Icelandic demersal f i s h stocks we w i l l also, partly to give reasons for our pa r t i c u l a r approach to this task, examine some important aspects of f i s h stock growth models in general. This chapter i s accordingly divided into two major sections. The f i r s t section contains a review and analysis of some widely u t i l i z e d f i s h stock growth theories. Although formulated in terms of f i s h resources the results derived are also relevant for modelling the growth of other replenishable resources. In the second part of the chapter the pa r t i c u l a r population dynamics of the Icelandic demersal stocks w i l l be considered and empirical estimates provided. 4.1 Some Fish Stock Growth Models. In t h i s part of chapter 4 we w i l l review a few f i s h stock growth models that have become standard in the b i o l o g i c a l and economic analysis of f i s h e r i e s . In addition we w i l l consider some simple ecological relationships and attempt to f i t these into the framework of a disaggregated population growth model of the Beverton-Holt type <1>. As indicated above, the purpose of this section i s to lay a theoretical basis for the empirical work and forward arguments for the selection of the pa r t i c u l a r population growth model employed in the second part of the present chapter. 37 4.1.1 Aggregative Population Growth Models. Adapting the Verhulst (1838) human population growth model to f i s h stocks, Graham (1935) proposed the following model to explain the growth of f i s h stocks: (1) x(t)=ax(t)(1-x(t)/b). Where x(t) i s the biomass of the f i s h stock at time t, x(t)=dx(t)/dt and the parameters, a and b, are the i n t r i n s i c growth rate of the biomass and i t s maximum sustainable size respectively. Equation (1) is the celebrated l o g i s t i c function the properties of which are well known and need not be l i s t e d here <2>. Its application to f i s h e r i e s y i e l d s a p a r t i c u l a r l y simple expression for equilibrium or sustainable catch as a function of fish i n g e f f o r t i f two additional assumptions, due to Schaefer (1954), are made: (2) x(t)=y(t), (3) y(t)=x(t)e(t) . Where y(t) represents the rate of catch and e(t) an appropriately normalized measure of fis h i n g e f f o r t <3>. The l o g i s t i c growth function has, since Graham's introduction, been extensively used both in fishery biology and fishery economics <4>. Some of the applications involve modifications to allow for nonsymmetric growth <5>, depensation 38 etc. The l o g i s t i c function and i t s r e l a t i v e s , mentioned above, belong to a general class of functions defined by: . (4) x(t)=G(x(t)), x(t)eX<=RJ. Where the function, G( ), may be assumed to be twice continuously d i f f e r e n t i a b l e and having the property that there exist x1 and x2 belonging to the set X and s a t i s f y i n g x2>x1>0 and such that G(x1)=G(x2)=0 and G(x)>0 for a l l xe(x1,x2). Several special cases of (4), e.g. the l o g i s t i c function, have been found to provide reasonable approximations to observed growth rates of various b i o l o g i c a l populations <6>. For th i s reason i t is perhaps not surprising that t h i s class of growth functions, especially i t s mathematically simpler variants, have become very popular in the economics of f i s h e r i e s . Certainly i t has dominated i t s theoret i c a l side <7>. However, when applied to most f i s h e r i e s , there i s a serious d i f f i c u l t y with this class of growth functions. We w i l l now b r i e f l y examine the nature of th i s di f f i c u l t y . Fish stocks, as in fact most replenishable resources, are in general not a homogenous mass. They are for instance t y p i c a l l y composed of a number of cohorts or yearclasses each of whom has i t s own growth rate <8>. Given t h i s , i t i s obviously of interest to investigate the conditions under which the growth of a multicohort f i s h population can be accurately described by a general aggregative growth function defined by (4). Consider a f i s h stock that is composed of I cohorts. For 39 sake of the argument, assume I>1. Assume, moreover, that each cohort, i , has a growth function, G ( i , x ( i , t ) ) , that i s twice continuously d i f f e r e n t i a b l e <9>. The existence of an aggregative growth function now requires the following: (5) x= Z x(i)= Z G(i,x(i))=G( Z x(i))=G(x), (I | (SI <•-/ where, for convenience of notation, an e x p l i c i t reference to the time dependence of the arguments has been suppressed. Notice that, apart from the obvious conditions: i (6) x(i)>0, a l l i and I x(i)=x, there are no r e s t r i c t i o n s on the cohort and aggregate biomass le v e l s . In other words, (5) must hold for a l l x(i) and x s a t i s f y i n g (6). Now, varying only one x(i) at a time and making the corresponding adjustment in x, as prescribed by (6), we f i n d : (7) G'(i,x(i))=G'(x), a l l i . So the slopes of the individual cohort growth functions must be i d e n t i c a l and equal the slope of the aggregative one. Moreover, since this property must hold for a l l x(i)>0, these slopes must be constant. More precisely, d i f f e r e n t i a t i n g (7) keeping x f ixed: (8) G" (i,x(i))=G'' (X)(1+ Z 3x ( j )/9x (i) ) =G " (x )-0=0 , a l l i . j Pi 40 Hence we conclude: (9) G(i,x(i))=a(i)+bx, a l l i , where a(i) and b are constants. (9) describes a necessary condition for the existence of an aggregative growth function. To see that (9) also provides a s u f f i c i e n t condition, write on the basis of (9): (10) L G(i,x(i))= I a(i)+b E x(i)=d+bx=G(x). i - i i n c - 1 We have thus arrived at the following proposition: Proposition 4.1 If a replenishable natural resource is composed of more than one cohort and the cohort biomass levels can vary independently an aggregative growth function for the resource exists i f and only i f a l l the cohort growth functions are linear with i d e n t i c a l slopes <10>. Since, according to proposition 4.1, the existence of an aggregative growth function implies exponential growth, positive or negative, of the aggregate biomass, which i s empirically untenable, we have the following c o r o l l a r y : 41 Corollary 4.1 For a multicohort replenishable natural resource a b i o l o g i c a l l y acceptable aggregative growth function generally does not ex i s t <11>. According to these results the growth process of t y p i c a l f i s h stocks can not, in general, be represented by an aggregative growth function of the class defined by (4). Hence the application of such functions to the empirical and th e o r e t i c a l analysis of f i s h stocks may easily lead to false conclusions. The errors involved in imposing an aggregative growth function on a multicohort population c l e a r l y depend on the actual v a r i a b i l i t y of the individual cohort biomasses as well as their growth functions. Being somewhat outside the main scope of t h i s study, examination of these issues is relegated to appendix 4.1.1-A. 42 Appendix 4.1.1-A Aggregative Growth Functions: Further Discussion. (1) When i s Aggregation Possible? In section 4.1 we found that for a multicohort replenishable natural resource, the existence of an aggregative growth function generally implied b i o l o g i c a l l y unacceptable r e s t r i c t i o n s on the shape of the individual cohort growth functions as well as the aggregative one. This result was arrived at by allowing certain, seemingly reasonable, variations in the r e l a t i v e biomasses of the cohorts to occur. It is conceivable, however, that extraneous factors, such as the harvesting process, somehow constrain the sequence of cohort biomasses over time to those that permit aggregation of b i o l o g i c a l l y plausible cohort growth functions. We w i l l now investigate this p o s s i b i l i t y . F i r s t we seek to uncover the r e s t r i c t i o n s on the combination of cohort biomasses, x ( i ) ' s , that, for given cohort growth functions, G ( i , x ( i ) ) ' s , allow an aggregative growth function. I . e : (A.1) I G(i,x(i))=G( 1 x(i))=G(x). Let A be the set of a l l cohort vectors, x * = ( x ( 1 ) , x ( 2 ) , . . x ( I ) ) , s a t i s f y i n g (A.1). ~ An obvious candidate for A are the vectors x* s a t i s f y i n g : (A.2) X * = ( F ( 1 , X ) , F ( 2 , X ) , . . . F ( I , X ) ) , where F ( i , x ) ' s are some functions indexed on i <12>. (A.2) simply states that i f each x(i) s a t i s f i e s the condition that i t can be uniquely inferred from the aggregate biomass, x, the aggregative growth function e x i s t s . This condition which i s s u f f i c i e n t for a l l G ( i , x ( i ) ) ' s irrespective of their form i s , however, too r e s t r i c t i v e . For our purposes i t i s s u f f i c i e n t to discover the set A for given G ( i , x ( i ) ) ' s , i . e . for each species. Given G ( i , x ( i ) ) ' s and assuming that an aggregative function, G(x), exists, (A.1) defines a I-dimensional hypersurface in the x ( i ) ' s . For (A.1) to hold, therefore, the set A must contain only vectors, x*, that l i e on this hypersurface. ~ How l i k e l y i s i t that harvesting w i l l be sel e c t i v e enough for x* to remain on the hypersurface defined by (A.1)? A p r i o r i , this~seems very unlik e l y . The hypersurface, (A.1), i s , at best, a very small subset of the space of nonnegative cohort biomasses. There does not seem to be any reason for the harvesting process to result in biomass vectors, x*'s, only on th i s hypersurface especially in the l i g h t of v a r i a b i l i t y in economic signals and the b i o l o g i c a l environment. The following 43 s i m p l e f i s h e r i e s e x a m p l e s e r v e s t o i l l u s t r a t e t h i s c o n t e n t i o n : C o n s i d e r a f i s h s t o c k c o m p o s e d o f I c o h o r t s . L e t t h e e q u i l i b r i u m p r o f i t f u n c t i o n <13> f o r i n d i v i d u a l f i r m s e n g a g e d i n a f i s h e r y b a s e d o n t h i s s t o c k b e d e f i n e d b y t h e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n : ( A . 3 ) p= Z P ( e ( i ) , X ( e ( i ) ) , s , i ) . W h e r e e ( i ) r e p r e s e n t s t h e f i s h i n g e f f o r t o n c o h o r t i . e,( i ) i s t h e ( 1 x ( i - 1 ) ) v e c t o r o f f i s h i n g e f f o r t o n c o h o r t s y o u n g e r t h a n i . X ( e v ( i ) ) r e p r e s e n t s t h e c o r r e s p o n d i n g b i o m a s s o f c o h o r t i . T h i s n a t u r a l l y d e p e n d s o n t h e p r e v i o u s f i s h i n g e f f o r t o n t h a t p a r t i c u l a r c o h o r t ( w h i c h d u e t o t h e e q u i l i b r i u m a s s u m p t i o n i s i n d i s t i n g u i s h a b l e f r o m t h e c u r r e n t f i s h i n g e f f o r t o n y o u n g e r c o h o r t s ) . T h e v e c t o r s , f i n a l l y , r e p r e s e n t s e c o n o m i c s h i f t v a r i a b l e s s u c h a s i n p u t a n d o u t p u t p r i c e s . I t i s a s s u m e d t h a t t h e p r o f i t f u n c t i o n i n c o r p o r a t e s t h e s h a d o w v a l u e o f t h e b i o m a s s g r o w t h c o n s t r a i n t t o t h e e x t e n t t h i s i s p e r c e i v e d b y t h e f i s h i n g f i r m s . N o w , a s s u m i n g p r o f i t m a x i m i z a t i o n , t h e c o h o r t s t h e f i s h i n g f i r m s s e e k t o h a r v e s t a r e t h o s e d e f i n e d b y t h e s e t I* w h e r e : ( A . 4 ) I * = { i : P e f o ( 0 , X ( e ( i ) ) , s , i ) > 0 } . T h e e l e m e n t s o f t h e o p t i m a l e f f o r t v e c t o r , e ( i ) * , a r e g i v e n b y : ( A . 5 ) E * = { e ( i ) : P e n ) ( e ( i ) , X ( e ( i ) ) , s , i ) = 0 } . A n d t h e c o r r e s p o n d i n g o p t i m a l b i o m a s s l e v e l s , x ( i ) * , a r e d e f i n e d b y t h e s e t : ( A . 6 ) X * = { x ( i ) : x ( i ) = X ( e ( i ) * ) } = H ( s , i ) , w h e r e e ( i ) * i s t h e ( 1 x i ) v e c t o r o f o p t i m a l f i s h i n g e f f o r t o n c o h o r t s ~ o f a g e i o r y o u n g e r a s d e f i n e d b y ( A . 5 ) a n d H ( ) i s s o m e f u n c t i o n . I n g e n e r a l I* i s a s u b s e t o f a l l t h e c o h o r t s i n t h e b i o m a s s a n d t h e e l e m e n t s o f E* a r e n o t i d e n t i c a l . I n o t h e r w o r d s , s e l e c t i v e f i s h i n g w i t h r e s p e c t t o c o h o r t s , e . g . v i a c h o i c e o f f i s h i n g g e a r , g r o u n d s e t c . , w i l l i n g e n e r a l b e o p t i m a l . M o r e t o t h e p o i n t , h o w e v e r , t h e s e t E* d e p e n d s o n s a n d h e n c e s o d o e s t h e c o r r e s p o n d i n g s e t o f e q u i l i b r i u m o r o p t i m a l b i o m a s s e s , X * . T y p i c a l e q u i l i b r i u m b i o m a s s l e v e l s f o r t h e c o h o r t s w i t h a n d w i t h o u t f i s h i n g a r e i l l u s t r a t e d i n f i g u r e 4 . 1 < 1 4 > . 44 In figure 4.1, the curve A,C depicts the cohort biomasses without harvesting. The curve, w ( i ) =Pea)( 0 ,X(.) , s , i ) =0 , defines the minimum biomass levels for harvesting to be optimal. I*, the corresponding set of exploited cohorts, comprises the cohorts [11,12]. The curve A,B,C,D indicates the resulting equilibrium cohort biomasses. Notice that these depend upon the economic parameters, s. It seems unlikely, indeed, that changes in these w i l l be r e s t r i c t e d to those that keep the vector of cohort biomasses, x, on the hypersurface defined by (A.1). (2) Empirical Estimation of Aggregative Growth Functions. As mentioned in section 4.1.1, the aggregative growth function has been extensively used in the economics of f i s h e r i e s . In theoretical applications this function has, not surprisingly, led to simple results that are easy to interpret. Empirical applications have, on the other hand, generally produced disappointing r e s u l t s . Most often i t i s found that the aggregative growth function simply does not explain much of the observed, or estimated, growth rates <15>. In the li g h t of the above this is readily understandable. Most f i s h stocks consist of a number of cohorts. An observed aggregate biomass may be composed of a wide range of individual cohort biomasses. At one extreme a given biomass may consist almost exclusively of the youngest cohort thus t y p i c a l l y exhibiting a r e l a t i v e l y high growth rate. At the other extreme the same biomass may, almost e n t i r e l y , consist of the oldest cohort that generally has a 45 negative growth rate. A given biomass may, in other words, map into a wide range of growth rates. The informational content of a certain aggregate biomass is simply not s u f f i c i e n t , in general, to deduce the aggregate growth rate. Since an aggregative biomass growth function for a multicohort fishery i s generally nonexistent, attempts to estimate the parameters of such a function usually involve a serious misspecification error. Attempts to improve the s t a t i s t i c a l properties of such estimates, retaining the aggregative functional form <16>, are consequently unlikely to be f r u i t f u l . It follows that recommendations based on such estimates of an aggregative biomass growth function may be far off target. Even i f the empirical estimates appear reasonable, due to accidental properties of the data, the resulting fishery policy w i l l , more often than not, change r e l a t i v e cohort biomasses considerably. Thus the harvesting policy i t s e l f w i l l generally invalidate the estimated growth function on which i s was based. To i l l u s t r a t e these problems consider the following estimates of an aggregative growth function for Icelandic cod.: Having time series data on the Icelandic cod fishery 1955— 1974, covering the yearly aggregate f i s h i n g e f f o r t , catch and biomass <17> as well as the biomass growth data for each cohort, i t was possible to estimate the aggregative growth function for cod in 3 d i f f e r e n t ways. F i r s t OLS techniques were employed to estimate a simple l o g i s t i c curve using the yearly catch/effort r a t i o as an estimator for the aggregate biomass. The resulting estimated growth curve i s l a b e l l e d "A" in figure 4.2. This standard method of estimating the aggregative biomass growth function is subject to serious s t a t i s t i c a l problems. One of these, discussed by Uhler (1978), is the "errors in variables" problem, due to the imperfect estimate of biomass by the catch-effort r a t i o . Having independent data on the aggregate biomass, however, these can be used in place of the catch/effort r a t i o . Hence i t is possible, in t h i s instance, to avoid the "errors in variables" problem while otherwise employing the same estimation technique as before. The graph of the resulting estimated growth function i s drawn as curve "B" in figure 4.2. Notice that t h i s estimate i s free of most of the s t a t i s t i c a l problems with which Uhler (1978) and Tierney (1978) are concerned. Hence this is the estimate to which the s t a t i s t i c a l improvements, recomended by these authors, s t r i v e . This curve, however, s t i l l incorporates the aggregation errors inherent in i t s very s p e c i f i c a t i o n . F i n a l l y , using the cohort disaggregated data, we were able to calculate a t h i r d aggregative growth function. This function was constructed as follows: The estimated vector of fi s h i n g mortalities for Icelandic cod in 1974 was l i n e a r i l y expanded and contracted thus r e f l e c t i n g a corresponding increase and decrease 46 in f i s h i n g e f f o r t . Meanwhile other b i o l o g i c a l factors were kept constant. The resulting equilibrium aggregate biomass and i t s growth l e v e l s were then taken to belong to the aggregative biomass growth function <18>. The graph of this function i s drawn as curve "C" in figure 4.2. Figure 4.2 Alternative Estimates of the Aggregative Growth Function. x G-rotvtli 0 f a S i As figure 4.2 c l e a r l y shows, the three aggregative growth functions are markedly d i f f e r e n t . Correspondingly so w i l l the respective "optimal" fishery p o l i c i e s be. In empirical work the use of disaggregated growth functions is often impossible due to lack of appropriate data. In such cases a r e s p e c i f i c a t i o n of the conventional aggregative growth function so as to r e f l e c t changes in r e l a t i v e biomasses of the d i f f e r e n t cohorts may improve matters. As suggested by figure 4.1 above, younger cohorts w i l l t y p i c a l l y constitute a greater proportion of the t o t a l biomass as fis h i n g e f f o r t increases and reverse. Hence current and lagged levels of fi s h i n g e f f o r t <19> may provide some information about the r e l a t i v e cohort composition of the current biomass and thus i t s growth rate. (3) The Aggregative Growth Function in Theoretical Work. F i n a l l y , the use of an aggregative growth function in th e o r e t i c a l analysis may lead to misleading r e s u l t s . This may be demonstrated with the help of a nonlinear version of a well known f i s h e r i e s model due to Clark and Munro (1975): 47 Let the p r o f i t function of the fishery at time t be defined by: (A.7) P(t)=Y(e(t),x(t))-C(e(t)), where Y( ) i s a smooth increasing j o i n t l y concave function and C( ) is a smooth increasing convex function. e(t) denotes fis h i n g e f f o r t and x(t) aggregate biomass both at time t. The resource growth constraint i s : (A.8) x(t)=G(x(t))-Y(e(t),x(t)), where G( ), the standard aggregative growth function, is assumed to be concave arid belonging to the class of functions defined by equation (4) in section 4.1.1. Maximizing the present value of the p r o f i t function yields the following equilibrium solution: (A. 9) G <+C eY x/P e=r, where r i s the rate of time discount. The second term of (A.9) i s c a l l e d the stock effect by Clark and Munro and. i s , as they point out, always positive within the framework of thei r , as well as t h i s , model. Thus the stock effect always works in the dire c t i o n of a lower equilibrium aggregate e f f o r t than would otherwise be the case. However, i f , as discussed above, higher e f f o r t l e v e l s produce upward s h i f t s in aggregative growth due to their effect on the r e l a t i v e cohort composition, an improved s p e c i f i c a t i o n of the aggregative growth function i s : (A.10) x(t)=G(x(t),e(t)), where G e( )>0. Solving the corresponding present value maximization problem, assuming G( ) to be j o i n t l y concave in x and e, yiel d s the maximal equilibrium condition: (A.11) G x+Y x(C e-G e)/P e=r. Conditions (A.9) and (A.11) d i f f e r with respect to the stock e f f e c t , which is smaller in the l a t t e r case and may be either positive or negative. So, taking the effect of fi s h i n g e f f o r t on the aggregative growth function into account indicates that the optimal equilibrium fishing e f f o r t i s indeed higher than that suggested by the standard aggregative population growth function. Keeping in mind our assumption as to the effect of fishing on the r e l a t i v e cohort configuration t h i s result i s readily understandable. 48 Appendix 4.1.1-B Estimates of Aggregative Growth Functions: An Example. We seek estimates of the c o e f f i c i e n t s a and b in the l o g i s t i c function: (B.I) x(t)=ax(t)+bx 2(t), where x(t) refers to biomass at time t. For a biomass that i s subject to harvesting and i s observed at discrete time intervals we may write as an approximation to (B.1): (B.2) x(t)-x(t-1)+y(t)=ax(t)+bx 2(t). Where y(t) represents the accumulative catch during [t-1,t] and x(t)=(x(t)+x(t-1))/2. The c o e f f i c i e n t s of (B.2) can be estimated with the help of the following data on the Icelandic cod stock: 49 Table B.1 Icelandic cod stock : Aggregative harvest ing and biomass < - Catch/ Est imated Biomass* Catch** e f f o r t b i oma s s Year 1000 tons 1000 tons index*** 1000 tons**** 1955 2.615 537 1 .28 2.826 56 2.429 482 1 .26 2.782 57 2.208 453 1 .00 2.208 58 2.089 51 1 0.98 2. 163 59 2.006 454 0.82 1.811 1 960 1.868 465 0.70 1 .546 61 1.745 376 0.56 1 .236 62 1 .635 386 0.59 1 .303 63 1 .505 409 0.58 1 .281 64 1 .480 435 0.52 1 . 148 65 1 .474 394 0.57 1 .259 66 1 .592 357 0.60 1.325 67 1 .846 344 0.58 1 .281 68 1 .959 380 0.87 1 .921 69 1 .994 405 1 .04 2.291 1970 1 .899 471 1 .09 2.407 71 1 .677 454 0.84 1 .855 72 1 .371 399 0.71 .1 .566 73 1 .319 380 0.70 1 .546 74 1 .205 375 0.79 1 .744 Sources: * These biomass estimates are based on VPA results using actual catch data to 1982. On the data see ICES, 1976, and Hafrannsoknarstofnun, 1983. ** Hafrannsoknarstofnun 26, 1982. *** ICES, 1968 and 1976. **** Catch/effort estimate of biomass normalized to be exact in 1957. Using these data the parameters of (B.2) were estimated in two d i f f e r e n t ways. F i r s t , (B.2) was estimated using the estimated biomass levels for x ( t ) . This we refer to as the standard method for estimating the aggregative growth function. Secondly, (B.2) was estimated using the "true" biomass levels for x ( t ) . This we refer to as the improved estimation method. The estimation technique in both instances was OLS <20>. The estimation results were as follows: 50 Table B.2 Estimates of Aggregative Growth Functions for Icelandic Cod. Estimation period: 1955-1974 Estimation technique: OLS Coefficient Estimates Implied M S y * a b R**2 1000 tons Standard method: 0.548 -.00018 .081 412 (4.4) (-3.1) Improved method: 0.425 -.00012 .007 369 (4.1) (-3.5) Numbers in brackets are t - s t a t i s t i c s . * MSY defined as -(a 2/4b). 51 4.1.2 Disaggregated Growth Models: The Beverton-Holt Model. Given the d i f f i c u l t i e s with the aggregative population growth models, discussed above, i t i s not surprising that fishery b i o l o g i s t s have, in recent years, increasingly turned to the use of models that are able to take account of the cohort structure of the biomass <21>. The most widely used model of this type in fishery biology is the one proposed by Beverton and Holt in 1957 and is usually associated with their name <22>. The Beverton-Holt model d i f f e r s from the aggregative growth models in at least two important respects. F i r s t l y , i t e x p l i c i t l y recognizes the fact that d i f f e r e n t cohorts of a species may have di f f e r e n t growth rates. In so doing the model avoids the misspecification inherent in the aggregative growth models. Secondly, the Beverton-Holt model i s ana l y t i c . It attempts to ide n t i f y and model separately the various elements of the growth process. The f i n a l expressions for catch and biomass are then obtained by e x p l i c i t l y aggregating these elements. Thus, while the aggregative growth models are bas i c a l l y "black-box" formulations mostly arrived at by means of induction, the Beverton-Holt model is analytic arrived at by means of deduction. The essential structure of the Beverton-Holt model may be described as follows: Consider a stock of f i s h consisting of I cohorts, where I>1. Its aggregate biomass at time t i s defined to be the sum of the biomasses of each cohort. The biomass of each cohort, on the other hand, i s simply the number of individual f i s h belonging to the cohort multiplied by their average weight. In other words: 52 (11) x(t)= Z x(i,t)= Z w ( i , t ) n ( i , t ) . C-l i = I Where x(t) represents the aggregate and x ( i , t ) cohort's i biomass at time t. w(i,t) and n ( i , t ) denote the average weight and and number of individuals in cohort i respectively. The aggregate volume of catch over a given period i s defined as the sum of the catch from each cohort during the same period: i i (12) y(t)= Z y( i , t ) = Z w ( i , t ) c ( i , t ) . Where y(t) denotes the volume of aggregate catch, y ( i , t ) the volume of catch from cohort i and c ( i , t ) the number of individuals caught during the period [ t - 1 , t ] . The number of individuals belonging to each cohort decreases over time according to their mortality rate. The t o t a l mortality rate i s decomposed into two components; natural and fis h i n g mortality. Fishing mortality represents the rate of deaths due to catching. Natural mortality represents a l l other deaths. Formally we may write: (13) n ( i , t ) / n ( i , t ) = - z ( i , t ) = - ( m ( i , t ) + f ( i , t ) ) , i=1,2,...I. where m(i,t) and f ( i , t ) represent, respectively, the instantaneous natural and fis h i n g mortality rates of cohort i at time t. Fishing mortality i s defined by: 53 (14) f ( i , t ) = c ( i , t ) / n ( i , t ) . F i n a l l y , the number of individuals in the youngest exploited (or exploitable) cohort, i=1, say, at t i s c a l l e d recruitment and denoted by r. So formally: (15) r(t)=n(1,t). These eight equations constitute the basic d e f i n i t i o n s or axioms of the Beverton-Holt model. It i s important to notice that, at least from a formal point of view, no b i o l o g i c a l hypotheses are involved so fa r . The model, at this stage, i s merely an a n a l y t i c a l truism. Now, integrating equation (13) over one period, a year, say, y i e l d s : (16) n(i,t)=n(i,t-1)exp(-z(i,t) 1) , i = 1,2,....I. Where z ( i , t ) stands for the average mortality rate during the year <23>. Substituting (16) into (14) and integrating over a year, keeping f i s h i n g mortality at i t s average value f ( i , t ) , say, we find <24>: (17) c ( i , t ) = n ( i , t - 1 ) ( 1 - e x p ( - z ( i , t ) ) ) f ( i , t ) / z ( i , t ) , i = 1,2, I. 54 Given the time path of the b i o l o g i c a l variables w(i,t), m(i,t) and r ( t ) , as well as the fishing mortality, f ( i , t ) , for a l l the cohorts i t is clear that the difference equations (16) and (17) together with the aggregation i d e n t i t i e s (11) and (12) define dynamic paths for aggregate biomass and catch. It i s convenient to write t h i s result in the following form: (18) x(t)=X({f},{m},{w},{r}), «*V <"!»/ *•* (19) y(t)=Y({f},{m},{w},{r}). Where x(t) and y(t) represent, as before, the aggregate,biomass and catch respectively. The dimensionality of the vectors, f, m and w, equals the number of exploited (or exploitable) cohorts and the curly brackets indicate that I values of the respective variables, covering the period [ t - I , t ] , are involved. Apart from the approximations involved in deriving (17), the results expressed by equations (18) and (19) are simply l o g i c a l deductions from a n a l y t i c a l i d e n t i t i e s . Thus, the whole theory, at this point, i s rather vacuous. Neither, due to the great number of independent variables involved, is the model p a r t i c u l a r l y useful to describe the history, l e t alone the future, of real f i s h e r i e s . In applied fishery biology, the Beverton-Holt model i s made operative by introducing assumptions that r e s t r i c t the number of independent variables in the model. At the same time these assumptions change the nature of the model from that of a general dynamic accounting system to that of a behavioural 55 theory. In applications i t i s commonly assumed that recruitment, r ( t ) , i s invariant with respect to time <25>. A t h e o r e t i c a l l y more sat i s f a c t o r y procedure, however, and the one proposed by Beverton and Holt (1957, pp. 44-67), i s to postulate that recruitment depends on the size of the spawning stock. I.e: (20) r(t)=R(s(t-u)). Where s(t-u). represents the size (usually biomass) of the spawning stock at time (t-u) and u, as before, i s the lead time from the time spawning occurs u n t i l the resul t i n g cohort i s recruited into the fishery. Applied fishery biology then usually proceeds by assuming that m and w are constant over time and, furthermore, that the elements of m are id e n t i c a l for a l l exploited cohorts <26>. Given these assumptions regarding r ( t ) , m(t) and w(t) equations (18) and (19) may be rewritten as functions of the time path of fis h i n g m o r t a l i t i e s , i n i t i a l cohort sizes and time invariant b i o l o g i c a l parameters <27> only: (21) x(t)=X({f};n(0),w,m), a l l t, (22) y(t)=Y({f};n(0),w,m), a l l t. Where n(0) i s the vector of cohort sizes at some arb i t r a r y i n i t i a l point of time, t=0. It should now be clear that the Beverton-Holt model i s 56 considerably more complex than the aggregative growth models discussed in the previous section. Not only i s i t mathematically much more involved, i t also requires far more data for i t s application. Its advantage i s that i t provides a more satisfactory description of the underlying b i o l o g i c a l processes, at least from a theoretical point of view. In spite of i t s theoreti c a l superiority over the aggregative growth models, the Beverton-Holt model s t i l l o ffers a very s i m p l i s t i c picture of the dynamics of f i s h stocks. Its functional structure involves, as pointed out above, at least one questionable simplifying assumption about the fi s h i n g process. Its application, moreover, requires, as we have seen, several additional assumptions. F i n a l l y , and perhaps most importantly, the Beverton-Holt model, as described above, largely ignores the ecological interactions of marine species and their dependence on the physical environment. 57 Appendix 4.1. 2-A Estimation of the Beverton-Holt B i o l o g i c a l Parameters. According to the standard Beverton-Holt model summarized in equations (21) and (22) above, aggregate catch and biomass at each point of time depend on current and previous mortality rates as well as weights by age. Of these variables only the f i s h i n g mortality i s con t r o l l a b l e by harvesting. The others, m and w to use the notation of the previous section, are regarded as b i o l o g i c a l parameters within the framework of the standard Beverton-Holt model. Therefore, to apply the Beverton-Holt model to actual f i s h e r i e s , the h i s t o r i c a l values of these parameters must be estimated and their future values predicted. In t h i s appendix, some of the more common techniques employed by applied fishery biology to estimate the b i o l o g i c a l parameters of the Beverton-Holt model w i l l be described <28>. The expected weight of individual f i s h belonging to a p a r t i c u l a r cohort i , w(i), i s generally estimated by the sample mean: (A.1) w(i) = L w(i,n)/N, n s i where A indicates an estimate, w(i,n) i s the weight of specimen n and N i s the sample s i z e . Applying (A.1) thus requires not only the weighing of individual f i s h in the sample but also the often d i f f i c u l t task of determining i t s age. Estimates of w(i) calculated on the basis of samples obtained in d i f f e r e n t years often d i f f e r s i g n i f i c a n t l y <29>. In formulating fishery policy, i t i s nevertheless generally assumed that w(i) is invariant with respect to time <30>. The natural mortality rates of individual cohorts are much less observable than individual w(i)'s and thus more d i f f i c u l t to estimate. Frequently i t i s assumed that m(i) i s invariant with respect to both cohorts and time provided the cohorts have been recruited to the fishery and are below a certain maximum age <31>. At the maximum age natural mortality i s assumed to jump to i n f i n i t y . Natural mortality before the recruitment age, on the other hand, is subsumed in the recruitment function <32>. Given these s i m p l i f i c a t i o n s , there are several techniques for estimating natural mortality. Most commonly they involve an assessment of t o t a l mortality over a period of time. This i s based on estimates of the r a t i o of the number of indviduals at the end of the period to that at the beginning of the period and application of the identity n(t)/n(0)=exp(-z(t)t) , where n(t) and n(0) represent the number of individuals at the end and beginning of the period respectively and z(t) represents the average t o t a l mortality during the period. Among the various 58 methods to estimate n(t)/n(0) are; direct counting, often with the help of electronic instruments, analysis of catch data, inference from tagging data etc. Once t o t a l mortality has been estimated, natural mortality i s calculated by subtracting from i t independent estimates of fi s h i n g mortality. Thus we see that since estimates of natural mortality generally depend upon preliminary estimates, which are of course subject to errors, they are usually not p a r t i c u l a r l y r e l i a b l e . In fact, of a l l the parameter estimates involved in the t y p i c a l application of the Beverton-Holt model, those of natural mortality are probably the least r e l i a b l e . Appendix 4.1.2-B V i r t u a l Population Analysis; VPA. Given estimates of natural mortalities and catch the size of the i n i t i a l cohorts, n(0), or for that matter the h i s t o r i c a l path of cohort sizes and f i s h i n g mortalities may be estimated with the help of the Beverton-Holt model i t s e l f . In section 4.1.2 we found: (16) n ( i , t ) = n ( i , t - 1 ) e x p ( - z ( i , t ) ) , i=u,u+1,..,I, (17) c ( i , t ) = n ( i , t - 1 ) ( 1 - e x p ( - z ( i , t ) ) ) f ( i , t ) / z ( i , t ) , i=u,u+1,..I, (13) z ( i , t ) = f ( i , t ) + m ( i , t ) , i = u,u+1,..,I, where we now only consider cohorts recruited to the fishery, i.e. of age u or older, and have, for convenience of notation, dropped the - from the tops of average variables. These 3 equations describe the stock and catch dynamics of each exploited cohort over any number of periods. Given catch data, c ( i , t ) , and estimates of natural mortality, m(i,t), equations (16), (17) and (13) provide two difference equations for cohort i in two unknowns, namely f ( i , t ) and n ( i , t ) . Hence we only need to obtain " s t a r t i n g " values for these variables to be able to calculate the remaining ones. In fact, due to the structure of the equations only " s t a r t i n g " values of either n ( i , t ) or f ( i , t ) are required. Rewriting (17) as (B.1) c ( i , t ) = n ( i , t ) ( e x p ( - z ( i , j ) ) - 1 ) f ( i , t ) / z ( i , t ) , the recursiveness of the system becomes obvious. It turns out that there are important advantages in taking the most recent f ( i , t ) ' s or n ( i , t ) ' s as " s t a r t i n g " values and l e t the difference equations trace out the path of n ( i , t ) and f ( i , t ) backwards in time. Moreover since for old cohorts f ( i , t ) i s generally estimated with more precision than n ( i , t ) the most recent f ( i , t ) ' s are usually taken as starting values. The calculations proceed roughly as follows: Let the 59 estimate of the most recent fi s h i n g mortality of cohort i be f( i , T ) i f the cohort i s s t i l l being fished from and f ( I , t ) i f the cohort has passed through the fishery. Given such a estimate or guess, f ( i , T ) , say, and knowledge of c(i,T) and m(i,T) calculate n(i,T) from (B.1). Having that estimate calculate n ( i , T ~ l ) from (16). With that result and knowlede of c ( i , T - l ) and m(i,T-1) calculate f(i,T-1) from (17). Now return to (16) and calculate n(i,T-2) etc. This method of obtaining estimates of h i s t o r i c a l f i s h i n g m o r t a l i t i e s and cohort sizes i s c a l l e d v i r t u a l population analysis or, in short, VPA <33>. The c r u c i a l property of the VPA technique <34> i s that the effects of errors in the "starting values, e.g. the f ( i , T ) ' s and f ( I , t ) ' s , on the calculated f ( i , t ) ' s and n ( i , t ) ' s become smaller with sucessive applications of equations (16) and (17), at least i f f i s h i n g takes place. In most cases, moreover, th i s convergence seems to be f a i r l y rapid. Thus, for the f i s h stocks considered in this study, simulations suggest that even r e l a t i v e l y large errors in the i n i t i a l guesses of f ( i , T ) and f ( I , t ) become i n s i g n i f i c a n t 4-5 years back. Table B.1 provides simulation results in support of thi s claim. The table shows the results of three d i f f e r e n t VPA estimates of the stock size of the 1968 cohort of Icelandic cod. These estimates are based on the same natural mortality estimates and catch data. They d i f f e r only with respect to the "s t a r t i n g " value of fis h i n g mortality for this cohort in 1981, i . e . f(68,8 1). The f i r s t set of estimates was calculated using the best available estimate of f(68,81). For the purpose of the convergence check, we may take this to be the true f i s h i n g mortality. The second set of estimates uses a 50% lower and the th i r d 50% higher f(68,8l). 60 Table B.1 VPA Convergence Test: Icelandic Cod; Cohort 1968. (1) (2) (3) 0.5xf(68,8l) 1.5xf(68,8l) Estimated n ( i , t ) * * True* Year Age n(i , t) n ( i , t ) % error n ( i , t ) % error 1981 1 3 . 1 00 .080 -20.00 . 161 61 .00 1980 1 2 .230 .206 -10.43 .305 32.61 1 979 1 1 .464 .435 -6.25 .557 20.04 1 978 10 .948 .912 -3.80 1 .062 1 2.03 1 977 9 3.083 3.039 -1.43 3.224 4.54 1976 8 9.571 9.517 -0.56 9.746 1 .83 1 975 7 22.201 22. 133 -0.31 22.416 0.97 1 974 6 42.901 42.817 -0.20 43.159 0.60 1 973 5 82.392 82.288 -0.13 82.709 0.38 1 972 4 133.110 132.983 -0.10 133.409 0.29 1971 3 176.960 176.804 -0.09 177.436 0.27 * I.e. the reference path. ** n ( i , t ) i s measured in mil l i o n s of individuals. As table B.1 shows, the effects of an error in star t i n g values die out r e l a t i v e l y quickly in these p a r t i c u l a r VPA calculations on the 1968 cohort of Icelandic cod. A 50% error in the i n i t i a l f i s h i n g mortality, f(68,81), result in less than a 10% error in calculated stock sizes after 2-4 years and less than 1% after 5-7 years. Thus, we may conclude that the VP-analysis is a powerful tool to estimate past fi s h i n g mortality rates and stock sizes for a multicohort f i s h stock, provided estimates of c ( i , t ) and m(i,t) are ava i l a b l e . However, i t should not be forgotten that (i) the VPA results are no better than the assumptions required to derive equations (16) and (17), ( i i ) the speed of error convergence to zero increases with the fis h i n g mortality and ( i i i ) the quality of the estimates depends c r i t i c a l l y on the accuracy of the estimated c ( i , t ) and m(i,t) <35>. Appendix 4.1.2-C Recruitment Functions. Let us assume that egg production i s proportional to the volume of the spawning stock. I.e: (C.1) n(0)=qS, 61 where n(0) denotes the number of f e r t i l i z e d eggs, S the volume of the spawning stock and q is a positive real constant. During the period from spawning to recruitment the mortality rate of the species i s by d e f i n i t i o n : (C.2) n(t)/n(t)=-m(t), te[0,u], where n(t) represents the number of individuals at time t, m(t) is the mortality rate and u is the recruitment age. It follows that recruitment i s given by: (C.3) r = n(u)=n(0) exp(- f"m(t)dt)=qS exp(- f wm(t)dt). o o (C.3) makes i t clear that the s p e c i f i c a t i o n of the recruitment function is equivalent to s p e c i f i c a t i o n of the natural mortality function during the pre-recruitment period. We now consider a few hypotheses of t h i s nature: H1: Mortality independent of spawning stock. From (C.3) i t follows immediately that on t h i s assumption the recruitment function i s the following simple linear function: (C.4) r=aS, where a=q exp(- J" am(t)dt). o H2: Density dependent mortality. (Beverton-Holt). This hypothesis, due to Beverton and Holt (1957), s p e c i f i e s natural mortality by the function: (C.5) m(t)=m1+m2n(t), where ml and m2 are positive constants, ml, the fixed mortality term, is supposed to r e f l e c t mortality that i s independent of both the spawning stock and the instantaneous cohort s i z e . This might e.g. r e f l e c t mortality due to predation by other species. The second term, m2n(t), represents natural mortality that i s increasing with the instantaneous cohort size. This might e.g. be due to food competition. Rewriting (C.5) as n(t)/n(t)=m2(-ml/m2-n(t)), and subsequently as (l/n(t)-l/(ml/m2+n(t)))n(t)=-m1, 62 yields upon integration and rearranging the Beverton-Holt recruitment function: (C.6) r=R(S)=aS/(b+cS). Where a=m1q, b=m1 exp(m1u), c=m2q(exp(m1u)-1). This mortality function i s monotonically increasing with R(0)=0 with an asymptotic maximum equalling a/c when S approaches i n f i n i t y . H3: Density dependent mortality. (Simple Beverton-Holt). Assume now that there i s no fixed mortality. I.e: (C.7) m(t)=m2n(t). The resulting recruitment function i s : (C.8) r=R(S)=aS/(1+cS). Where a=q and c=qm2u. (C.8) has the same general shape as (C.6) with an asymptotic maximum of a/c=l/m2u. H4. Stock dependence. (Ricker-Foerster). Ricker and Foerster (1948) hypothesized that pre-recruitment mortality depends on the parent stock. I.e: (C.9) m(t)=(m1+m2S), where ml and m2 are, as before, positive constants. The fixed mortality term, ml, may be interpreted as above. The variable one, m2S, may represent the predation of the parent stock on i t s offspring. A l t e r n a t i v e l y , r e c a l l i n g equation (C.1), m2S may be taken to represent the effects of food supply r e l a t i v e to the cohort size at a very early stage on the mortality rate of the cohort. Solving (C.2) with (C.9) imposed y i e l d s : (C.10) r=R(S)=qS-exp(-(m1+m2S)u). This function, often c a l l e d the Ricker recruitment function, i s dome shaped with a maximum at S=l/m2, R(0)=0 and r asymptotically converging to zero as S approaches i n f i n i t y . 63 Appendix 4.1.2-D E q u i l i b r i a of the Beverton-Holt Model. According to section 4.1.2, the Beverton-Holt model describes population dynamics in terms of nonlinear difference equations of, possibly, a very high order. The resulting dynamic paths may consequently be very complex. In pa r t i c u l a r there is no guarantee that (i) a nontrivial equilibrium exists <36> and ( i i ) i f one exis t s , i t i s globally stable <37> and ( i i i ) w i l l be reached in reasonable, or even f i n i t e , time. However, for those species of f i s h whose population dynamics seem reasonably stable <38> i t may be useful to summarize important aspects of their growth c h a r a c t e r i s t i c s with the help of equilibrium rel a t i o n s h i p s . The purpose of th i s appendix is to define two such equilibrium relationships; the sustainable y i e l d function and the biomass per rec r u i t function <39>. The sustainable y i e l d function, for a multicohort fishery, describes the relationship between a certain family of fishing mortality vectors and the resulting equilibrium, or sustainable, aggregate catch. More precisely, on the basis of equation (19) in section 4.2.2 we define t h i s function as: (D.1) y=Y(f;m,w,r), f=af, aeRi. /•w ~* t** Where - on top of variables denotes that their equilibrium values, perhaps also dependent on f, are involved. The mu l t i p l i c a t i v e scalar, a, may be interpreted as a measure of fishi n g e f f o r t . Thus b a s i c a l l y , (D.1) defines a mapping from a measure of fishi n g e f f o r t into the resulting equilibrium catch. This i s what is meant by the t r a d i t i o n a l sustainable y i e l d function. Consider now, as an example, a pa r t i c u l a r case with fixed m and w vectors as well as recruitment, r. In t h i s simple case, the sustainable y i e l d curve is defined as: (D.2) y= I y(i)= Z w ( i ) n ( i , t - 1 ) ( 1 - e x p ( - z ( i ) ) ) a f ( i ) / z ( i ) . Where n ( i , t-1)=?• exp(- Z z ( j ) ) , j - u. and z(i)=af(i)+m(i), a l l i . The cohort biomass function is defined as the aggregate biomass of a cohort i n i t i a l l y of equilibrium recruitment size that i s not subject to fis h i n g at d i f f e r e n t ages of i t s l i f e t i m e . More precisely, for cohort i : _ _ t -1 (D.3) x(i)=w(i)n(i)=w(i)r exp(- Z m(j)), 64 where x(i) denotes the cohorts biomass at age i . Dividing through (D.3) by r we obtain the more common biomass per recruit function: (D.4) x(i)=x(i)/?=w(i)•exp(- Z m(j)), i=u,u+1,...I. where x(i) represents biomass per r e c r u i t . 65 4.1.3 Ecological Considerations. It i s axiomatic that a l l resources, replensihable as well as nonreplenishable, are imbedded in an ecosystem. By an ecosystem we mean a c o l l e c t i o n of organic and nonorganic e n t i t i e s which are causally related to each other <40>. Thus, according to t h i s d e f i n i t i o n , the state of the ecosystem i s , in general, one of the determinants of the growth of any given natural resource belonging to i t . It follows that the economics of natural resource u t i l i z a t i o n depend not only on the par t i c u l a r resource of direct interest but also on the ecosystem in which they are imbedded. Given a pa r t i c u l a r resource, the scope of the relevant ecosystem i s c l e a r l y important from the point of view of research strategy. Above the ecosystem was defined in terms of the interdependence of i t s elements <41>. Elements that neither adjust to changes in the remainder of the ecosystem nor exert influence on i t can, without loss of information, be dropped from the analysis of the ecosystem. This si t u a t i o n corresponds to a complete decomposabi1ity <42> of the so-called community matrix to be defined below. A l t e r n a t i v e l y , an element may be independent of the remainder of the ecosystem while exerting i t s influence upon i t . In thi s case, this p a r t i c u l a r element is p a r t i a l l y decomposable from, or exogenous to, the ecosystem proper. Perhaps i t may be said to belong to the environment of the ecosystem. It i s , on the other hand, not at a l l obvious that, in any given empirical s i t u a t i o n , the set of elements s a t i s f y i n g complete or p a r t i a l decomposability, in terms of the community matrix, i s non-empty. The d e f i n i t i o n of an ecosystem, 66 given above, may well be too broad for that. In practice, however, comparatively i n s i g n i f i c a n t relationships are ignored in the interest of keeping the scope of the research within manageable bounds. In recent years, there has been a great surge in ecological modelling <43>. Due to the complexity of the subject, however, progress has been slow, the models proposed diverse <44> and reliance on computer simulation models heavy <45>. Nevertheless, t h i s research has demonstrated, quite convincingly, that ignoring ecological factors in resource management may lead to substantial errors <46>. In t h i s subsection, we w i l l b r i e f l y review some basic ecological ideas concerning f i s h resources. In the next subsection some simple ecological modifications of the Beverton-Holt model w i l l be suggested. Following Lackey (1975) two aspects of a f i s h stock ecosystem may be distinguished: (i) The habitat, which includes hydrographic conditions, currents, s a l i n i t y , temperature, basic nutrients etc., and ( i i ) the aquatic biota, which comprises the organisms of the ecosystem. Our focus in thi s study w i l l be on the aquatic biota. The reason has to do with our assumed structure of the relevant ecosystem. In deep-sea f i s h e r i e s , l i k e the Icelandic demersal ones, i t stands to reason that the habitat i s only to a r e l a t i v e l y small extent influenced by the aquatic biota <47>. The habitat i s primarily determined by geographical and meteorological factors that are largely exogenous of the remainder of the ecosystem. It i s , to use our previous 67 terminology, p a r t i a l l y decomposable from the ecosystem. Hence, in modelling the growth processes of the Icelandic demersal f i s h stocks, i t seems plausible to regard the habitat as exogenous. Following ecological practice <48>, the dynamics of the aquatic biota may be described by the following set of equations: Where x(i) i s a vector of relevant population c h a r a c t e r i s t i c s of species i belonging to the aquatic biota. G ( i , ) is the corresponding vector of growth functions. K stands for the number of species in the system and x i s the (1xK) vector of x ( i ) ' s . . F i n a l l y , u i s a vector of habitat c h a r a c t e r i s t i c s independent of x. Let us now, for purposes of exposition, select a single population c h a r a c t e r i s t i c for each species, namely i t s aggregate biomass. Denote th i s c h a r a c t e r i s t i c and the corresponding growth function for species i by x*(i) and G*(i,x*,u) respectively. We have thus defined the following subset of (23): (23) x(i)=G(i,x,u), i=1,2 T K. (24) x*(i)=G*(i,x*,u), i=1,2 r • « • i K. The (KxK) Jacobian matrix of (24) with respect to x* (25) J=[G*(i,j)]=[3G*(i ; • i • )/3x*(j)], i s c a l l e d the community matrix. It entries describe the direct 68 effects each species in the ecosystem has on others. The signs of the p a r t i a l derivatives in J are often used to characterize these relationships as follows <49>: (1) G*(i,j)>0, G*(j,i)>0 : Symbiotic relationship. (2) G*(i,j)>0, G*(j,i)=0 : Commensalism. (3) G*(i,j)>0, G*(j,i)<0 •: Predator-prey relationship. (4) G*(i,j)=0, G*(j,i)=0 : Direct independence. (5) •G*(i,j)=0, G*(j,i)<0 : Amenalism. (6) G*(i,j)<0, G*(j,i)<0 : Predator-predator relationship <50>. Usually some a p r i o r i r e s t r i c t i o n s on the structure of the community matrix, J, may be in order. F i r s t l y , since the ecosystem is available for observation, i t may well be l o c a l l y stable. In that case the eigenroots of J have negative real parts. Secondly, J should not be completely decomposable <51>. Otherwise more than one ecosystem i s implied. The aggregative and disaggregated growth models discussed in the previous section are but very special cases of these general ecological models. The aggregative growth functions are only concerned with a single c h a r a c t e r i s t i c of one species, namely i t s biomass. They thus correspond to equations (24) above with a diagonal community matrix imposed. The Beverton-Holt model corresponds, on the other hand, to the more general ecological model expressed by equations (23). Within this framework, however, the standard version of the Beverton-Holt model, as described in subsection 4.1.2, also i m p l i c i t l y assumes 69 a diagonal or, at best, a block diagonal community matrix. 4.1.4 An Ecological Extention of the Beverton-Holt Model. The Beverton-Holt model was o r i g i n a l l y only designed for a single stock of f i s h . However, as suggested by Schaaf (1975) and, in fact, Beverton and Holt themselves (1957, pp. 165-72) the model can be extended to incorporate some f a i r l y general ecological interactions. As described in section 4.1.2 above, the Beverton-Holt model decomposes biomass growth into four seperate elements; (i) the fishing mortality rate, ( i i ) the natural mortality rate, ( i i i ) the weight increase by age and (iv) the recruitment process. Of these, only the fis h i n g mortality i s controllable by means of the harvesting a c t i v i t y . The other elements of the growth process are, presumably, determined by the inhererent genetics of the species and external ecological forces. Thus, within the basic framework of the Beverton-Holt model, ecological e f f e c t s may, at least to some extent, be accounted for by appropriately specifying the determinants of these elements of biomass growth. A simple approach i s to make the instantaneous change <52> of the three natural elements of biomass growth dependent on a l l the c h a r a c t e r i s t i c s of the ecosystem. Formally: 70 (26) z= m w =G(z,y,u)=G(x,u). /V ««v Where m and w are the column vectors of natural mortality and individual weights for a l l cohorts of a l l the species belonging to the ecosystem. Si m i l a r l y , r i s the column vector of the recruitment of a l l the species in the ecosystem, y i s a vector of b i o l o g i c a l c h a r a c t e r i s t i c s of the ecosystem, other than m, w and r, and u i s the vector of habitat c h a r a c t e r i s t i c s <53>. Equations (26) in combination with the standard Beverton-Holt model c l e a r l y comprise f a i r l y general ecological interactions. In particular t h i s formulation includes, as special cases, a l l the major types of ecological relationships defined in terms of the community matrix in section 4.1.3 above. The properties of system (26) are determined by the functions G( ). Depending on the nature of the ecosystem, a p r i o r i knowledge may be u t i l i z e d to r e s t r i c t the set of allowable functional forms for t h i s function. T y p i c a l l y , however, very l i t t l e empirical knowledge about ocean ecosystems is available <54>. Empirical applications of system (26) must therefore frequently be r e s t r i c t e d to simulations using more or less informed guesses as to the s p e c i f i c a t i o n of the functions, G( ). The habitat variables, u, are exogenous to the ecology. Moreover, they are generally non-controllable and their future values subject to uncertainty. For these reasons they might, in 71 applications, be represented by stochastic variables having the appropriate d i s t r i b u t i o n function. 72 4.2 Icelandic Demersal Fish Stocks. In t h i s part of chapter 4 b i o l o g i c a l c h a r a c t e r i s t i c s of three commercially exploited demersal f i s h stocks off Iceland, namely cod, haddock and saithe, w i l l be considered <55>. As t h i s study is primarily concerned with e f f i c i e n t harvesting paths, the discussion w i l l focus on the growth processes of these f i s h stocks and their response to harvesting. The Icelandic continental shelf, the sea above i t and the various organisms inhabitating t h i s space, may be taken to constitute an ecosystem <56>. Being members of a common ecosystem the stock dynamics of cod, haddock and saithe may be expected to be interdependent. In fact, the available q u a l i t a t i v e b i o l o g i c a l information suggests the existence of a relationship of this nature. Quantitative data on the intensity of these relations are, however, very limited. For this reason, much of the descriptive material of this section w i l l deal with the growth processes of cod, haddock and saithe without reference to their ecological context. In section 4.2.5.1, however, an attempt w i l l be made to specify some s i m p l i s t i c ecological relationships between these three species and extract, from the available data, estimates of the magnitudes involved. By r e s t r i c t i n g the analysis to the demersal subsystem of cod, haddock and saithe, important aspects of the relations of these species to the remainder of the ecosystem are also doubtlessly ignored. Extending the analysis in this respect would, on the other hand, be rather spurious from an empirical point of view since so l i t t l e i s known about the relevant 73 structure of the ecosystem. Nevertheless, treating the remainder of the ecology as exogenous to the demersal subsystem, we w i l l take some formal account of i t s influence on the demersal subsystem by including seemingly appropriately sp e c i f i e d stochastic elements in the growth functions of the demersal species as suggested in section 4.1.4 above. The discussion w i l l proceed broadly as follows: In the f i r s t subsection the habitat of the Icelandic f i s h stocks w i l l be described. In the next four subsections the individual growth processes of the three demersal species, without reference to their ecological interdependence, are examined. This discussion w i l l be contained in the framework of the simple Beverton-Holt model focussing on the Beverton-Holt growth elements; fi s h i n g and natural m o r t a l i t i e s , weight by age and recruitment <57>. In section 4.2.5, however, the analysis is extended to include simple ecological relationships between the species and the effects of thi s extention on the respective growth processes i s examined. F i n a l l y , we w i l l b r i e f l y consider the way in which exogenous ecological effects on the demersal subsystem may be represented in the model. 4.2.1 The Habitat. As a result of generally favourable environmental conditions, the sea around Iceland i s unusually r i c h in marine l i f a . The environment consists of the hydrographic, geographic and meterological conditions that ecologists c a l l habitat <58>. Of the various aspects of the habitat that create these 74 hospitable conditions, the most important one i s probably the system of currents off the coast. A branch of the Gulf stream, bringing with i t r e l a t i v e l y warm sea from the Gulf of Mexico, almost surrounds the island. The Gulf stream is met, off the North-west and North coast, and to a lesser extent, off the East and South-east coast, by the r e l a t i v e l y cold East-Greenland and East-Iceland currents flowing from the A r c t i c . This i s i l l u s t r a t e d in figure 4 . 3 . Figure 4 .3 The System of Currents Around Iceland. Where these currents meet, an upswelling and mixing of seawater takes place, creating extremely favourable conditions for pythoplankton and subsequently zooplankton. Another important factor in the v e r t i c a l mixing of seawater 75 is the seasonal v a r i a b i l i t y in surface temperatures. Lying close to the Arct i c c i r c l e the sea around Iceland i s r e l a t i v e l y cold. During winter and early spring the sea close to the surface becomes increasingly colder and heavier. Hence, during t h i s period, surface sea is continuously sinking and being replaced by colder and lig h t e r sea, carrying with i t nutrients from the bottom <59>. Combined with increased sunlight t h i s process results in a blooming period for pythoplankton in spring and early summer. A t h i r d important factor in bringing nutrients to the surface layers i s the combination of an extensive continental shelf and frequent high winds. Large areas of the continental shelf are less than 100 meters deep. In these, r e l a t i v e l y shallow, waters strong winds are often s u f f i c i e n t to plow nutrients from the bottom to the surface. The continental shelf, as such, also provides a favourable environment for marine l i f e . Being of r e l a t i v e l y recent volcanic o r i g i n , i t offers a varied environment; rugged l a v a f i e l d s providing excellent protection for small f i s h , extensive sand and clay areas, that are favoured by several species and deep canyons that cut far into the continental shelf, thus allowing the species to select their optimal depth and temperature while s t i l l being within easy reach of their food supply. Those, and other, c h a r a c t e r i s t i c s of the marine habitat combine to make Icelandic waters among the richest fishing grounds in the world <60>. 76 4.2.2 The Cod (Gadus Morhua). The Icelandic cod i s a medium sized demersal species <61>. Mature individuals are generally 70-100 cm. and weigh 5-7 kg. In the absence of the current heavy exploitation, however, 140 cm cod weighing 20 kg. or more would not be uncommon <62>. The l i f e cycle of the cod is broadly as follows: Spawning takes place in late spring every year primarily off the South-west coast of Iceland. The eggs d r i f t with the currents close to the surface for a period of 2-3 weeks before they hatch. The larvae or fry remain pelagic for the next 4-5 months. The pelagic stage permanently ends after 4-5 months, i . e . in the autumn and the fry revert to the bottom. The very young and juvenile cod, preferring the security provided by the bottom growing sea-weed, l i v e in r e l a t i v e l y shallow waters, mostly off the North coast. As they grow larger they gradually seek deeper waters and at the average age of about three years they enter, or are recruited to, the fishing grounds. At the average age of about seven years the cod becomes sexually mature and i t s annual migrations to the spawning grounds begin. According to standard o f f i c i a l estimates of the cod's natural mortality rate <63>, even without f i s h i n g , less than 25% of the cod that reach the nursery grounds, w i l l l i v e to become sexually mature and less than 5% w i l l reach 15 years of age. The cod i s b a s i c a l l y a carnivore. The fry feed, as already mentioned, on zooplankton. The young and juvenile cod (5 months to 3 years) feed mostly on bethnic animals such as worms and small crabs. The adolescent and mature cod prefer larger prey, such as bigger crabs, s h e l l f i s h , sandeel and capelin <64>. In 77 addition to human predation, the cod i s preyed upon by whales, seals and sharks. For mature cod, however, th i s predation i s not believed to be s i g n i f i c a n t . During the f i r s t few months of the cods's l i f e , before i t reverts to the bottom, the larvae and fry are eaten in great quantities by birds and pelagic f i s h . The cods habitat i s a l l around Iceland. The greatest number of mature cod, however, are found off the North and North-west coasts, at a depth of about 100-300 m., where feeding conditions are favourable. During the spawning season, however, the mature cod migrate to the South-west coast. 4.2.2.1 B i o l o g i c a l Parameters. According to the Beverton-Holt model the volume of f i s h stocks and catch are defined by past and present values of fis h i n g mortality, natural mortality, weight by age and the recruitment function. Applying the techniques outlined in appendices 4.2.2-A and B above, fishery b i o l o g i s t s have provided estimates of the f i r s t three of these elements. These are l i s t e d in table 4.1. The estimation of an appropriate recruitment function is considered in the next subsection. 78 Table 4.1 Estimates of B i o l o g i c a l Parameters for Cod. Fishing Natural Weight Age mortality* mortality** kg. 1 0.000 NA 0.22 2 0.000 NA 0.70 3 0.038 0.20 1.17 4 0. 1 97 0.20 1 .70 5 0.322 0.20 2.59 6 0.446 0.20 3.73 7 0.650 0.20 5 . 1 8 8 0.771 0.20 6.33 9 0.770 0.20 7.34 10 0.817 0.20 8.56 1 1 0.803 0.20 10.28 12 0.910 0.20 1 1 .99 1 3 1 .450 0.20 14.19 14 1 .380 0.20 17.94 15 NA i n f . NA NA: Not ava i l a b l e . i n f . = posit i v e i n f i n i t y . Sources: * VPA results based on catch data per 1982. Average for 1976-8 ** See ICES, 1976 *** Hafrannsoknarstofnun, unpublished data. Average for 1970-80. 4.2.2.2 The Recruitment Function. As indicated in section 4.1.2 above, annual recruitment i s a fundamental determinant of the dynamics of f i s h populations. Consequently this component of the cod's growth function must now be examined somewhat more c l o s e l y . The cod becomes sexually mature at an average age of 7 years <65>. Early each year mature cod migrate to the spawning gounds, the most important of which are in the r e l a t i v e l y warm sea off the South-west coast of Iceland. These, as well as the migrating routes are i l l u s t r a t e d in figure 4.4. 79 The spawning period l a s t s from late March to early May. During t h i s period the stock is unusually concentrated and fis h i n g is heavy. The average cod lays about 2-3 m i l l i o n eggs which, having been f e r t i l i z e d , f l o a t to the surface where they d r i f t northwards with the currents. The eggs hatch in about 16-20 days and soon thereafter the fry start searching for their own food which i n i t i a l l y is almost exclusively zooplankton. The survival of the f r y , at t h i s stage, thus depends c r i t i c a l l y on the a v a i l a b i l i t y of zooplankton in their immediate surroundings. This, in turn, depends on the supply of pythoplankton and hence on the general state of the ecological habitat as outlined in section 4.2.1 above. During t h i s period, moreover, the cod i s pelagic and becomes, in great quantities, the prey of seabirds and pelagic species of f i s h , such as herring. 80 Four to five months after spawning, the f r y , now about 4-6 cm., begin to seek the bottom. At this stage i t i s c r u c i a l that the currents have brought them to r e l a t i v e l y shallow waters. If not, they w i l l now perish. Those individuals that happen to find themselves above favourable nursery grounds, which are mostly to the North off Iceland, w i l l seek shelter in the seaweed at a depth of about 40-80 m. Three years l a t e r , measuring about 40 cm. and weighing a l i t t l e over 1 kg., the survivors of the cohort are recruited to the fi s h i n g grounds. It should be clear from this description that ecological factors are probably c r u c i a l in determining the cod's recruitment. The state of the habitat, e s p e c i a l l y currents, winds and sea temperture that determine both the d r i f t of the eggs and larvae and the location and timing of the blooming period for pythoplankton, probably plays the central role in this respect. Also important i s the state of the remainder of the aquatic biota, especially the population of species that prey on the cod's f r y . Compared to these ecological factors the effects of the size of the spawning stock, provided i t is above a certain minimum, may be i n s i g n i f i c a n t . After a l l , in terms of egg production, less than 200 spawning individuals are needed to produce an average cohort. On the other hand, the spawning stock has never, during the l a s t 30 years, counted less than 25 m i l l i o n individuals. Hence there seems, in the case of the Icelandic cod, l i t t l e reason to expect a strong r e l a t i o n between the size of the spawning stock and the resulting recruitment. We now turn to the empirical estimation of recruitment 81 relationships for cod. Unfortunately almost no h i s t o r i c a l data on the environmental determinants of recruitment are available. Hence, in t h i s study, we have no choice but to r e s t r i c t estimation of recruitment functions to the possible direct relationship between the size of the spawning stock and the subsequent recruitment. Our approach w i l l be to apply the . available data to the recruitment hypotheses outlined in appendix 4.1.2-C. Denoting recruitment at time t, by r ( t ) , the corresponding spawning stock by S(t-3), a stochastic disturbance term by e(t) and parameters by a, b and c <66>, these hypotheses may be expressed as follows: HI . Linear Recruitment Function. r(t)=aS(t-3)e(t). H2. Beverton-Holt Recruitment Function. r( t ) = (aS(t-3)/(b+cS(t-3)))e(t) . H3. Simple Beverton-Holt Recruitment Function. r(t) = (aS(t-3)/(1+cS(t-3))e(t) . 82 H4. Ricker Recruitment Function. r(t)=aS(t-3) exp(-b-cS(t-3))e(t) . Now, given that ecological forces are important determinants of early natural mortality of cod, one may reason as follows: F i r s t , i f the spawning stock i s above a certain minimum i t s effect on the subsequent recruitment is i n s i g n i f i c a n t compared with the o v e r a l l ecological -factors. Second, given the survival of the cod stock over m i l l i o n s of years and i t s observed r e l a t i v e s t a b i l i t y during the l a s t decades the ecological forces tend to generate a certain mean recruitment. In the face of lack of data on the ecological factors, these ideas may be translated into the following simple recruitment hypothesis: H5. Mean Random Recruitment. r(t)=a e ( t ) , where a i s the fixed mean recruitment and e(t) is a stochastic error term as before. F i n a l l y , to complete our empirical version of these 5 recruitment hypotheses, the d i s t r i b u t i o n of the stochastic term, e ( t ) , must be s p e c i f i e d . In short we assume that the e(t)'s are 83 log-normally distributed <68>. More precisely we assume: l n ( e ( t ) ) = u ( t ) ~ NIID(0,a), a l l t <69>. Applied fishery biology provides data on r ( t ) and S(t-3) for the period 1955-77 <70>. With the help of these data we w i l l now attempt to reject some of the above hypotheses and estimate the relevant c o e f f i c i e n t s . • We f i r s t notice that in these recruitment hypotheses no more than two parameters are i d e n t i f i a b l e . In p a r t i c u l a r the simple Beverton-Holt recruitment function, H3, is observationally equivalent to the Beverton-Holt recruitment function, H2. Thus from an empirical point of view, we have only four recruitment hypotheses, namely H1, H3, H4 and H5. We also notice that H1, H4 and H5 are nested in the more general recruitment function: (27) lnr(t)=a1+a2 lnS(t-3)+a3S(t-3)+u(t), where a1, a2 and a3 are c o e f f i c i e n t s . H4 is obtained from (27) by imposing the r e s t r i c t i o n a2=1. H1 adds the r e s t r i c t i o n a3=0. H5 i s obtained by setting a3=0 and r e s t r i c t i n g a2=0. Thus the appropriate s t a t i s t i c a l tests of these hypotheses are l i k e l i h o o d r a t i o tests or th e i r equivalents <71>. To compare the two nonnested recruitment hypotheses; H3 and H4, is somewhat more complicated. For this we w i l l rely on three dif f e r e n t c r i t e r i a : (i)'The maximal values of the respective 84 l i k e l i h o o d functions. ( i i ) The s t a t i s t i c a l properties of the residuals as measured by appropriate tests on their normality and lack of autocorrelation <72>. ( i i i ) The b i o l o g i c a l p l a u s i b i l i t y of the estimated functions. For b i o l o g i c a l reasons, i t is clear that S(t-3) is s t a t i s t i c a l l y independent of r ( t ) . Hence i f we assume that S(t) is measured without errors, the maximum li k e l i h o o d estimator of the unknowns in H1, H4 and H5, assuming these hypotheses to be true, i s the OLS estimator applied to their logrithmic form. H3, on the other hand, is nonlinear in parameters. To obtain comparable estimates of this function we employ a two-step procedure. The f i r s t step consists of a nonlinear estimation of both parameters. The second step consists of applying OLS to the equation with the estimate of c r e s t r i c t e d to that in the f i r s t step thus rendering the equation linear in the unknown parameter. The estimation results are given in table 4.2: 85 Table 4.2 Estimation of Recruitment Functions for Icelandic Cod. Estimation technique: Maximum Likelihood Data period: 1955-1977. Likelihood values S t a t i s t i c a l propert of residuals <73> DW Chi * les B i o l o g i c a l p l a u s i b i l i t y Implied max imum recr.m. Eq (27): -5.79 H1: -24.3 H3: -6.07 H4: -7.64 H5: -6.08 0.61 2. 18 1 .93 2.18 2.21 *(3)=4.1 Z\ 3 ) = 1 . 3 2^ 2) = 3.8 & 3 ) = 1 . 3 A 1 ) = 1 . 2 Poor large S Poor small S ok ok ok i n f . 228. 273. 227. 240. * On HO: Residuals are log-nomally d i s t r i b u t e d . ** The sample mean i s 227.2 m i l l i o n individuals. The r e s t r i c t i o n s in H1 r e l a t i v e to eq. (27) are c l e a r l y rejected. The relevant test s t a t i s t i c i s F(2,20)=40.0. This hypothesis also has poor s t a t i s t i c a l properties and is b i o l o g i c a l l y implausible at high levels of spawning stock. The r e s t r i c t i o n in H4 r e l a t i v e to eq. (27) i s , on the other hand, not rejected on the 5% l e v e l . The relevant s t a t i s t i c i s F(1,20)=3.5. On s t a t i s t i c a l grounds there i s l i t t l e to choose between H3, H4 and H5 although H3 and H5 have s l i g h l t y better ov e r a l l properties in terms of the data. H5, on the other hand, is b i o l o g i c a l l y untenable at very low levels of spawning stock. A scatter diagram of the h i s t o r i c a l spawning stock and recruitment pairs during the data period as well as the estimated Beverton-Holt, Ricker and mean random recruitment functions are presented in figure 4.5. 86 Figure 4.5 Icelandic Cod: Recruitment and Recruitment Functions, RECRUITMENT (M. INDIVIDUALS) 600-400 200-OBSERVED MEAN RICKER BH ~ i — i — r " i — i — r ~ i | i i i r 500 1000 1500 SPAWNING STOCK (1000 TONS) 2000 As figure 4.5 indicates, none of the estimated recruitment functions f i t the data very well. In fact, as we have seen, only the Beverton-Holt recruitment function manages to do s l i g h t l y better than the mean random recruitment hypothesis. This lends strong support to the contention that, at least over the spawning stock range experienced during 1955-77, factors other than the spawning stock biomass have been the main determinants of recruitment. 87 4.2.2.3 Equilibrium Relationships. Equilibrium relationships are often used to describe the growth c h a r a c t e r i s t i c s of f i s h stocks and a s s i s t in formulating fishery p o l i c y <74>. In appendix 4.1.2-D two standard relationships of thi s nature, the sustainable y i e l d and biomass per recruit functions, were defined. Having estimates of the relevant b i o l o g i c a l parameters and functions for cod, we are now in a position to present empirical estimates of these functions. The sustainable y i e l d function defines the equilibrium catch as a function of the vector of fishing m o r t a l i t i e s . In the case of Icelandic cod thi s vector i s 12-dimensional. To reduce the dimensionality of the problem we resort to the t r i c k of considering only scalar multiples of a fixed vector of fishing m o r t a l i t i e s . For this purpose we pick the average fishing mortality vector 1976-8 given in table 4.1. C a l l t h i s vector f*. The set of allowable f i s h i n g mortality vectors i s thus defined by: (28) f=ef*, a l l e>0. A standard argument in applied fishery biology is that an appropriately defined f i s h i n g e f f o r t i s in fact l i n e a r i l y related to f i s h i n g mortality <75>. Hence we may identify the scalar e with t h i s kind of fis h i n g e f f o r t . Since we are not able to conclusively distinguish between recruitment hypotheses H3, H4 and H5 above, we calculate sustainable y i e l d functions for each of these. However, i t 88 should not be forgotten that H5 i s b i o l o g i c a l l y untenable at very low spawning stock levels or, equivalently, high f i s h i n g e f f o r t l e v e l s . Also i t i s worth repeating that these sustainable y i e l d functions can only be graphed in 2 dimensions on the r e s t r i c t i v e assumption expressed by (28). Any other ray of fish i n g m ortalities w i l l result in diff e r e n t sustainable y i e l d functions. The empirical sustainable y i e l d functions are depicted in figure 4.6 below. Figure 4.6 Sustainable Yi e l d Functions: Icelandic Cod. SUSTAINABLE YIELD (1000 TONS) 600 — i 400 — £00 — 0 — ft — MEAN RICKER BH \ \ A . I I I l | I I I l | I 1 I I | I I i i | i i i i | i i i i i i i C 1 2 3 4 5 6 7 EFFORT (MULTIPLES OF 1975-3 FISHING MORTALITIES) On the basis of the data in table 4.1 i t i s also easy to calculate the corresponding biomass per re c r u i t function as defined in appendix 4.1.2-D. This i s depicted in figure 4.7. 89 Figure 4.7 Biomass per Recruit Function: Icelandic Cod. BIOMASS KG. 3—| 2 4 6 8 10 12 14 IB AGE. YEARS The cohort biomass curve in f i g . 4.6 largely conforms with a p r i o r i b i o l o g i c a l considerations. It increases with age up to a certain point and then decreases. Only for very old cod, above 13 years, i s th i s relationship broken. This, however, may be due to errors in either weight or natural mortality estimates, or both. After a l l , a 5% underestimate of natural mortality for 13 and 14 year old cod would explain t h i s shape of the calculated curve. An error of t h i s magnitude i s not at a l l unlikely as sample properties of cod of th i s age are poor due to the fact that nowadays so few are caught. Also, determining the exact age of old individuals is a delicate process involving considerable uncertainty. The biomass per rec r u i t function is analogous to the so-ca l l e d stump value function in forestry l i t e r a t u r e <76>. If, in 90 fact, the output price per unit biomass of cod i s constant, the two functions are conceptually i d e n t i c a l . It follows that the cohort biomass function has similar economic implications, e.g. with respect to the optimal age of harvesting. 4.2.2.4 Simulation in the Data Period. Some measure of the a p p l i c a b i l i t y of the Beverton-Holt model and our estimates of i t s parameters to the Icelandic cod fishery may be obtained by simulations in the data period, i . e . by comparing catch and biomass values predicted by the model with alternative estimates. Given estimates of cohort sizes at the beginning of 1955, weight and mortality estimates 1955-77 <77> and the recruitment functions estimated in the section 4.2.2.2, model "predictions" of aggregate catch and biomass can be calculated using equations (11), (12), (16) and (17) in section 4.1.2. These calculated values may then be compared with the "actual" ones as recorded by fishery s t a t i s t i c s <78> and marine research i n s t i t u t e s <79>. Since the catch data i s simply a measure of the weight of t o t a l landings of Icelandic cod, t o t a l l y independent of the structure of the Beverton-Holt model, comparison of t h i s variable with the catch predictions of the model provides a joint check on the a p p l i c a b i l i t y of the model and i t s estimated parameters. The biomass data published by the marine research i n s t i t u t e s i s , on the other hand, based upon the Beverton-Holt model using, apart from recruitment, the same b i o l o g i c a l parameters as the simulations above. For annual recruitment 91 these o f f i c i a l estimates use actual values as produced by the VPA method whereas the simulations use the endogenous recruitment estimated in section 4.2.2.2. Hence, comparing these o f f i c i a l biomass values with those predicted by our model provides e s s e n t i a l l y a check on the s u i t a b i l i t y of our estimated recruitment functions. In assessing the simulation r e s u l t s , i t should be noticed, that due to the dynamic structure of the Beverton-Holt model, an error in the estimate of the size of one cohort in a given year w i l l be c a r r i e d over to ensuing years. Thus, comparatively small errors generally have an accumulative effect on subsequent predictions. The main results of the simulations are given in the table 4.3. Table 4.3 Comparison of Simulated to "Actual" Values 1955-77: Cod. Mean Percentage Standard Error* Recruitment premises: Biomass Catch 1. Actual Recruitment 2. Beverton-Holt recruitm. H3 3. Ricker recruitment H4 4. Mean random recruitment H5 0.3% 7.4% 8.8% 9.7% 12.7% 18.0% 9.4% 9.9% * Defined as 100*MSE**0.5/x(act,mean), where MSE=( £ ((x(pred)-x(act))**2)/(n-1), where x(pred) denotes predicted values, x(act) actual values and x(act,mean) the actual sample mean, n is the number of periods, 23 in t h i s case. F i n a l l y , to i l l u s t r a t e further the a p p l i c a b i l i t y of the estimated model, we present, in f i g . 4.8, a graph of the actual and predicted catch during the simulation period. More 92 precisely, catch predictions generated by the estimated model based on (i) the estimated Beverton-Holt recruitment function and ( i i ) the estimated Ricker recruitment function are compared with the actual volume of catch during 1955-77. Since recruitment predictions of the mean random recruitment function are p r a c t i c a l l y indistinguishable from those of the Beverton-Holt function during the simulation period <80> so also are the respective catch predictions. Thus the simulated Beverton-Holt catch path may be taken to represent the one that would have been generated by the mean random recruitment function as well. Figure 4.8 Catch Simulations 1955-77: Cod. CATCH (1000 TONS) I i i i i I i i i i p—i i i — i — i — r — i — i — i — i — i — i — i — i — i 55 60 65 70 75 80 YEARS 93 4.2.3 The Haddock (Melanoqrammus Aegelfinus) <8l>. The Icelandic haddock i s a considerably smaller demersal species than cod. Mature individuals are generally 60-70 cm. and weigh 3-5 kg. Only very rarely are individuals over 80 cm. and 6 kg. caught. The l i f e cycle of the haddock i s similar to that of the cod. Spawning takes place in spring each year in the r e l a t i v e l y warm sea off the South-west coast of Iceland. The eggs and larvae d r i f t with the currents northwards. The fry are i n i t i a l l y pelagic but revert to the bottom in late summer. Their preferred nursery grounds are the sand and clay areas off the West and South coasts. The haddock matures more quickly than cod but has a shorter l i f e span. It i s recruited to the f i s h i n g grounds at an age of 2 years. Sexual maturity is reached at an average age of 4 years. Very few individuals become more than 10 years old. The haddock, l i k e the cod, i s b a s i c a l l y a carnivore. It feeds mostly on bethnic animals such as worms, crabs and s h e l l f i s h , but also on small f i s h such as sandeel and f i s h eggs such as herring roe. The haddock i s preyed upon, in turn, by larger f i s h such as halibut and sharks. During the pelagic stage, the larvae and fry are eaten in great numbers by seabi rds. As the cod, the haddock i s found a l l around Iceland. Its greatest concentrations, however, are off the South and West coasts. The haddock prefers s l i g h t l y shallower waters than the cod or 30-150 m. It i s , moreover, much more s t r i c t l y demersal, seldom moving from the bottom. The haddock is also more lo c a l i z e d than the cod and, since i t s usual feeding grounds are 94 not far from the spawning grounds, does not undertake migrations on anything l i k e the same scale. 4.2.3.1 B i o l o g i c a l Parameters. Fishery biology provides the following estimates of the Beverton-Holt b i o l o g i c a l parameters for haddock: Table 4.4 Estimates of B i o l o g i c a l Parameters for Haddock. Fishing Natural Weight*** Age mortality* mortality** kg. 1 0.000 NA 0.27 2 0.006 0.20 0.62 • 3 0.048 0.20 0.90 4 0.188 0.20 1.29 5 0.410 0.20 1.94 6 0.691 0.20 2.72 7 0.903 0.20 3.46 8 1.110 0.20 4.06 9 1.033 0.20 4.56 10 1.000 0.20 5.15 11 NA i n f . NA NA: Not available. i n f . = positive i n f i n i t y . Sources: * VPA results based on catch data per 1982 Average for 1976-8. ** ICES, 1976. *** ICES, 1976. Hafrannsoknarstofnunin 1983. Average for 71-81. 95 4.2.3.2 The Recruitment Function. A key factor in the stock dynamics of haddock i s i t s annual recruitment. We w i l l now b r i e f l y examine th i s process. In late winter each year mature haddock migrate to the spawning grounds. These are exclusively off the South-west coast of Iceland in deeper and somewhat warmer waters than those of the cod. The spawning period usually begins in late A p r i l and ends in early June, 2-3 weeks later than the cod's spawning period. Apart from these differences in location and timing, the spawning pattern and the development of the eggs and larvae are si m i l a r . The average haddock lays 1-2 m i l l i o n eggs. The f e r t i l i z e d eggs d r i f t with surface currents in a generally northerly d i r e c t i o n . The eggs hatch in 12-14 days. During the f i r s t week the larvae are fed from a yolk-sack. When i t i s finished their survival depends on the a v a i l a b l i t y of zooplankton in their immediate surroundings. Haddock frys mature more quickly than those of cod. Consequently, in spite of the former being hatched l a t e r , they revert to the bottom at about the same time, i . e . in August. The nursery grounds are d i s t i n c t , however, with the haddock frys preferring deeper waters (70-80 m.) and clay or sandy bottom. Two years later adolescent haddock are recruited to the normal f i s h i n g grounds. Most of our discussion about recruitment functions for cod in section 4.2.2.2 applies to haddock as well. Consequently we w i l l now examine how the same four recruitment hypotheses, i . e . H1 , H3, H4 and H5, f i t the data. The available data cover the spawning stock and the resulting recruitment from 1960 to 1977 <82>. 96 The main results of the estimation exercise are given in table 4.5. Table 4.5 Estimation of Recruitment Functions for Haddock, Estimation Technique: Maximum Likelihood. Data period: 1960-1977. S t a t i s t i c a l properties Implied jiduals B i o l o g i c a l maximum Chi-square* p l a u s i b i l i t y recr.m** Li kelihood of values DW H1 : -21.3 0.58 H3: -17.7 0.70 H4: -17.9 0.71 H5: -17.9 0.68 Eq (27): -17.7 0.70 X(3)=2.2 Poor high S i n f . Z\3)=7.6 ok 121. X\2)=3.0 ok 123. £(3)=9.6 poor low S 93. #11)=7.6 ok 113. * On HO: Residuals are log-normally d i s t r i b u t e d . ** Sample mean i s 99.2 m i l l i o n individuals. According to the results in table 4.5, the r e s t r i c t i o n in H1 r e l a t i v e to H4 i s rejected. The relevant test s t a t i s t i c i s F(1,16)=6.7. This hypothesis also has poor s t a t i s t i c a l properties and is b i o l o g i c a l l y untenable at high levels of spawning stock. The r e s t r i c t i o n in H4 r e l a t i v e to eq. (27) i s , on the other hand, not rejected. The relevant test s t a t i s t i c i s F(1,15)=0.2. In the case of H5, the s t a t i s t i c a l assumption that the disturbances are log-normally di s t r i b u t e d must be rejected on the 2.5% l e v e l <83>. Otherwise, there i s very l i t t l e to choose between H3, H4 and H5 in terms of s t a t i s t i c a l properties although H3 f i t s the data a l i t t l e better than the others. It must be emphasized, however, that none of the estimated recruitment functions f i t the data very well. As table 4.5 shows 97 the Beverton-Holt and Ricker functions f i t the data only marginally better than the mean recruitment hypothesis. This, of course, supports the contention that, at least during the data period, factors other than the spawning stock biomass have been the main determinants of the recruitment of haddock. The DW s t a t i s t i c s , moreover, indicate the existence of a s i g n i f i c a n t positive f i r s t order autocorrelation of disturbances. This suggests that, r e l a t i v e l y slowly changing, ecological factors may have been a s i g n i f i c a n t determinant of the haddock's recruitment during the data period <84>. A scatter diagram of the data on spawning stock and recruitment during the data period is presented in figure 4.9 below. Also drawn in f i g . 4.9 are the graphs of the estimated Beverton-Holt and Ricker as well as the mean random recruitment funct ions. 98 Figure 4.9 Recruitment and Recruitment Functions: Icelandic Haddock. RECRUITMENT (M. INDIVIDUALS) 600—1 400 — OBSERVED MEAN RICKER BH 200 — 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 j 0 100 200 300 400 500 600 SPAWNING STOCK (1000 TONS) 4.2.3.3 Equilibrium Relationships. Following the approach in section 4.2.2.3 on cod, we now proceed to calculate, on the basis of our parameter estimates presented above, the two standard descriptive equilibrium relationships for haddock, i . e . the sustainable y i e l d and biomass per rec r u i t functions. F i r s t we present estimates of sustainable y i e l d functions. As for cod, these are derived on the assumption that the allowable f i s h i n g mortality vectors are those obtained by l i n e a r i l y contracting or expanding the average fishing mortality vector during 1976-8. Denote th i s vector by f*. Its elements are given in table 4.4 above. The set of allowable f i s h i n g 99 mor t a l i t i e s is thus defined by the relationship: (29) f=ef*, a l l e>0. Since we are not able to conclusively d i s t i n g u i s h between recruitment hypotheses H3, H4 and H5 above, we calculate sustainable y i e l d functions for each of these. However, i t should be kept in mind that the s t a t i s t i c a l properties of H5 are somewhat i n f e r i o r to those of H3 and H4. Moreover, H5 is b i o l o g i c a l l y untenable at very low spawning stock levels or, equivalently, high f i s h i n g e f f o r t l e v e l s . The calculated sustainable y i e l d functions are depicted in figure 4.10. 1 00 Figure 4.10 Sustainable Y i e l d Functions: Icelandic Haddock. SUSTAINABLE YIELD (1000 TONS) 150—, 100 — 50 — MEAN BH RICKER Y " ^\ I I I I | I I I I | I i I 1 | I I I I | I I I I | I I i I | I I I I | 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 EFFORT (MULTIPLES OF 1976-8 FISHING MORTALITIES) With the help of the data in table 4.4 we can also calculate the biomass per rec r u i t function. This is shown in f igure 4.11. 101 Figure 4.11 Biomass per Recruit: Icelandic Haddock. BIOMASS KG. 1.5—1 AGE. YEARS 4.2.3.4 Simulation in the Data Period. The Beverton-Holt model for Icelandic haddock that has now been estimated may be used to simulate catch and biomass during the data period. As discussed at some length in section 4.2.2.4 above, comparison of simulation results with actual catch and biomass data <85> provides a measure of the a p p l i c a b i l i t y of the model to the haddock fishery. The simulations are caried out in the same manner as for cod in section 4.2.2.4 above. The starting stock levels are the o f f i c i a l VPA estimates of the cohort stock sizes in 1960 <86>. The annual catch and aggregate biomass le v e l are then calculated 1 02 with the help of the actual VPA fis h i n g mortality levels <87>, the estimates of the b i o l o g i c a l parameters in table 4.4 and the alternative recruitment functions discussed in section 4.2.3.2. A summary of the results i s given in table 4.6. Table 4.6 Comparison of Simulated with "Actual" Values, 1960-77; Haddock. Mean Percentage Standard Error* Recruitment premises: Biomass Catch 1 . Actual Recruitment 0.6% 50.4% 70.3% 44.4% 19.0% 49.9% 66.7% 47.0% 2. Beverton-Holt recruitm. H3: 3. Ricker recruitment H4: 4. Mean random recruitment H5: * For d e f i n i t i o n of th i s concept see table 4.3. For further insight we also present, in figure 4.12, graphs of the predicted and actual catch paths. 103 Table 4.6 and figure 4.12 show a substantial difference between predicted and actual catch l e v e l s . More seriously, perhaps, the difference i s one-sided, indicating a systematic prediction bias. This suggests that important determinants of the catch dynamics have been l e f t out of the model. These might be ecological forces working on the recruitment relationship or individual weights or both. Closer examination of the errors showed, in fact, that they were caused by a combination of overestimate of individual weights in the early part of the period <88> and consistent overprediction of recruitment during the middle part. The effects of these errors are exacerbated by the dynamic structure of the model. The former item, the average weight drop during 1960-66, may perhaps be explained by r e l a t i v e 104 food shortage, as two exceptionally large adjacent cohorts were moving through the fishery during t h i s period. If t h i s proves to be the case, i t provides a good example of ecological constraints compensating for abnormal cohort s i z e s . The l a t t e r source of errors, the overprediction of annual recruitment, is to a small extent explained by the overestimated spawning stock lev e l s during the i n i t i a l years. The greater part of the difference must, however, be attributed to some unobserved ecological factors a f f e c t i n g recruitment as, in fact, the estimation of recruitment functions in section 4.2.3.2 suggests. 4.2.4 The Saithe (Pollachius Virens) <89>. The saithe is an average sized demersal species. Mature individuals are usualy 70-110 cm. and weigh 4-10 kg. Very few individuals become more than 14 years old, even in the absence of f i s h i n g . Occasionally, however, individuals weighing 15 kg. or more are caught. Although the saithe is b a s i c a l l y a demersal species, i t i s less s t r i c t l y so than cod and, in p a r t i c u l a r , haddock, frequently moving to the surface in search of prey such as herring and capelin. The saithe i s found a l l around Iceland. The greatest concentrations, however, are encountered off the South and West coasts as i t seems to prefer the warmer waters of these regions. The saithe feeds on very much the same species as does cod. On the bottom i t preys on crabs, sandeel, f i s h frys etc. At the surface, i t preys on capelin and herring. The saithe, in turn, i s preyed on by whales, seals and sharks. Also, seabirds eat 105 great quantities of saithe frys which, in contrast to cod and haddock fr y s , remain largely pelagic throughout their f i r s t year. The saithe's growth rate i s similar to that of cod. Young saithe i s recruited to the f i s h i n g grounds at an average age of 3 years and becomes mature at an average age of about 6 years. 4.2.4.1 B i o l o g i c a l Parameters. Fishery biology provides the following estimates of the Beverton-Holt b i o l o g i c a l parameters for saithe. Table 4.7 Estimates of B i o l o g i c a l Parameters for Saithe. Age F i shing mortality* Natural mortality ** Weight*** kg. 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 0.000 0.000 0.008 0.110 0.240 0.340 0.402 0.476 0.388 0.427 0.411 0.452 0.653 0.505 NA NA NA 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 inf . 0.33 0.53 1.01 1 .78 3.05 4.34 5.37 6.46 7.76 8.72 9.41 9.78 10.31 11.73 NA NA: Not avai l a b l e . i n f . = posit i v e i n f i n i t y . Sources: *VPA results based on catch data per 1982. Average for 1976-78. ** ICES, 1978. *** Average 1974-80. See ICES, 1977 og 1980. 1 06 4.2.4.2 The Recruitment Function. 1 The saithe's spawning behaviour i s similar to that of cod and haddock. Its preferred spawning grounds are in the r e l a t i v e l y warm sea off the South-west coast of Iceland. The saithe, however, spawns much e a r l i e r in the year than cod and haddock or in February to March. The average saithe lays an even greater number of eggs than those of the other two demersal species or 2-5 m i l l i o n . The f e r t i l i z e d eggs float close to the surface and d r i f t with currents primarily northwards from the spawning grounds. The hatching of the eggs and the development of the fry i s very similar to that of cod. In late June the fry are about 5 cm. Their preferred habitat, at t h i s age and for the remainder of their f i r s t year, is in shallow waters very close to the shore. As a result, they become in great numbers the prey of seabirds. As i t grows older, the saithe gradually moves to deeper waters and at the age of 2-3 years i t enters the adult f i s h i n g grounds. Sexual maturity i s reached at an average age of 6 years. The general discussion about recruitment functions for cod in section 4.2.2.2 above applies equally well to saithe. We can therefore move straight to the empirical estimation of these recruitment functions for saithe. The available data cover the spawning stock and resulting recruitment during 1960-77 <90>. Again, there is no data on the possible environmental determinants of recruitment. The relevant results of the estimation exercise are given in table 4.8. 1 07 Table 4.8 Estimation of Recruitment Functions for Saithe. Estimation Technique: Maximum Likelihood. Data period: 1960-77. Likelihood values S t a t i s t i c a l properties Implied of residuals B i o l o g i c a l maximum DW Chi-square* p l a u s i b i l i t y recr.m* Eq (27): H1 : H3: H4: H5: -20.9 -12.8 -10.0 -12.8 -9.9 0.31 0.60 0.76 0.60 0.78 £(3)=2.2 #J3 ) = 11.0 *(2)=1.2 ft 3 ) = 1 1 . 0 XI 1 ) = 1.2 poor high S poor low S ok ok ok in f . 52. 64. 52. 72. * On HO: Residuals are log-normally d i s t r i b u t e d . ** Sample mean is 51.3 m i l l i o n individuals. According to the results in table 4.8, the r e s t r i c t i o n s in H1 r e l a t i v e to eq. (27) are conclusively rejected. The relevant test s t a t i s t i c i s F(2,15)=18.1.This hypothesis also has poor s t a t i s t i c a l properties and is untenable at high levels of spawning stock. The r e s t r i c t i o n in H4 r e l a t i v e to eq. (27) i s not rejected. The relevant test s t a t i s t i c i s F(1,15)=0.2. Although H4 f i t s the data considerably better than H5, the r e s t r i c t i o n s in the l a t t e r are not rejected r e l a t i v e to eq. (27). The relevant test s t a t i s t i c i s F(2,15)=2.9. However, in terms of ov e r a l l s t a t i s t i c a l properties, H5 and, in fact, H3 too are considerably i n f e r i o r to H4, i . e . the Ricker recruitment function. However, as before, none of the estimated functions exhibits a good f i t to the data and in a l l cases the hypothesis of no f i r s t order autocorrelation of residuals must be rejected. This, as previously discussed, may be taken as evidence of slowly changing environmental factors a f f e c t i n g the annual l e v e l 108 of recruitment <91>. A scatter diagram of the observations on the spawning stock and recruitment during the data period is presented in figure 4.13. Also drawn in that figure are the estimated Ricker recruitment and, for comparison, the Beverton-Holt recruitment curves. Figure 4.13 Recruitment and Recruitment Functions: Icelandic Saithe, RECRUITMENT (M. INDIVIDUALS) lOO—i 75-50 — 25 — f / 200 400 600 800 SPAWNING STOCK (1000 TONS) " OBSERVED - MEAN -• RICKER - BH 1000 4.2.4.3 Equilibrium Relationships. As for cod and haddock previously, we now proceed to calculate sustainable y i e l d and biomass per rec r u i t functions for saithe. F i r s t we present estimates of sustainable y i e l d functions. 109 These are derived in the same way as for cod in section 4.2.2.3 above. In p a r t i c u l a r , f i s h i n g mortality vectors are r e s t r i c t e d to those obtained by l i n e a r i l y contracting or expanding the average f i s h i n g mortality vector during 1976-8 as l i s t e d in table 4.7. Otherwise the calculations are based on the b i o l o g i c a l parameter estimates presented in table 4.7 and the estimated Ricker recruitment function in section 4.2.4.2. In addition, primarily for comparative purposes, we present a sustainable y i e l d function based on the estimated Beverton-Holt recruitment function as well as the mean recruitment hypothesis. Figure 4.14 Sustainable Yi e l d Functions: Icelandic Saithe. SUSTAINABLE YIELD (1000 TONS) 150-100 — 50-MEAN BH RICKER , - 1 i 1 i 1 i 1 r 0 2 4 6 8 1 1 I r 1 10 12 14 EFFORT (MULTIPLES OF 1976-8 FISHING MORTALITIES) The biomass per re c r u i t functions for saithe are shown in figure 4.15. 1 10 Figure 4.15 Biomass per Recruit: Icelandic Saithe. BIOMASS KG. 3 - 1 2 4 6 8 10 12 14 16 AGE. YEARS 4.2.4.4 Simulations in the Data Period. The b i o l o g i c a l model for Icelandic saithe that has now been constructed may be used to simulate catch and biomass levels during the data period. As discussed at some length in section 4.2.2.4 above, comparison of the simulation results with actual catch and biomass data <92> provides some indications as to the a p p l i c a b i l i t y of the model to the saithe fishery. The simulations are ca r r i e d out in the same manner as for cod in section 4.2.2.4 above. The i n i t i a l stock sizes are o f f i c i a l VPA estimates of the cohort stock lev e l s in 1960 <93>. Given th i s the subsequent catch and biomass levels are calculated on the basis of the actual annual VPA fis h i n g 111 mortality le v e l s <94>, the b i o l o g i c a l parameters contained in table 4.7 and the alternative recruitment functions estimated in section 4.2.4.2. A summary of the results is given in table 4.10 below: Table 4.9 Comparison of Simulated with "Actual" Values 1960-77: Saithe. Mean Percentage Standard Error* Recruitment premises: Biomass Catch 1. Actual Recruitment 2. Beverton-Holt recruitm. H3 3. Ricker recruitment H4 4. Mean random recruitment H5 0.2% 25.1% 28.1% 16.3% 33.3% 26.9% 25.2% 16.2% * For d e f i n i t i o n of this concept see table 4.3 For further information we also present, in figure 4.16, graphs of the actual and predicted catch paths. The l a t t e r are drawn only for the Ricker and the Beverton-Holt recruitment funct ions. 112 Figure 4.16 Catch Simulations 1960-77: Icelandic Saithe. CATCH (1000 TONS) 150—j 100 — \ 5 0 — . -BH RICKER ACTUAL 0 — " I I I I | I I 1 I | I 1 1 1 1 1 r — l 1 1 60 65 70 75 80 YEARS . As table 4.9 and figure 4.16 show, the predicted catch and biomass leve l s deviate substantially from the actual ones. This, as before, must be primarily attributed to: (i) Inaccurate recruitment predictions by the model. The degree of this inaccuracy i s suggested by comparing the entries in the f i r s t column of table 4.9. ( i i ) Errors in the estimates of the b i o l o g i c a l parameters of table 4.7, especially individual weights. The degree to which this i s the case may be gauged from the two entries in the f i r s t l i n e of table 4.9. It is interesting to notice, in table 4.9, that, as the simulated catch errors are generally less than the biomass errors, these two sources of errors seem to counteract each other to some extent in the simulations. 1 1 3 F i n a l l y note that the Ricker recruitment function, in spite of having better s t a t i s t i c a l attributes in terms of the recruitment data, generates the worst simulations. This i s ba s i c a l l y due to the dynamic nature of the saithe fishery. I n i t i a l errors in estimated recruitment are carr i e d over to ensuing years resulting in incorrect spawning stock predictions and thus ultimately further recruitment errors. Since the Ricker recruitment process i s , in an i n t u i t i v e sense, less stable than the other recruitment processes considered, t h i s recruitment s p e c i f i c a t i o n tends to be more sensitive to errors of thi s nature. 4.2.5 Ecological Relationships. The above description of the stock dynamics of Icelandic cod, haddock and saithe, while in accordance with standard applied biology <95>, is based on the rather questionable hypothesis of ecological independence <96>. By thi s we mean that, in the b i o l o g i c a l formulation presented so far, the growth process of each species is assumed to be independent of the state of the other species as well as the ecology as a whole. In this section we propose to reconsider this hypothesis in the li g h t of the available evidence and, when appropriate, attempt to obtain numerical estimates of the relevant ecological parameters. However, since the available data are poor the empirical results w i l l be rather unreliable. Our approach is to include ecological relationships in the basic Beverton-Holt model along the lines suggested in section 1 1 4 4.1.4 above. More precisely, the essential a n a l y t i c a l framework is as follows: The b i o l o g i c a l parameters of the simple Beverton-Holt model are natural mortality, m ( i , j ) , individual weights, w(i,j) and annual recruitment, r ( j ) , where the indices i and j refer to age and species respectively. In the context of an ecological model these parameters should in general depend on a l l the stock variables in the system <97>. Thus, following the approach of section 4.1.4, we may specify <98>: (30) m(i,j)=M(x,u), a l l i , j =1,2,3, (31) w(i,j)=W(x,u), a l l i , j = 1,2,3, (32) r(j)=R(x,u), j =1,2,3. Where x represents a vector of a l l endogenous and u a l l exogenous c h a r a c t e r i s t i c s of the ecosystem. The following discussion i s arranged in two subsections. In the f i r s t subsection, we w i l l regard the demersal subsystem as constituting an ecology and consider the nature and magnitude of the relationships involved. In the second subsection, we w i l l consider how the remainder of the complete ecosystem may affect the population dynamics in the demersal subsystem. No attempt w i l l be made to model the complete ecology. 1 15 4.2.5.1 The Demersal Subsystem as an Ecology. If we r e s t r i c t our attention to the demersal subsystem, the endogenous stock variables in equations (30)-(32) are the number of individuals belonging to each cohort at a par t i c u l a r point of time. Thus, rewriting equations (30)-(32) accordingly, we obtain: (33) m(i,j)=M(n(1),n(2),n(3),u), a l l i , j=1,2,3, (34) w(i,j)=W(n(l),n(2),n(3),u) , a l l i , j = 1 ,2,3, (35) r(j)=R(S(j),n(l),n(2),n(3),u), a l l i , j=1,2,3. *\S m+ —' —' Where the vectors n(l)r ",(2) a n <3 ",(3) represent the number of individuals in each cohort of species 1, 2 and 3, respectively, and S(j) is the spawning stock for r ( j ) . The f i r s t set of equations may represent predation re l a t i o n s . Positive p a r t i a l derivatives, 9m(i,j)/9n(k,1), may for instance suggest that cohort k of species 1 preys on cohort i of species j <99>. The second set of equations may represent either food competition or predation or both. Negative p a r t i a l derivatives of (34), for instance, suggest food competition. Positive p a r t i a l derivatives on the other hand may indicate predation. The p a r t i a l derivatives of (35) are open to similar interpretations. The main point i s that the ecological extention expressed by equations (33)-(35) allows the Beverton-Holt model to incorporate a l l the categories of ecological i n t e r r e l a t i o n s h i p s defined in section 4.1.3. 1 16 We w i l l now proceed to discuss each of these functions in turn: (i) Natural Mortality Functions. As discussed in appendix 4.1.2-A, good estimates of natural mortality for marine species are generally hard to come by. In fact, of a l l the Beverton-Holt b i o l o g i c a l parameters, natural mortality i s the most d i f f i c u l t to measure. In the case of the Icelandic demersal species, fishery b i o l o g i s t s have not presented any numerical estimates of h i s t o r i c a l v a r i a t i o n in natural mortality at a l l . Consequently, for these species, estimation of natural mortality functions of the type defined by (30) or (33) i s not fea s i b l e . The available q u a l i t a t i v e evidence <100>, does not support the hypothesis of s i g n i f i c a n t predator-prey relations between cod, haddock and saithe. After a l l , these species are of similar size so any predation would have to take place between d i f f e r e n t age-groups. These, however, tend to be s p a t i a l l y segregated as pointed out in previous sections. Nevertheless, i t may be of some value to incorporate, i f only formally, ecological natural mortality relationships in the b i o l o g i c a l model under construction. This w i l l , at the minimum, allow us to investigate, later on, the potential e f f e c t s of such ecological interactions on e f f i c i e n t fishery paths. A particular hypothesis as to the e x p l i c i t form of (33) i s : 1 1 7 (36) m(i,j)=m*(i,j)+ Z Z a(k,1)n(k,1), a l l i , j . Where m*(i,j) may be interpreted as that part of natural mortality that i s independent of the stock size of the three species being considered. a ( k , l ) , on the other hand, measures the marginal e f f e c t of the size of cohort k of species 1 on m(i,j) . This linear s p e c i f i c a t i o n of (34) implies that each predator k i l l s a fixed proportion, namely a ( k , l ) , of the available number of the prey, n ( i , j ) . Several other sp e c i f i c a t i o n s of (33) are of course equally plausible <101>. Now, applied fishery biology generally provides estimates of average h i s t o r i c a l natural mortality <102>. C a l l these estimates m ( e s t , i , j ) . The m(est,i,j)'s presumably depend on the state of the ecosystem during the observation period. Thus, assuming th i s estimate to be unbiased, we may write i t as: (37) m(est,i,j)=m*(i,j)+ Z Z a(k,1)n(k,1) , a l l i , j . Where n(k,l) i s the "appropriate" average n(k,l) during the sample period. From (36) and (37) i t follows: (38) m(i,j)=m(est,i,j)+ Z Z a(k,1)(n(k,1)-n(k,1)), a l l i , j . K Z Thus, given the b i o l o g i c a l estimate of average natural mortality, the a ( l ) ' s are the only unknown parameters. We continue by i l l u s t r a t i n g the potential e f f e c t s of ecological mortality relations on the formulation of fishery 1 18 po l i c y : Take the population c h a r a c t e r i s t i c s of cod and haddock as described in sections 4.2.2 and 4.2.3 above. Assume, contrary to the available b i o l o g i c a l knowledge, that cod preys on haddock and l e t this mortality relationship be described by (38). This means i . a . that the a(k,l)'s where 1 refers to cod are nonzero. In the interest of expositional s i m p l i c i t y add the r e s t r i c t i o n that a l l the a(k,l)'s are i d e n t i c a l . C a l l them a ( l ) . In other words: (39) m(k,2)=m(est,k,2)+a(1)•L(n(k,1)-n(k,1)), k where the indices 1 and 2 refer to cod and haddock respectively. Now, define the joint sustainable y i e l d function of the two species as the sum of the respective sustainable y i e l d functions <103>: (40) y=y(1)+y(2)=Y(f(1),m(1),w(1), r" (1))+Y(fj2),m(2),w(2),r(2)). Where the natural mortality function for haddock, nj(2) , depends now partly on the sustainable stock size of cod as specified by (39). Thus, according to equation (D.1) in appendix 4.1.2-D, m(2) depends on the fi s h i n g mortality of cod, as well as the predation c o e f f i c i e n t , a ( l ) , and other b i o l o g i c a l parameters in the system represented by the vector z. I.e: m(2)=M(f(1);a(1 ) ,z) . 1 1 9 As in sections 4.2.2.3 and 4.2.3.3, we r e s t r i c t the fishing mortality vectors to linear multiples of the estimated ones during 1976-8. As pointed out in these sections, these m u l t i p l i c a t i v e factors, c a l l them e(l) and e(2) respectively, may be regarded as measures of fis h i n g e f f o r t . It should now be clear that we have sp e c i f i e d the following joint sustainable y i e l d function for cod and haddock: (41) y=Y(e(1) , e ( 2 ) ; a ( 1 ) , z ) , where, as before, e ( l) and e(2) represent 1976-8 fishing mortality multiples and the vector z represents equilibrium b i o l o g i c a l parameters other than a(1). Given our estimates of the b i o l o g i c a l parameters in sections 4.2.2 and 4.2.3 and adopting the mean recruitment hypothesis, we can now calculate the joint sustainable y i e l d of cod and haddock for d i f f e r e n t values of the ecological mortality parameter, a(1). Sustainable y i e l d contours for two such values are drawn in figure 4.17 below. The f i r s t map i s drawn for a(l)=0, i . e . no predation by cod on haddock. The second is drawn for a(1)=0.0003. This value corresponds roughly to unitary mean e l a s t i c i t y of natural mortality of haddock with respect to the stock size of cod, i . e . that a doubling the size of the cod stock r e l a t i v e to i t s h i s t o r i c a l average during 1955-77 w i l l also double the h i s t o r i c a l estimate of the natural mortality of haddock. 1 20 Figure 4,17 Predation Relationship. Joint Sustainable Yi e l d Contours. Species 1: Cod. Species 2: Haddock. Figure (i) Figure ( i i ) No predation; i . e . ad)=0. Predation; a(l)=0.0003 SPECIES l: EFFORT UNITS ' SPECIES 1: EFFORT UNITS By i l l u s t r a t i n g the s h i f t s in the joint sustainable y i e l d contours, figure 4.17 demonstrates the ef f e c t s predation relationships may have upon the s p e c i f i c a t i o n of fishery p o l i c i e s . As stated more precisely in table 4.10 below, the lev e l of f i s h i n g e f f o r t on cod corresponding to maximum joint sustainable y i e l d i s about 30% higher under predation than i t would be with no predation. This result provides a measure of the risks taken when possible ecological relationships are ignored in the formulation of fishery p o l i c i e s . 121 Table 4.10 Maximum Joint Sustainable Yield E f f o r t Levels <104>. Fishing E f f o r t corresponding to Maximum Sustainable Joint Y i e l d Ecological mortality parameter: Cod Haddock a d )=0; a( 1 ) = .0003; No Predation: Predat ion: 0.56 0.73 0.85 0.76 ( i i ) Ecological Weight Functions. A p r i o r i , i t seems almost axiomatic that the average weight of individual cod, haddock and saithe should be eco l o g i c a l l y influenced. After a l l , i t is a well established b i o l o g i c a l fact that the growth rate of individual f i s h depends on the food supply and for most f i s h species the t o t a l food supply i s an integral part of the ecosystem. The r e l a t i v e food supply, i . e . the food supply per indi v i d u a l , depends, moreover, on the number of close food competitors. Close food competitors are species inhabitating the same general area and having similar food or feeding preferences. Thus, individuals of the same species are usually close food competitors. Given t h i s , i t seems that, as far as the demersal subsystem is concerned, the problem is rather to estimate the form and magnitude of ecological weight functions than to determine their existence. In fact, fishery b i o l o g i s t s have long suspected the existence of s i g n i f i c a n t ecological weight relationships for Icelandic f i s h stocks, especially within each species <105>. Recently some quantitative but incomplete data have emerged to 1 22 support t h i s proposition <106>. We now address the problem of estimating ecological weight functions for the demersal subsystem. We f i r s t consider the proper s p e c i f i c a t i o n of equations (34). Unfortunately, b i o l o g i c a l theory provides very l i t t l e additional information on the form of these equations. Nevertheless, there are a few basic functional r e s t r i c t i o n s the ecological weight functions should apparently s a t i s f y . In the f i r s t place they should be continuous. Secondly, weight should converge asymptotically to zero as the population of food competitors approaches i n f i n i t y <107>. Thirdly, as the number of food competitors approaches zero, the weights of individuals at a given age should converge to a certain upper l i m i t . A s p e c i f i c a t i o n of (34) s a t i s f y i n g these three conditions i s : J 2 (45) w(i,j)=exp(a(i,j)- Z I b ( i , j ) n ( i , j ) * * c ( i , j ) ) u ( i , j ) , J i where a l l the c o e f f i c i e n t s , i . e . the a( )'s, b( )'s and the c( )'s, are assumed to be p o s i t i v e . In (45), exp(a(i,j)) represents the upper weight l i m i t . The e l a s t i c i t y of weight w.r.t. stock size is given by: E ( w ( i , j ) , n ( i , j ) ) = - b ( i , j ) c ( i , j ) n ( i , j ) * * c ( i , j ) and is negative for a l l posi t i v e n ( i , j ) . If c(i,j)>1, the second derivative of w(i,j) w.r.t. n ( i , j ) i s negative for s u f f i c i e n t l y 1 23 low n ( i , j ) and positive after that <108>. Otherwise this derivative is uniformly p o s i t i v e . We would now l i k e to obtain estimates of (45) for the demersal subsystem. As usual, however, the available data l i m i t s our a b i l i t y to do so. As already pointed out, VPA-estimates of the n ( i , j ) ' s for a l l three species are only available annually from 1960 onwards. Since the VPA-estimates are unreliable after 1977, the data series is limited to 18 observations. In (45), however, there are up to 65 unknown c o e f f i c i e n t s in each equation <109>. Some aggregation over cohorts i s thus unavoidable i f estimation i s to be possible. In addition, there is also a problem of unobserved variables. Systematic b i o l o g i c a l measurements on the w(i,j)'s are unavailable p r i o r to 1975 and not available at a l l for saithe. Since these variables are not observed for the complete data period, we are forced to resort to indirect estimates. Before proceeding i t i s convenient to adopt formally the following notation: Let the label "est" refer to estimated and "act" to actual variables. Thus, for instance, x ( e s t , i , j ) refers to the estimated l e v e l and x ( a c t , i , j ) the actual l e v e l of variable x for cohort i of species j . The aggregative structure chosen here is to represent the weight and stock variables by one aggregated variable for each species. Imposing this aggregative structure on equation (45) leaves at most 7 c o e f f i c i e n t s to be estimated. The aggregated stock variable i s defined as: I n(j)= I n ( e s t , i , j ) w ( e s t , i , j ) , i 1 24 where n ( e s t , i , j ) is the VPA-estimate of the number of individuals of cohort i of species j at some given time and w(est,i,j) i s the average weight estimate of cohort i of species j as reported in tables 4.1, 4.4 and 4.7 above. Thus, n(j) represents a certain standardized biomass measure, i . e . one with fixed individual weights but varying cohort sizes. In the current context this standardization seems natural since the food demand of each cohort may be expected to increase proportionally with i t s standardized biomass. Due to the unobserved w(act,i,j)'s the construction of an aggregate measure for species weights i s somewhat more involved. Let us now consider t h i s issue. The actual volume of catch of species j as recorded by the catch s t a t i s t i c s must s a t i s f y the id e n t i t y : V(act,j)= I c ( a c t , i , j ) w ( a c t , i , j ) , i where c ( a c t , i , j ) stands for the actual number of individuals of cohort i and species j caught. Define the "estimated" volume of catch as: V(est,j)= I c ( e s t , i , j ) w ( e s t , i , j ) , where w(est,i,j) is as defined above and c ( e s t , i , j ) is the number of f i s h caught as recorded by the fishery s t a t i s t i c s . Now, assume that c ( e s t , i , j ) = c ( a c t , i , j ) and define the var iable 1 25 x x "a ( i , j )=c (act, i , j )/( Z c (act, i , j )) =c (est, i , j )/( Z c (act, i , j ) ) . <• i. So a ( i , j ) i s simply the share of cohort i in the t o t a l catch (in numbers) of species j . Clearly j Z a(i,j)=1, a l l j . c F i n a l l y , construct the r a t i o : R(j)=V(act,j)/V(est,j), which according to the above d e f i n i t i o n of a ( i , j ) equals: i j R(j)= Z a ( i , j ) w ( a c t , i , j ) / Z a ( i , j ) w ( e s t , i , j ) . Now, the numerator of this r a t i o i s simply the weighted average of individual weights of species j <110>. It i s , in other words, an aggregative measure of the kind we seek. The denominator of R(j) is a similar average weight measure but standardized, i . e . with fixed w ( e s t , i , j ) . R(j) thus only d i f f e r s from the required aggregate average weight measure by a positive scalar multiple <111>. We have thus arrived at the following estimable version of (45) : (46) R(j)=exp(a(j)+ L b ( j ) n ( j ) * * c ( j ) ) ) u , a l l j , j where we do not impose nonnegativity r e s t r i c t i o n s on the 126 c o e f f i c i e n t s , a ( j ) , b(j) and c ( j ) . The average individual weights depend obviously on previous levels of r e l a t i v e food supply as well as the current one. It follows that an appropriate empirical version of (46) involves a di s t r i b u t e d lag structure of the explanatory variables, n ( j ) . Modifying (46) accordingly y i e l d s : j (47) R(j,t)=exp(a(j)+ Z b ( j , 1 ) L ( j , 1 ) n ( j , t ) * * c ( j ) ) u ( t ) , t=1,..l8. J where L ( j , l ) represents the appropriate polynomial lag operator <112> and the t arguments have been included to emphasize the dynamic nature of the formulation. The guiding p r i n c i p l e of the estimation procedure was, as before, to arrive at the simplest formulation not contradicted by the data. This led us, ultimately, to r e s t r i c t the lag structure to 2nd order Almon di s t r i b u t e d lag polynomials <113> with 3 lags at the most and r e s t r i c t a l l the c ( j ) ' s to c(j)=2 <114>. The main estimation results are l i s t e d in tables 4.11 and 4.12: 1 27 Table 4.11 Ecological Weight Functions: Estimation r e s u l t s . Data period: 1960-77. Estimation procedure: OLS. Eq' s Ecological Weight Matrix (Total m u l t i p l i e r s ; Z o b ( j , t - l ) ) Cod Haddock Saithe Degrees of fr.dom R Variance of res' s (1) Cod: -.59-10 0* -.13 10"^ 8 .95 .0006 L(3,2)** L(3,2)** (2) Had: 0* -.191 0 " s 0* 12 .90 .002 L(3,2)** (3) Sai : 0* -. 1 1 • 1 0"s -.69 • 10 - 4 10 .95 ' .002 L(1,0)** L(3,2)** * Restricted to zero. ** L(h,k)denotes an Almon polynomial lag structure of order k and maximum lag of h. The corresponding s t a t i s t i c a l results are l i s t e d in table 4.12: Table 4.12 Ecological Weight Functions: S t a t i s t i c s and Tests. Ecological Weight Matrix Tests of Significance of Z. b( j , t-1) ' s** Exclusion*** Auto Norm-Cod Haddock Saithe r e s t r i c t i o n s c o r r e l * a l i t y Cod: #(3) = 15.1 " *(3)=29.0 2(3)=3.8 2.5 £(1)=3.7-Had: - *(3) = 64.4 a - z\6) = 7.6 2.4 **(1) = 2.;3. Sai: - tdO)=2.5 X(3) = 104. & 3) = 1 .2 2.7 X.\3)=8.,l * The numbers reported are Durbin-Watson s t a t i s t i c s . ** On HO: Zb(j,t-1)=0. *** .I.e. the zero r e s t r i c t i o n s in table 4.11. From a s t a t i s t i c a l point of view, the results in tables 4.11 and 4.12 seem f a i r . According to the autocorrelation and 1 28 normality tests, the OLS estimator may not be too far removed from the maximum l i k e l i h o o d one. Consequently some f a i t h in the other s t a t i s t i c s reported may be j u s t i f i e d . The central result reported in tables 4 . 1 1 and 4 . 1 2 i s that, for the demersal subsystem, ecological weight relationships"seem to be highly s i g n i f i c a n t . As expected, own weight e f f e c t s , i . e . the diagonal terms in the estimated ecological weight matrix (hereafter referred to as the W-matrix), seem to be strongest. The hypothesis that interspecies effects are nonexistent, i . e . that a l l the off-diagonal terms in the W-matrix are- simultaneously zero, is also conclusively rejected. The corresponding test s t a t i s t i c i s X 2 ( 1 8 ) = 6 5 . 6 . There are no positive entries in the W-matrix. In fact, the i n s i g n i f i c a n t elements were generally found to be negative. Thus there is no evidence of s i g n i f i c a n t predation between the three species. The estimated structure of the W-matrix i s interesting. It is recursive, meaning that there are no s i g n i f i c a n t two-way relationships. This feature may be c a l l e d s t r i c t dominance with respect to food competition. Cod, in p a r t i c u l a r , i s s t r i c t l y dominated by saithe with regard to food competition. Saithe in turn i s s t r i c t l y dominated by haddock and the average weight of haddock i s not s i g n i f i c a n t l y affected by the stock sizes of the other two species. The magnitude of the relationships are perhaps better appreciated when cast in terms of e l a s t i c i t i e s . The respective e l a s t i c i t i e s calculated at sample means are shown in table ( 4 . 1 3 ) below. 1 29 Table 4.13 Ecological Weight Relationships. E l a s t i c i t i e s at Sample Means Cod Haddock Saithe Cod: -0.301 0* -0.067 Haddock: 0* -0.274 0* Saithe: 0* -0.164 -0.347 * Not s i g n i f i c a n t According to these results, the ecological weight e f f e c t s , estimated at sample means, are not only s i g n i f i c a n t but of r e l a t i v e l y high order of magnitude. For instance, doubling of the stock size of cod leads, c e t e r i s paribus, to roughly 30% reduction in i t s average individual weights. This demonstrates the potential errors of ignoring ecological relationships in formulating f i s h e r i e s p o l i c i e s . ( i i i ) Ecological Recruitment Functions. According to equation (C.3) in appendix 4.1.2-C, recruitment, once the spawning stock is given, depends only on the pre-recruitment mortality rate. Hence any ecological e f f e c t s on recruitment must work through pre-recruitment mortality. Before turning to the available empirical evidence, i t should be made clear, that the q u a l i t a t i v e b i o l o g i c a l knowledge about the recruitment process outlined in sections 4.2.2.2-4.2.4.2 above does not suggest the existence of s i g n i f i c a n t early mortality effects between cod, haddock and saithe. Although the spawning patterns of these species are broadly similar, the spawning .time and spawning areas do not coincide. 1 30 The nursery grounds favoured by these three species are, moreover, di f f e r e n t and largely d i s t i n c t from the grounds normally inhabited by the older cohorts. The q u a l i t a t i v e knowledge, in other words, does not indicate the geographical proximity between individuals of the required size d i f f e r e n t i a l to support a hypothesis of s i g n i f i c a n t predation. A simple version of ecological pre-recruitment mortality for a certain cohort of species j during the pre-recruitment period i s : (47) m(j)=m(j)*+ Z Z a ( i , j ) n ( i , j ) + b ( j ) S ( j ) + c ( j ) n ( j ) , where i i s an index for already recruited cohorts. (47) i s , of course, understood to hold at each point of time during the pre-recruitment period. m(j)* represents that part of early mortality that is independent of the state of the demersal subsystem. The a ( i , j ) n ( i , j ) terms represent ecological mortality e f f e c t s . S(j) represents the spawning stock for the cohort in question and n(j) denotes i t s size at each point of time during the pre-recruitment period. The last two terms of (47) thus represent the Ricker and Beverton-Holt recruitment hypotheses respectively. b(j) measures the marginal effect of the spawning stock on pre-recruitment mortality and c ( j ) the own density dependent mortality. Integrating (47) as in appendix 4.1.2-C we derive the corresponding recruitment function. Adopting in turn the Beverton-Holt and Ricker r e s t r i c t i o n s , i.e. b(j)=0 and c(j)=0 and making further simplifying assumptions explained in appendix 131 4.2.5-A, the respective ecological recruitment functions may be written as: (48) r=aS/(b+cS) (49) r=dSexp(e+fS). where the parameters a, b, c, d, e and f are rather lengthy functions of the c o e f f i c i e n t s in (47) <115>. Now, given appropriate data, i t i s in p r i n c i p l e possible to estimate the parameters of equations (48) and (49). However, in this p a r t i c u l a r study, only 23 observations on the cod's recruitment and 18 on the recruitment of haddock and saithe are available. The number of unknown parameters in (48) and (49), on the other hand, is 36. Thus, to obtain any estimates at a l l , these equations must be s i m p l i f i e d in some way. For th i s purpose, we selected to reduce the number of cohorts, i . e . the n(i,k)'s, by r e s t r i c t i n g the attention to two groups of cohorts for each species, namely immature and mature individuals <116>. These we denote by n(1,k) and n(2,k) respectively. This leaves us with 9 recruitment c o e f f i c i e n t s for each species, i . e . 6 ecological interaction c o e f f i c i e n t s , a ( i , k ) ' s , fixed mortality, m(j)*, the egg production parameter, q, and b(j) or c ( j ) as the case may be. With these modifications (48) and (49) may be estimated using e s s e n t i a l l y the techniques employed in estimating the simple recruitment functions in sections 4.2.2.2-4.2.4.2 above. This, however, may not be a p a r t i c u l a r l y e f f i c i e n t way of 1 32 estimating these functions. The b i o l o g i c a l considerations outlined in sections 4.2.2.2-4.2.4.2 as well as the residuals from single equation estimates of (47) and (48) <117> indicate the influence of omitted, and unobserved, habitat variables, e.g. sea-temperature, currents etc., on annual recruitment. Given the s i m i l a r i t y of the recruitment processes of the three species, i t seems plausible that their recruitment may be s i m i l a r i l y affected by these factors. If th i s is indeed so, the estimated single equation residuals of the three recruitments functions w i l l not have zero covariance. In that case, i t i s well known <118>, that single equation estimation techniques do not provide e f f i c i e n t estimates of the recruitment parameters. More e f f i c i e n t estimates are obtained by applying the GLS technique to a l l three functions simultaneously <119>. We w i l l now b r i e f l y describe the estimation process and i t s resu l t s . As already mentioned the available data cover the period 1960-77; altogether 18 observations on the 12 variables included in the modified recruitment functions. These data are l i s t e d in Appendix 1: Data at the end of the thesis. The estimation procedure consists of three steps. In each step we retain the stochastic assumptions concerning the error term stated in sections 4.2.2.2-4.2.4.2 above. In the f i r s t step, equations (48) and (49) were estimated for each species separately. The purpose was to determine which s p e c i f i c a t i o n , the Beverton-Holt or Ricker one, was more appropriate in terms of the data. Employing the same c r i t e r i a as 133 in sections 4.2.2.2-4.2.4.2 above, the conclusion was to adopt Beverton-Holt s p e c i f i c a t i o n for cod and haddock and the Ricker s p e c i f i c a t i o n for saithe. The aim of the second step was to test the hypothesis that a l l ecological mortality c o e f f i c i e n t s are simultaneously zero. More precisely we tested the hypothesis: HO: a(i,j)=0, i=1,2; j=1,2,3. For t h i s purpose the three equations were j o i n t l y estimated according to the Zellner procedure. The resulting test s t a t i s t i c was F(6,30)=1.14. Consequently, the n u l l hypothesis could not be rejected on conventional significance l e v e l s . We must therefore conclude that the data does not contradict the hypothesis of no ecological pre-recruitment mortality effects between the three species. In the f i n a l step we reestimated the system imposing the HO r e s t r i c t i o n of the second step. This, of course, largely reproduced our previous estimation results of sections 4.2.2.2-4.2.4.2 with s l i g h t l y improved s t a t i s t i c a l properties <120>. and with added information about the covariance of the residuals from each equation. The implied co r r e l a t i o n matrix of the residuals with t - s t a t i s t i c s is presented in table 4.11: 1 34 Table 4.14 Estimated Recruitment Functions: Correlation of Residuals. Correlation* of residuals Cod Haddock Saithe Cod: 1 .00 Haddock: 0.35 (1.6) -.11 (0.4) 1 .00 Saithe: -.59 (3.62) 1 .00 * Defined as r=Cov(e(j),e(k))/(Var(e(j))Var(e(k)))**0.5 where e(j) represents the error vector of species j . ** t - s t a t i s t i c s in brackets calculated as t=r((18-2)/(1-r*))**0.5 These results can hardly be regarded as very conclusive. There i s some weak evidence of favourable recruitment conditions for cod and haddock occuring simultaneously and somewhat stronger evidence that good recruitment conditions for saithe coincide with poor conditions for haddock. There i s , however, no evidence of a strong c y c l i c a l movement in the environmental factors that af f e c t the recruitment of a l l three species simultaneously. 4.2.5.2 Environmental Influences. The demersal subsystem, as pointed out above, i s only a part of the more comphrehensive ecosystem encompassing a l l the aquatic biota of Icelandic waters as well as their habitat. Although the environment of the demersal subsystem i s regarded as exogenous in thi s study, there i s no reason to ignore i t s 135 potential effects on dynamics of cod, haddock and saithe. In fact, as suggested by the empirical results above, especially those on the recruitment process, such a procedure might s i g n i f i c a n t l y misrepresent the true s i t u a t i o n . Unfortunately, however, p r a c t i c a l l y no data on the environmental variables are available. Given t h i s , a reasonable alternative seems to be to represent the environmental influences by including appropriately s p e c i f i e d stochastic variables in the demersal submodel. To formulate t h i s idea more precisely, rewrite the general ecological equations, (30)-(32), as: (50) m(i,j)=M(x,u), a l l i , j=1,2,3, ueAO). (51) w(i,j)=W(x,u), a l l i , j = 1 ,2,3, ueA(2). (52) r( j)=R(x,u), . j=1,2,3, ueA(3). Where the environmental variables, u, are now drawn from di f f e r e n t stochastic d i s t r i b u t i o n s , denoted by A(1), A(2) and A(3). Since, as we have discussed above, there are no observations on m(i,j) and few on w ( i , j ) , there i s hardly an empirical basis for estimating the probability d i s t r i b u t i o n s A(1) and A(2). On the other hand, examples of the effects of a r b i t r a r i l y s p e c i f i e d A(1) and A(2) on the population dynamics of the demersal subsystem may be calculated. As regards A(3), however, we are in a d i f f e r e n t position. 136 Our estimation of recruitment functions in previous sections •included estimates of a stochastic error term and i t s d i s t r i b u t i o n . Regarding these as s u f f i c i e n t l y good estimates of the c o l l e c t i v e effects of environmental forces we may extract from them estimates of the underlying stochastic process. The vector u in equation (52) may be written as: u=(u(1),u(2),u(3)), where the indices refers to cod, haddock and saithe respectively. The empirical estimation of the recruitment functions for these three species described in sections 4.2.2.2-4.2.4.2 yielded estimates of the u( )'s, or rather their natuaral logs, for the period 1960-77 <121>. A further empirical investigation indicated a certain s t r u c t u r a l regularity in these error terms, namely a degree of autocorrelation <122> and some cross-species covariances <123>. On the basis of these data and indications we now estimate the system: ud,t)= Z a( 1 , i ) u ( i , t ) + Z b( 1 , i )u(i ,t-1 )+e( 1 ,t) , s u(2,t)= Z a(2 , i ) u ( i , t ) + Z b(2,i)u(i,t-1)+e(2,t), u(3,t)= Z a( 3 , i ) u ( i , t ) + Z b(2,i)u(i,t-1)+e(3,t). Where, as mentioned, the u ( i , t ) ' s are the natural logs of the 1 37 estimated recruitment residuals during 1960-77 and the e ( i , t ) ' s are supposedly the remaining white noise residuals. The main aim of the estimation of thi s system i s thus to discover r e g u l a r i t i e s in the recruitment residual's and estimate the corresponding c o e f f i c i e n t s in order to be able to represent the u( )'s in terms of a systematic part and a purely stochastic part, i . e . the e( )'s. The estimation results are summarized in the following table: Table 4.15 The Structure of the Recruitment Residuals, Estimation Technique: 2SLS (Instruments are lagged values of the u ( j , t ) ' s ) . <124> Data period: 1961-77* Dis t r i b u t i o n of e(i) Normality Test of test Zero r e s t r i c t i o n s zero Chi-Eq's on c o e f f i c i e n t s r e s t r i c t i o n s Var DW square Cod : a(1,3),b(1,1),b(1,3) F(3,11)=0.48 0.06 1.61 z\2)=1.9 Had : a(2,3) ,b(2, 1 ) ,b(2,3) F(3,11)=0.35 0.37 1.42 7C\ 2 ) = 1 . 4 Sai : On a l l except b(3,3) F(4,11) = 0.36 0.13 1.69 ^ 3 ) = 1.9 * One observation lost because of the lag structure. These results seem encouraging. The recruitment residuals have been represented by a rather simple systematic structure and a stochastic term that seems to be close to being white noise. The estimated stochastic processes are as follows <125>: 138 (53) u(1,t)=-0.136u(2,t-1)+1.257e(2)+4.084e(1), (54) u(2,t)=0.697u(2,t-1)+l0.02e(1)+4.084e(2), (55) u(3,t)=0.672u(3,t-1)+e(3) . According to these results, the recruitment of cod and haddock are affected in the same way by the environmental factors represented by the stochastic terms e(1) and e(2). The recruitment of saithe seems to depend on other environmental factors than that of cod and haddock. The estimated recruitment residuals for haddock and saithe are s i g n i f i c a n t l y autocorrelated suggesting perhaps the effects of slowly moving environmental factors. The negative term in the residual equation for cod means that the recruitment of cod i s adversely affected by unusually good and p o s i t i v e l y by unusually poor recruitment of haddock one year previously. This may perhaps be attributed to food competition between juvenile and adolescent haddock and the younger and smaller cod and dominance w.r.t. food competition of the l a t t e r group by the former. 139 Appendix 4.2.5-A Derivation of Ecological Recruitment Functions. Let the n(i,k)'s in equation (42) be r e s t r i c t e d to cohorts already recruited to the f i s h e r y . In that case i t seems b i o l o g i c a l l y plausible to take the n(i,k)'s to be independent of the pre-recruitment stock size, n ( j ) . Moreover, for even greater s i m p l i c i t y l e t the n(i,k)'s be represented by their average values during the pre-recruitment period of n ( j ) . Hence these variables are not only independent of n(j) but fixed during this period. The Beverton-Holt version of (42) i s : (A.1) m(j)=m(j)*+ L L a ( i , k ) n ( i , k ) + c ( j ) n ( j ) , where, i t w i l l be remembered, n(j) i s the pre-recruitment stock size of a p a r t i c u l a r cohort of species j . Since the n(i,k)'s are by assumption fixed during the pre-recruitment period of n ( j ) , we may rewrite (A.1) for a given species and at time t as: (A.2) m(t)=m1+m2n(t), where m1=m(j)*+ L L a ( i , k ) n ( i , k ) . m2=a(j). 1 * Now, (A.2) has exactly the same mathematical structure as (C.5) in appendix 4.1.2-C. It follows that the corresponding recruitment function i s given by (C.6) in appendix 4.1.2-C. The Ricker version of equation (42) i s : (A.3) m(j)=m(j)*+ I Z a ( i , k ) n ( i , k ) + b ( j ) S ( j ) . i H Again, since the n(i,k)'s are fixed, we have: (A.4) m(t)=m1+m2S(t), where ml i s given above and m2=b(j). Since (A.4) i s i d e n t i c a l to (C.9) in appendix 4.1.2-C, i t follows that the corresponding ecological Ricker recruitment function i s given by (C.10) in the same appendix. 1 40 Footnotes. 1. See Beverton and Holt, 1957. 2. See e.g. Clark, 1976, pp. 11-12. 3. Notice that the functional form of (3) i s very r e s t r i c t i v e . Regarding (3) as a production function, i t belongs to the CES class of such functions with unitary e l a s t i c i t y of substitution, equal imputed shares and i s homogeneous of degree 2. For more de t a i l s on t h i s see e.g. Allen, 1973, pp 49-55. 4. Among the b i o l o g i c a l applications we may mention Baerends, 1947, Schaefer, 1954 and 1957, and Parsons and Parsons, 1975. The economic applications include Gordon, 1954, C r u t c h f i e l d and Zellner, 1962, Neher, 1974, and Clark, 1976. 5. See e.g. Gulland, 1961, and Pell a and TomHnson, 1969. 6. For examples consult e.g. Emlen, 1973. 7. For references see two recent textbooks in the f i e l d ; Clark, 1976, and Howe, 1979. Also see the i n f l u e n t i a l papers by Smith, 1968, Plourde, 1970, Clark and Munro, 1978, and Clark, Clarke and Munro,1979. 8. Although cohorts are taken here as an example, any s i g n i f i c a n t heterogeneity w.r.t. growth could play the same role. 9. This degree of smoothness of the G ( i , )'s i s not at a l l necessary for the conclusions below but s i m p l i f i e s the argument considerably. 10. Analogous results are well known in economic theory. In demand theory, for instance, exact aggregation of demand functions requires individual Engel-curves to be l i n e a r . For d e t a i l s see e.g. Deaton and Muellbauer, 1980, pp. 149-53. 11. It may be noted that describing biomass growth in terms of cohort biomasses instead of the aggregate biomass s t i l l involves aggregation over individuals belonging to each cohort. Hence, thi s degree of disaggregation does not e n t i r e l y avoid the aggregation problem. The requirements in proposition 4.1, i . e . those of i d e n t i c a l l y sloping individual growth functions, however, are much more plausible within a cohort than over a l l the cohorts. Moreover, referring again to proposition 4.1, the v a r i a b i l i t y in individual biomasses within a cohort is generally much less than the v a r i a b i l i t y in cohort biomasses. For similar reasons the aggregation of t y p i c a l economic relationships, such as demand functions, normally involves aggregation errors of much lesser magnitude than the aggregation over cohort growth functions. 12. I.e. the functions F( ) need not be i d e n t i c a l . 141 13. I.e. the p r o f i t function where cohort biomasses are in equi1ibr ium. 14. For a more detailed discussion of these types of cohort biomass relations see Clark, Edwards and Friedlander, 1973. 15. See e.g. Gulland, 1961, Parsons and Parsons, 1975, Wilen, 1976, and the estimation results in appendix 4.1.1-B, below. 16. See e.g. Schnute, 1975, and Uhler, 1978. 17. The data are l i s t e d in appendix 4.1.1 —B. 18. An equation defining this sustainable growth relationship i s given in section 4.1.2. 19. Or, i f that i s not available, r e l a t i v e input and output prices . 20. OLS, of course, may not be the maximum l i k e l i h o o d estimator in t h i s case. This, however, i s beside the point. 21. Thus the respective ICES (International Council for the Exploration of the Sea) working groups for the North A t l a n t i c area adopted disaggregated population growth models ( i . e . the Beverton-Holt model) for cod and haddock in 1970, saithe in 1977 and redfish in 1978. 22. As Beverton and Holt point out, however, several elements of their model had previously been suggested by other authors. (See Beverton and Holt, 1957, pp 13, 26 and 30. 23. More precisely, z = / 1 z ( i , t ) d t . 24. It is worth emphasizing that (17) i s derived on the assumption that f i s h i n g mortality i s constant over the period of integration. I f , in p a r t i c u l a r , fishing mortality i s a nonmonotonic function of time, within t h i s period, which, i s probably the case in many f i s h e r i e s , the catch level must be represented by a much more complex expression. 25. See e.g. ICES, 1976. 26. See e.g. the various ICES reports in the l i s t of references. 27. Fishery biology offers various techniques for the estimation of these parameters. For a discussion of these see appendix 4.1.2-A. A more complete account can be found in Gulland, 1969 and especially Ricker, 1978. 28. As suggested in section 3.2.1 above, natural mortality, individual weights and recruitment are generally endogenous variables within a more complete ecological framework. Here, however, we are taking a more limited view, b a s i c a l l y describing standard practice in fishery biology (see e.g. the ICES working 1 42 groups reports in the l i s t of references) which t y p i c a l l y regards these variables as parameters i f not constants. 29. See e.g. Schopka, 1972, Jakobsson, 1978, and Hafrannsoknarstofnun, 1983. We w i l l have more to say on this in section 4.2 below. 30. See e.g. the various ICES publications in the l i s t of references. 31. See the various ICES publications in the l i s t of references. 32. Denoting the number of individual fry produced by the spawning by n(0), the subsequent recruitment i s given by r=n(0) exp(-f um(t)dt) , o where u i s the recruitment age. 33. The method was f i r s t proposed by Gulland, 1965. 34. See e.g. Pope, 1971. 35. See e.g. Ulltang, 1976. 36. I.e. ones involving pos i t i v e biomass. 37. Beverton and Holt, 1957, pp. 55-61, consider i n s t a b i l i t y due to the recruitment process. See also Clark, 1976, pp. 211-15. Within the context of a general ecological framework, natural mortality and weight relationships may become the source of a similar kind of i n s t a b i l i t y . 38. E.g. many demersal species such as those considered in t h i s study. 39. For examples of the use of these equilibrium functions in applied fishery biology see e.g. ICES 1974, 76, 77, 78 and 80a. 40. This d e f i n i t i o n is similar to the one given by Southwick, 1 976. 41. The dependence does not, of course, have to be d i r e c t , instantaneous or continuous. 42. For d e f i n i t i o n s of decomposability see e.g. Takayama, 1974, p. 370. 43. See e.g. Southwick, 1976, pp. xv-xvi. 44. See Schaaf, 1973. 45. See Russel, 1975, and Levin, 1975. 46. In f i s h e r i e s , examples are provided by Riffenburg, 1969, and 143 Jakobsson, 1978. Riffenburg argues that the collapse of the P a c i f i c sardine stock in the 1950's was, in addition to heavy fis h i n g , due to food competition with anchovies and the predation of hake. Jakobsson argues that the collapse of the Icelandic spring spawning herring was, at least partly, due to deteriorating environmental conditions on the top of continual heavy f i s h i n g . 47. In a small ecosystem l i k e a pond or a lake t h i s may well be di f ferent. 48. See e.g. May, 1973, and Patten, 1975. 49. As in equation (25), G*(i,j) denotes the f i r s t derivative of the growth function of species i with respect to the biomass of species j . Notice, that the signs of these derivatives may change when the functional arguments change. 50. The names given to the relationships are according to May, 1973. 51. J may, on the other hand, be p a r t i a l l y decomposible implying e.g. some recursiveness in the biology. 52. The continuous time formulation i s for expositional convenience only. A t r a n s i t i o n to a discrete time Beverton-Holt model i s , of course, straight forward. 53. It i s interesting to note that the system in (26) i s s t r u c t u r a l l y i d e n t i c a l to the well known price adjustment formulation in neoclassical price theory (see e.g. Shone, 1975, p. 291-99). In short, z corresponds to a price vector and the function, G( ), to an excess demand system. The vectors, y and u, correspond to exogenous variables in price theory7 e.g. tastes, technology, government regulations etc. Thus i t seems l i k e l y that results from the theory of equilibrium prices and their s t a b i l t y may be brought to bear upon t h i s ecological system. 54. One of the f i r s t large scale projects to obtain empirical information of t h i s nature was recently embarked upon by the International Council for the Exploration of the Sea (see ICES 1980b). Its aim is to discover the direct food links between several commercial f i s h stocks in the North Sea. 55. As already pointed out in section 3.2.1, those three species have in recent years accounted for over 80% of the t o t a l demersal catch and over 90% of i t s value. 56. This p a r t i c u l a r demarcation of the ecosystem i s , of course, f a i r l y a r b i t r a r y . 57. This, incidently, i s the standard practice in applied fishery biology. See e.g. ICES, 1976, 1977, 1978 and 1980a. 144 58. See section 4.1.3 above. 59. The mass of a given volume of water is at a maximum close to the temperature of 4°C. The bottom temperature in temperate and cold seas i s t y p i c a l l y below 4°C, while the surface layers are generally warmer than 4°C. Hence, as the surface layers become colder during winter and spring, v e r t i c a l mixing takes place. This i s one of the reasons why temperate and cold seas are generally more f e r t i l e than warm seas. 60. According to one estimate, Graham and Edwards, 1961, the y i e l d of demersal species per acre of Icelandic f i s h i n g grounds is roughly three times that of the North Sea, two and a half times that of the Grand Banks and twice that of the Barents Sea. 61. The q u a l i t a t i v e descriptive content of this section i s to a considerable extent based on Saemundsson, 1926, and Jonsson, 1983. Other general sources are various publications by Hafrannsoknarstofnunin and ICES, see the l i s t of references. 62. In fact there are records of 50 kg cod being caught. This, however, i s exceptional. 63. See ICES, 1976. 64. See Palsson, 1983. 65. The age of sexual maturity depends in fact on the cod's rate of growth which varies with environmental conditions. In addition to Saemundsson, 1926, see Jonsson and Schopka, 1973, Hafrannsoknarstofnun, 1982, and Jonsson, 1983, on t h i s . 66. These r e f l e c t fixed and variable natural mortality c o e f f i c i e n t s as well as the egg production c o e f f i c i e n t . For d e t a i l s see appendix 4.1.2-C. 67. It i s in fact easy to show that a f a i r l y invariant food constraint would tend to produce this r e s u l t . 68. Given that recruitment i s r e s t r i c t e d to nonnegative numbers thi s is a natural assumption. 69. "NIID" is a abbreviation for "normally, i d e n t i c a l l y and independently d i s t r i b u t e d " . 70. These data are given in Appendix 1: Data (section 4.2.5) at the end of the thesis. 71 For the theory on t h i s type of tests see e.g. Silvey, 1975, and Schmidt, 1976. 72. For t h i s purpose we employ a standard chi-square test and the well known Durbin-Watson test. As suggested by Nerlove, 1963, and Hendry and Anderson, 1977, the l a t t e r may be interpreted as a test of misspecification. 145 73. These s t a t i s t i c s are defined in most elementary econometrics and s t a t i s t i c s textbooks. 74. See e.g. the various ICES publications in the l i s t of references. 75. See e.g. ICES, 1976. 76. See e.g. Pearse, 1967. 77. I.e. the weight and natural m o r t a l i t i e s in table 4.1 and VPA estimates of fishing m o r t a l i t i e s during 1955-77 in appendix: data (chapter 7.1) at the end of the thesis. 78. See Hafrannsoknarstofnunin, 1983. 79. See ICES, 1976, and Hafrannsoknarstofnunin, 1980-83. 80. For v e r i f i c a t i o n , consult section 4.2.2.2 especially figure 4.5. 81. The q u a l i t a t i v e descriptive biology of t h i s section is largely based on Saemundsson, 1927 and Jonsson, 1983. 82. These data are l i s t e d in appendix: data (section 4.2.5). 83. This probably explains the negative bias in the corresponding recruitment prediction in table 4.5. 84. This l i n e of inquiry is pursued further in section 4.2.5 below. 85. As estimated by the respective marine research i n s t i t u t e s , i. e . Hafrannsoknarstofnunin, 1983 and ICES, 1976. 86. See the ICES, 1976, and the various publications by Hafrannsoknarstofnunin in the l i s t of references. 87. See Appendix 1: Data (section 7.1). 88. According to estimates in ICES, 1976, the average individual weights of Icelandic haddock during t h i s period was up to a t h i r d below normal. 89. The q u a l i t a t i v e descriptive biology of this section is largely based on Saemundsson, 1927, and Jonsson, 1983. 90. These data are l i s t e d in appendix: data. 91. This w i l l be further discussed in section 4.2.5 below. 92. As estimated by the respective marine research i n s t i t u t e s , i. e . Hafrannsoknarstofnunin, 1983, and ICES, 1976 and 1978. 93. See ICES, 1980. 1 46 94. See Appendix 1: Data. 95. As presented by the various ICES working group papers in the l i s t of references. 96. In addition, as we have seen, the model assumes fixed mortality rates during each time period. 97. See section 4.1.3. 98. The ecological relationships in 4.1.4. were defined in d i f f e r e n t i a l form. The s p e c i f i c a t i o n here may be j u s t i f i e d either on the grounds of s i m p l i c i t y or very rapid adjustment to equilibrium values. 99. This derivative i s of course open to other interpretations. 100. See e.g. Saemundsson, 1927, and Jonsson, 1983. 101. Another plausible s p e c i f i c a t i o n , for instance, is that each predator k i l l s a fixed number of the prey irrespective of i t s stock s i z e . 102. This i s c e r t a i n l y true for Icelandic cod, haddock and saithe. 103. See appendix 4.1.2-D. 104. The results in table 4.10 are readily explainable. The maximum joi n t sustainable y i e l d e f f o r t on cod increases when cod i s assumed to prey on haddock because the resulting drop in the sustainable y i e l d of cod i s i n i t i a l l y exceeded by the corresponding increase in sustainable y i e l d of haddock due to lower natural mortality. The f i s h i n g e f f o r t on haddock corresponding to maximum joint sustainable y i e l d i s lower in the predation case because, with a smaller stock of cod, natural mortality of haddock is reduced and the maximum y i e l d harvesting age goes up. The only way to increase the mean harvesting age in t h i s model, however, is to reduce o v e r a l l e f f o r t . 105. See e.g. Schopka, 1972, and Jakobsson, 1978. 106. See Hafrannsoknarstofnun, 1983. 107. Of course, individual f i s h cannot be expected to survive the experience. But that i s another issue. 108. In that case the function may be v i s u a l i z e d as having the same general shape as the LHS of a normal density function. 109. In addition to a, there are 12+8+12 b ( i , j ) ' s , the same number of c ( i , j ) ' s . 110. Notice that the a ( i , j ) ' s vary over time e.g. with changes in the cohort composition of the catches. However, during the 147 data period t h i s remained f a i r l y constant. 111. The reason for dividing by V(est,j) in R(j) i s to eliminate the t o t a l number of f i s h caught, c ( a c t , i , j ) , from the average weight measure. 112. This lag operator i s formally defined by: b(l)L(l)x(t)=b(0)x(t)+b(1)x(t-1)+ + b ( l ) x ( t - l ) . 113. See Almon (1965). 114. A sci-square s t a t i s t i c on these last r e s t r i c t i o n s yielded the result X 2(3)=10.7. 115. For d e t a i l s see appendix 4.2.5-A. 116. Remember that the immature group consists only of cohorts already recruited to the fishery. 117. See also the empirical results in 4.2.2.2-4.2.4.2. 118. See e.g. Schmidt, 1976, pp. 64. 119. See e.g. Zellner, 1962. A system estimation of the Zellner kind, however, r e s t r i c t s the sample period to the shortest one, i . e . that of haddock and saithe. Hence the cost of applying this technique i s the loss of several observations on the recruitment of cod. For cod, therefore, t h i s approach may not improve ef f ic iency. 120. This i s in accordance with the properties of the Zellner method. The results for cod, however, were s l i g h t l y altered due to the truncated sample period. 121. These estimates, as i s well known (see e.g. Schmidt 1976, p. 65), w i l l , in fact, be unbiased under f a i r l y standard s t a t i s t i c a l assumptions. 122. See e.g. sections 4.2.3.2 and 4.2.4.2. 123. See table 4.14. 124. It should be pointed out at t h i s stage and for future reference that since 2SLS i s only guaranteed to have good s t a t i s t i c a l properties asymptotically small sample results are are unreliable. 124. The interdependent equations for cod and haddock solved together. 1 48 5. The Economic Submodel. We now turn our attention to the commercial aspects of the u t i l i z a t i o n of the three Icelandic demersal species whose biology was discussed in the previous chapter. Our chief objective i s to obtain numerical estimates of the economic components of the value function or discounted p r o f i t function defined in section 3.1 <1>. According to section 3.2.2, the major components of t h i s function are the processing and harvesting cost functions, a revenue function mapping catch into values of f i n a l products, a c a p i t a l function to describe the dynamics of the harvesting c a p i t a l , a c a p i t a l adjustment cost function and a discount factor. The chapter is arranged broadly as follows: In section 5.1 the general c h a r a c t e r i s t i c s of the demersal f i s h i n g industry w i l l be described. The next two sections deal with the cost functions of the processing and harvesting sectors respectively. These are followed by section 5.4 on the revenue function of the f i s h i n g industry and 5.5 on the f i s h i n g c a p i t a l dynamics. The f i n a l section provides a brief discussion on the discount factor and the appropriate rate of discount. 149 5.1 The Demersal Fishing Industry: A General Description. The u t i l i z a t i o n of demersal f i s h stocks in Iceland may, somewhat a r b i t r a r i l y , be divided into two a c t i v i t i e s ; harvesting and processing. The harvesting sector is engaged in the actual catching of demersal species. The catch i s sold to the processing sector which transforms i t into f i s h products and exports the f i n a l i z e d commodities to world markets <2>. In this section the basic structure of the harvesting and processing sectors as well as their interactions w i l l be described. The demersal f i s h i n g f l e e t comprises over 600 vessels of a s i g n i f i c a n t size <3>. In size t h i s fleet ranges from about 12 tons vessels (about 12 f t . ) to about 1000 tons vessels (over 250 ft . ) with the average size being close to 160 tons. The vessels also d i f f e r widely in design. Some are b u i l t for trawling. Others are specialized purse-seiners, g i l l - n e t t e r s or long-l i n e r s . Some of these vessels, especially the purse-seiners, only p a r t i c i p a t e in the demersal fishery during certain periods, especially the spawning season <4>. The great majority of the f l e e t , over 500 vessels, are, however, primarily engaged in demersal f i s h e r i e s throughout the year. The f i s h i n g vessels are run by fish i n g firms, most of whom are r e l a t i v e l y small, the average holding of each firm being well under 3 vessels. In terms of size, there are no dominant firms. The largest ones hold less than 4% of the t o t a l tonnage of the f l e e t . Ownership, however, is somewhat more concentrated than these numbers suggest since i t i s not uncommon for one agent to hold a c o n t r o l l i n g interest in more than one f i s h i n g f i rm. 150 The crew, or the fishermen, are generally hired labour neither sharing in the ownership of the vessel nor i t s operating costs. Their remuneration i s a combination of shares in the t o t a l value of the catch and a fixed salary, independent of the catch l e v e l . The value of the former has t r a d i t i o n a l l y constituted the greater part of the fishermen's earnings. In recent years, however, the trend has been towards increased fixed salary. The f i s h i n g f l e e t i s d i s t r i b u t e d among 60 fi s h i n g ports a l l around the island. This r e f l e c t s the fact that good f i s h i n g grounds are, broadly speaking, to be found anywhere off Iceland although the richest grounds are primarily located off the West coast. Figure 5.1 indicates the location of the f i s h i n g ports (crosses) and the demersal fi s h i n g grounds (dots). Each dot in figure 5.1 refers to the position of a specialized demersal trawler at 3 day intervals during 1975-6 <5>. This class of vessels accounted in these years for about 40% of the t o t a l tonnage of the demersal fishing f l e e t and almost 60% of the catch. Hence, with the reservation that f i s h i n g within the enclosed areas of the map was r e s t r i c t e d , figure 5.1 should provide a reasonably accurate indication as to the location and r e l a t i v e importance of the demersal fi s h i n g grounds <6>. 151 Figure 5.1 Fishing Ports ( + ) and Fishing Grounds (.) 1975-6. Most of the f i s h i n g ports are located in small v i l l a g e s <7> whose economy i s primarily based upon the f i s h e r i e s . In each of these v i l l a g e s there i s at least one f i s h processing firm. In 1977 the t o t a l number of processing firms was 134. Hence, while on average there are just over two processing firms per each fi s h i n g port many only have one. The size of the processing firms varies greatly. The processing capacity of some firms i s only a few hundred tons of demersal species a year while others are able to process over 20 thousand tons. The processing sector employs 4 basic processes; (a) frozen f i s h production, (b) s a l t f i s h production, (c) stockfish production and (d) f i s h meal and f i s h o i l production. Each of these processes i s generally operated with the help of 1 52 speci a l i z e d units of c a p i t a l which we may as well c a l l plants. The processing firms frequently operate more than one plant. Very few firms, on the other hand, operate more than 4 plants, i . e . one for each basic process. Only very rarely do processing firms run plants in more than one v i l l a g e . Reliable data on the d i s t r i b u t i o n of market power in the demersal f i s h i n g industry are not available. The above discussion, however, does not suggest s i g n i f i c a n t horizontal integration within the harvesting and processing sectors. The degree of v e r t i c a l integration, on the other hand, is an e n t i r e l y d i f f e r e n t matter. A s u p e r f i c i a l investigation revealed that in 1977 at least 60% of the t o t a l tonnage of the demersal fis h i n g f l e e t was, in fact, owned by f i s h processing firms <8>. Thus the degree of monopoly power of the processing firms r e l a t i v e to the harvesting firms i s considerable. The fact that most fi s h i n g v i l l a g e s have only one or two processing firms but more than a dozen fis h i n g vessels serves to enhance this power. The essentials of the harvesting process may be described as follows: From the fi s h i n g ports, at which they are based, the f i s h i n g vessels embark on their fishing t r i p s . Those last from about 12 hours to over 2 weeks, depending i . a . on the type and size of the vessel. Needless to say, so does the choice of f i s h i n g grounds, with the larger vessels frequently harvesting distant f i s h i n g grounds. During the fishing t r i p the vessels are not exclusively engaged in f i s h i n g . Considerable time i s spent s a i l i n g to and from the f i s h i n g grounds and searching for promising f i s h concentrations. The crew also undertakes the f i r s t stages of f i s h processing by preserving the catch and 1 53 preparing i t for subsequent processing ashore. At the end of the fish i n g t r i p the vessels normally return to their port of departure where their catch is sold to the l o c a l processing firms <9>. This does not suggest that there generally exist well developed f i s h markets in the fi s h i n g v i l l a g e s . On the contrary. Most often the catch is simply unloaded to the processing firm that either owns the vessel in question, runs the only processing plants in the v i l l a g e or with which the owner of the vessel has made an a p r i o r i landings contract <10>. During the fishing season, the price of a unit of catch i s fixed and uniform for a l l fi s h i n g ports <11>. This price i s set by a previous agreement arrived at by the representatives of the processing and the harvesting sectors, the fishermen and the Government. This agreement, which i s reached via an elaborate i n s t i t u t i o n a l procedure in which the Government plays the key role, i s designed to prevent disruption of the fis h i n g a c t i v i t y and provide simultaneously an equitable d i s t r i b u t i o n of income between the parties concerned. This agreement is p e r i o d i c a l l y (usually t h r i c e a year) revised in the l i g h t of changed economic conditions. Given these predetermined prices, the value of the catch depends only on i t s species composition, volume and quality <12>. Given the vessel, these variables depend primarily on the choice of fishery, f i s h i n g grounds, f i s h i n g gear and the length of the fishing t r i p . Consequently, these are the most important short run decision variables available to the fis h i n g f irms. The essentials of the processing process may be described as follows: The processing firms purchase a certain quantity of 1 54 catch from the harvesting firm at the predetermined unit pr i c e . This transaction i s i n s t i t u t i o n a l l y arranged in one of the following ways: (i) The vessel in question actually belongs to the processing firm. ( i i ) The processing firm has a catch purchase contract with the vessel, ( i i i ) The processing firm is the only one in the v i l l a g e and hence t r a d i t i o n a l l y obliged to accept a l l the catch supplied to i t . (iv) The processing firm simply deems i t p r o f i t a b l e to purchase catch from the vessel in question. Of these a l t e r n a t i v e s , the f i r s t two are probably the most common and the fourth least so. Having received a given quantity of catch or landings, the short run decision variables of the processing firms are largely limited to a l l o c a t i n g the landings to alternative production processes. In t h i s , however, the firms are constrained by the c h a r a c t e r i s t i c s of the catch <13> and the capacity of their plants. In fact, as many firms only run one process they have l i t t l e choice in this respect. At the end of the production process the products enter the inventory of finished products u n t i l shipped to markets. One of the primary objectives of t h i s work is to ident i f y e f f i c i e n t harvesting programs for the Icelandic demersal f i s h e r i e s . In so doing, market prices w i l l , for the most part, be taken as measuring correct shadow prices. In the l i g h t of the above discussion, the prices of demersal catch as inputs to the processing industry are an obvious exception. The way in which these prices are determined makes i t clear that they need not take much account of market forces. In fact, the evidence suggests that they have not done so for long periods of time <14>. Therefore, although our main interest is in the e f f i c i e n c y 155 of the harvesting sector, t h i s lack of a good measure of i t s output value makes i t unattractive to r e s t r i c t the analysis to this sector only. The export price of demersal f i s h products i s , on the other hand, probably quite competitive, there being many world producers and several close substitutes in consumption. Thus, the d i f f i c u l t y of not observing the correct output price for the harvesting sector, and the corresponding input price to the processing sector, i s bypassed by regarding these two sectors as one process. In fact, t h i s i s the main rationale for including the processing sector in the analysis. The central idea of thi s argument may be made s l i g h t l y more e x p l i c i t with the help of the following example <15>: Let the objective be to maximize the p r o f i t s in a harvesting sector. I.e: Max Prof(1)=p(fish)h-C(h), h where Pr o f ( l ) denotes the p r o f i t s , p(fish) stands for the true unit value of catch, h represents the harvesting l e v e l and C(h) is the harvesting cost function. The solution to thi s problem i s given by h s a t i s f y i n g the equation: (1) p ( f i s h ) = C ( h ) , where ' denotes the f i r s t derivative of the respective function. 1 56 T h i s e q u a t i o n i s n o t v e r y h e l p f u l , h o w e v e r , i f p ( f i s h ) i s u n k n o w n . T h u s , f o l l o w i n g t h e i d e a o u t l i n e d a b o v e , c o n s i d e r m a x i m i z i n g p r o f i t s i n t h e f i s h i n g i n d u s t r y a s a w h o l e . I . e : M a x P r o f ( 2 ) = p ( e x p ) Q ( h ) - C ( Q ( h ) ) - C ( h ) , h w h e r e P r o f ( 2 ) i s t h e n e w , a g g r e g a t e d p r o f i t f u n c t i o n , p ( e x p ) i s t h e e x p o r t p r i c e o f f i s h p r o d u c t s , Q ( h ) i s t h e f i s h p r o c e s s i n g p r o d u c t i o n " f u n c t i o n a n d C ( Q ( h ) ) i s t h e c o r r e s p o n d i n g c o s t f u n c t i o n . T h e p r o f i t m a x i m i z i n g l e v e l o f h c o r r e s p o n d i n g t o t h i s p r o b l e m i s g i v e n b y h * s a t i s f y i n g : (2) ( p ( e x p ) - C ' ( Q ( h * ) ) ) Q ' ( h * ) = C ( h * ) . T h u s , g i v e n k n o w l e d g e o f p ( e x p ) , t h e p r o c e s s i n g p r o d u c t i o n f u n c t i o n a n d t h e p r o c e s s i n g a n d h a r v e s t i n g c o s t f u n c t i o n s , t h e o p t i m a l h a r v e s t i n g l e v e l h * c a n b e c a l c u l a t e d . N o t i c e t h a t t h i s i s f e a s i b l e w i t h o u t a n y k n o w l e d g e , w h a t s o e v e r , o f t h e t r u e u n i t p r i c e o f c a t c h . H o w e v e r , a s a b y p r o d u c t , t h i s a p p r o a c h i m p l i c i t l y d e f i n e s t h e s h a d o w p r i c e o f c a t c h a s t h e d e r i v a t i v e : (3) 3 P r o f ( 3 ) * / 9 h = ( p ( e x p ) - C ( Q ( h * ) ) Q ' ( h * ) , w h e r e P r o f ( 3 ) * i s t h e m a x i m a l l e v e l o f p r o f i t s i n t h e p r o c e s s i n g s e c t o r d e f i n e d b y : 1 57 Prof(3)*=p(exp)Q(h*)-C(Q(h*)). Moreover, from (1) and (2) i t follows that h=h* i f f p(fish)=(p(exp)-C'(Q(h*)))Q'(h*). The solution to the harvesting maximum problem, given by (1), i s , in other words, optimal i f and only i f the unit catch price equals the shadow price defined by (3). 5.2 Processing Cost Functions. The processing sector comprises 4 basic production processes <16>. Each processing firm runs one or more of these processes. The combination of processes operated by the firms shows a d e f i n i t e pattern, however. Thus f i s h meal and o i l production i s , invariably, c a r r i e d out by single process firms. S a l t f i s h and stockfish production i s , on the other hand, generally run concurrently <17>. As regards process combinations there are 4 types of processing firms in the data <18>. (i) Fish meal and o i l production firms. ( i i ) Frozen f i s h production firms. ( i i i ) S a l t f i s h and stockfish production firms. (iv) Frozen f i s h , s a l t f i s h and stockfish production firms. 158 The supply of catch to the processing firms consists of a considerable variety of species, demersal and pelagic as well as crustaceans <19>. Moreover, the processing cost data is not distinguished according to the species processed. Hence, in estimating processing cost functions for the demersal species, we have to take appropriate account of the input of other species as well. Each process transforms the input of each species into one or more f i n a l products. The number of outputs i s thus considerably greater than the number of species. This transformation of catch inputs into outputs is considered in d e t a i l s in section 5.4 below. As far as processing cost functions are concerned, on the other hand, the transformation issue i s not central, since i t turns out that, during the sample period at least, t o t a l production costs are adequately explained by the quantity of inputs. 5.2.1 Processing Cost Functions; Theoretical Formulation. Consider a t y p i c a l processing firm. As for any other firm, i t s t o t a l costs during a period equal the value of the inputs i t consumes during the period. Write this in vector notation as: (4) C(w,x)=wx, where w i s the (1x1) vector of input prices, imputed prices i f required, and x is the (1x1) vector of inputs. I, of course, represents the t o t a l number of inputs. 159 Now, since I is usually a rather high number, (4) i s not very convenient for empirical work. However, by appealing to the firm's p r o f i t maximizing behaviour, the number of variables in (4) may be reduced. Express the p r o f i t maximization problem facing the firm by: (5) Max H(p,w,x)=pQ(x)-wx. Where H(.,.,.) i s the firm's p r o f i t function, p is a (1xJ) vector of output prices. Q(.) i s the corresponding column vector of outputs and w and x are as defined above. Usually only a small subset of the 2I+J variables in H( ) are controllable by the firm. In the p a r t i c u l a r case of the Icelandic demersal processing sector- we maintain that of these variables only some of the elements in the input vector, x, are, in fact, c ontrollable by the processing firms. F i r s t , considering the input and output price vectors, p and w, we notice that, as p r a c t i c a l l y the entire output is exported, the output prices are, e s s e n t i a l l y , world market prices for f i s h products on which neither each processing firm nor even a l l the Icelandic processing firms in unison can be expeced to have any measurable influence. The unit price of catch i s , as explained in section 5.1, set by a c o l l e c t i v e bargaining process dominated by the Government and c e r t a i n l y not influenced by the decisions of any single firm. The l e v e l of wages is also determined on a national basis by c o l l e c t i v e bargaining between the representatives of labour and c a p i t a l as 1 60 a whole on which any single processing firm has negligible influence. The remaining input prices can largely be reduced to wage costs and import prices and are consequently not responsive to the actions of individual processing firms either. Thus, as an empirical proposition, i t seems safe to assume that p and w are not endogenous to the maximization procedure of any single processing firm. Several of the elements of the input vector, x, may also be taken to be exogenous or at least predetermined. Among these are the stock variables of the firm, e.g. their l e v e l s of physical c a p i t a l etc. So are also, we would l i k e to argue, the quantity of catch inputs to the various production processes. Let us now investigate t h i s more c l o s e l y . As explained in section 5.1, a t y p i c a l processing firm, having decided to operate a certain number of i t s own f i s h i n g vessels and enter a certain number of catch purchase contracts with independent harvesting firms, has very l i t t l e scope for c o n t r o l l i n g the actual quantitiy of catch i t receives. Given these a p r i o r i decisions, the actual catch supplied to the processing firms depends on the size of the f i s h stocks, their migratory behaviour, the weather and numerous other exogenous variables. Since, secondary markets for unprocessed f i s h are, moreover, i n s i g n i f i c a n t , partly because of the geographical i s o l a t i o n of many fishing v i l l a g e s , what control the processing firms have over their quantity of f i s h inputs, is almost ent i r e l y of an a p r i o r i nature. Therefore, the quantity of catch inputs may be taken to be exogenous to the firm or at least predetermined. 161 The scope of the processing firms in a l l o c a t i n g the catch received to alternative processes i s also severly r e s t r i c t e d . F i r s t l y , many of the firms, over a t h i r d of our sample, operate only one process. Secondly, many of the multiprocess firms are geared to one favourite process, the others being primarily run to meet temporary excess supply of catch. These firms, thus, have a tendency towards a lexiographic ordering of inputs to processes. Hence, to the extent t h i s holds, the a l l o c a t i o n of catch to processes i s as exogenous as the t o t a l supply of catch. Thirdly, a l l o c a t i o n of inputs to the various processes i s also r e s t r i c t e d by the quality of the catch, with the highest quality generally allocated to freezing and the lower q u a l i t i e s to s a l t f i s h , stockfish and f i s h meal and o i l production, in that order. On the basis of this discussion we conclude that, as a matter of empirical fact, the price vectors, w and p, in (5) and some of the elements of the vector x, namely the inputs of catch to processes, are not controllable by the t y p i c a l Icelandic demersal processing firm, at least not in the short run <20>. Now, p a r t i t i o n the vector x as follows: X=/X(1)\, where X(1) represents the (I(1)x1) vector of noncontrollable inputs and £(2) the (l ( 2 ) x l ) vector of controllable ones <21> and rewrite the p r o f i t maximization problem, (5), as: 162 (6) Max H(p,w,x)=pQ(x)-wx. x(2) The solution to this problem, i f i t exis t s , is given by: x*(2)=X(p,w,x(1)). ~ ~ ~ ~ ~ And the processing cost function may thus be rewritten as: id) lz) L w(i)x(i)+ L w(i)X(i;p,w,x(1)), <--•> i--Mi) + L " ~ ~ where X(i,•.,.,.) represents the i - t h element of the vector X(p,w,x(1)). For later reference i t i s useful to note the following properties of (7): (i) It i s homogenous of degree one in p and w <22>. ( i i ) It i s additive in processes. As the inputs may, at least in theory, be named according to the process they are used i n , this property follows immediately from the form of (7). ( i i i ) It i s not, in general, linear in the elements of x^ ( 1 ) . As the factor demand functions, X(p,w,x(l)), are -~ ~ ~ arguments in (7), thi s result i s again immediate. (7) C(p,w,x(1) )=w / x(1) x*(2), 1 63 5.2.2 Processing Cost Functions: Empirical Estimation. Our aim, in this section, i s to determine the functional form of the cost function defined by (7) above and obtain e f f i c i e n t estimates of i t s parameters. This inference must be based on the available quantitative information about the operations of the fishing industry in previous years <23> as well as the general q u a l i t a t i v e information provided in previous sect ions. The Icelandic processing sector comprises four basic processes; f i s h meal and o i l , frozen f i s h , s a l t f i s h and stockfish production. In terms of these processes there are observations on 4 types of processing firms: (i) 23 specialized f i s h meal and o i l production firms, ( i i ) 18 specialized freezing firms, ( i i i ) 26 specialized s a l t f i s h and stockfish producing firms and (iv) 41 frozen f i s h , s a l t f i s h and stockfish producing, or multiprocess, firms. As far as estimation is concerned, therefore, there seem to be two alte r n a t i v e s . One i s to emphasize processes rather than firms and use the available data to estimate one cost function for a l l types of firms, irrespective of whether they are specialized or multiprocess firms. The alternative is to emphasize firm types and estimate a cost function for each basic type. Compared with the f i r s t a l t e r n a t i v e , the advantage of th i s disaggregated procedure i s that i t may highlight technological differences between the dif f e r e n t types of firms <24>. The cost, on the other hand, l i e s in the potential loss of precision in parameter estimates due to reduced sample sizes <25>. The estimation w i l l be carr i e d out in two stages. In the 1 64 f i r s t stage, we w i l l attempt to determine the functional form of the factor demand functions defined in the previous section. In the second stage, we w i l l use the appropriate s t a t i s t i c a l techniques to estimate the parameters of the processing cost functions given the functional form arrived at in the f i r s t stage <26>. The empirical data contain annual observations on the operating costs and consumption of a few inputs for a l l the 108 processing firms in the sample during 1974-76 as well as input and output price indices in these years <27>. The observed inputs consist of the volume of catch allocated to each process. Observations on other noncontrollable inputs, such as c a p i t a l , are not available. This combination of time series and cross section data i s c a l l e d panel data <28>. Now, given the nature of the data, write the stochastic version of (7) as: id) (8) c(s,t)= Z w(t)x(i,s,t)+ Z w(t)X(i;p(t),w(t),x(s,t))+u(s,t), a l l s,t. Where c(s,t) denotes the t o t a l costs of firm s in year t. w(t) is the input price and p(t) the output price index in year t. x( i , s , t ) represents the consumption of input i by firm s in year t. :x(s,t) i s the corresponding (1x1(1)) vector. X(i;.,.,.) is the factor demand function for input i and u(s,t) i s the stochastic disturbance term for firm s in year t <29>. Now, as mentioned in the previous section, the factor 165 demand functions are possibly nonlinear in their arguments. To explore t h i s p o s s i b i l i t y we approximate (8) by the following power transformation <30>: (9) c(s,t)/w(t)= I a ( i ) x ( i , s , t ; q ( i ) )+bw(t;r1 )+dp( t; r.2)+u(s,.t) . Where the a( i ) ' s, b and d are c o e f f i c i e n t s , x(0,.,.;.) = 1 and the q ( i ) ' s , r1 and r2 indicate the Box-Tidwell power transformations of the respective variables defined by: y (.; z ) = (y ( . f - 1 ) /z . As explained in appendix 5.2-C, the values of the Box-Tidwell transformation parameters indicate the appropriate functional form of the factor demand functions r e l a t i v e to the data. In pa r t i c u l a r , a unitary estimate of these parameters suggests that the factor demand functions are, in fact, linear <31> and the respective variable should appear l i n e a r i l y in (8). Using the available data, the parameters of (9) may now be estimated. As, in t h i s preliminary stage, we are primarily interested in obtaining unbiased estimates that may allow us to r e s t r i c t the set of possible functional forms for the factor demand functions we w i l l not be overly anxious about the structure of the variance-covariance matrix of the u(s,t)'s. Estimating the t o t a l cost function for each type of firm separately and including an annual dummy variable to pick up the year s p e c i f i c disturbances <32>, makes i t reasonable to assume that the resulting error term i s s u f f i c i e n t l y close to being 166 white noise for our present purposes. Our aim is to test the following two n u l l hypotheses: I. r1=r2=q(i)=1, a l l i . I I . b=d=0. The results were as follows: Table 5.1 Processing Cost Functions: Tests of Functional Forms. Likelihood Ratio Tests Firms Hypothesis I* Hypothesis II Fish meal + O i l firms: X(3)=3.70 £(2)=3.87 Freezing firms: Xz( 4 ) = 1 0 . 84 2 l ( 2 ) = 1.55 S a l t f i s h + Stockfish: X1 (2)= 2.12 ** Multi-process firms: 6) = 1 5.46 ** * Due to numerical d i f f i c u l t i e s i t was not not possible to estimate the Box-Tidwell transformation parameters for a l l the respective variables, especially not those less s i g n i -f i c a n t , in each case. The actual number estimated is i n d i -cated by the respective degrees of freedom. ** The test s t a t i s t i c could not be calculated due to numerical d i f f i c u l t i e s . According to the results in table 5.1, the n u l l hypotheses that the processing cost functions are linear in the inputs are generally not rejected by the data <33>. Moreover, in the cases where the appropriate test s t a t i s t i c could be calculated, the input and output price indices were not found to be s i g n i f i c a n t . Thus, adopting the linear s p e c i f i c a t i o n of the processing cost functions, we now proceed to obtain e f f i c i e n t estimates of i t s parameters. For this purpose write a l l the observations on the cost function as follows: 167 (10) c/w=Xb+u. Where c/w i s a (3Sx1) vector of observations on c(s,t)/w(t). S i s , as before, the number of p a r t i c u l a r type of firms in the sample and 3 represents the number of years in the sample. X i s a matrix of observations on the explanatory variables, x ( i , s , t ) , including a constant term, b is a vector of unknown parameters and u i s a (3Sx1) vector of stochastic disturbances. We assume the following d i s t r i b u t i o n for u: (11) u~N(0,U) , where the structure of the (3Sx3S) covariance matrix, U, i s defined in appendix 5.2-A. For the s p e c i f i c a t i o n described by (10) and (11), i t i s well known <34> that the maximum l i k e l i h o o d estimator for b i s defined by the so-called generalized least squares estimator, GLS, as follows: (12) b=(X'U-' X)"1 X'U-'c/w. However, the parameters of U are not known. Hence the GLS estimator i s not f e a s i b l e . The problem, however, can be solved by simultaneously choosing values of b and the unknown parameters of U so as to maximize the corresponding l i k e l i h o o d function <35>. As before, we estimated processing cost functions for each class of firms seperately. The main results were as follows: 1 68 (1) Fish Meal and O i l Producing Firms. Number of firms in the sample: 23 <36>. Number of years: 3 Total number of observations: 69. Form of estimated cost function: c/w=a(0)+a(1)y(1)+a(2)y(2)+a(3)y(3)+u, where c/w i s t o t a l costs divided by the input price index. a ( i ) , i = 0, 1,2,3 denote parameters. y( l ) i s the input of demersal species excluding redfish. y(2) i s the input of redfish. y(3) i s the input of pelagic species. u i s the stochastic disturbance term. The most important s t a t i s t i c a l results are reported in table 5.2 169 Table 5.2 Fish Meal and O i l Processing Cost Function: Estimation Results Est imated coef f ic ients a(0)=4.26 a( 1 ) = . 0018 a(2)=.0046 a(3)=.0025 R-square=0.95 s t a t i s t i c s E l a s t i c i t y at means 2.4 10.8 6.4 33.4 0.21 0.06 0.64 Total: 0.91 Tests for constancy of parameters over time F(2,57)* 7.5 11.8 30.5 5.4 X (8)=75.1** * On HO: parameters are constant over time <37> ** A l i k e l i h o o d r a t i o test on HO: A l l the parameters are constant over time <38>, On the basis of these res u l t s , we conclude that the estimation procedure seems to have produced reasonably well determined estimates of the c o e f f i c i e n t s , as judged by the t-s t a t i s t i c s . Moreover, since the estimation procedure i s , on the assumptions made, equivalent to the GLS technique, we may be j u s t i f i e d in having considerable f a i t h in these s t a t i s t i c s . The results are not en t i r e l y s a t i s f a c t o r y , however. The F-tests as well as the chi-square test on the constancy of the parameters over time suggest that the n u l l hypothesis should, in a l l instances, be rejected. This result may indicate that the input price index does not f u l l y account for the price changes over time. An alternative explanation is that, during the sample period, some technological and structural changes took place in the f i s h meal and o i l industry. In fact, there is some qu a l i t a t i v e evidence of this in connection with the capelin 1 70 summer fishery. (2) Specialized Freezing Plants. Number of plants: 18. Number of years: 3. Total number of observations: 54. Form of the estimated cost function: c/w=b(0)+b(1 )x(1)+b(2)x(2)+b(3)x(3)+b(4)x(4)+u, where b(i),i=0,1,2,3,4 denote parameters. X(1) is inputs of demersal species excluding f l a t f i s h . x(2) is inputs of f l a t f i s h . x(3) is inputs of pelagic species. x(4) is inputs of crustaceans. The other variables are as defined above. The most important s t a t i s t i c a l results are l i s t e d in table 5.3. 171 Table 5.3 Specialized Freezing Processing Cost Functions: Estimation Results. Estimated coef f ic ients b(0)=9.1 b(1)=.0185 b(2)=.029 b(3) = .0l 19 b(4)=.0723 R-square=0.92 s t a t i s t i c s E l a s t i c i t y at means 1 .6 21.3 0.2 1 .7 3.1 0.89 .003 0.04 0.03 Total 0.963 Tests for constancy of parameters over time F(2, 39)* 2.3 11.9 5.0 5.9 12.4 Pi( 10) = 59.2** * On HO: parameters are constant over time. ** A l i k e l i h o o d r a t i o test on HO: A l l the parameters are constant over time. These estimation results seem f a i r . For panel data, the o v e r a l l f i t and the t - s t a t i s t i c s are reasonably good. As in the case of the f i s h meal and o i l production, however, the n u l l hypothesis of constancy of the estimated parameters over time are in most cases rejected although not as conclusively <39>. For possible explanations of t h i s apparent v a r i a b i l i t y of the parameters over time, the reader is referred to the corresponding speculations concerning the f i s h meal and o i l processing firms above. 1 72 (3) S a l t f i s h and Stockfish Producing Plants. Number of plants in the sample: 26 Number of years: 3. Total number of observations: 78. Form of the estimated cost function: c/w=b(0)+b(5)x(5)+b(6)x(6)+b(7)x(7)+u, where b(0), b ( i ) , i=5,6,7, denotes parameters, x(5) i s the input of demersal species to the s a l t f i s h process. x(6) i s the input of demersal species to the stockfish process. x(7) i s the input of pelagic species to the s a l t f i s h process. The other variables are as defined above. The most important s t a t i s t i c a l results are l i s t e d in table 5.4: 173 Table 5.4 S a l t f i s h and Stockfish Cost Functions: Estimation Results. Estimated coef f ic ients b ( 0 ) = 3 . 9 l b ( 5 ) = . 0 l 0 9 b ( 6 ) = . 0 l 4 b ( 7 ) = . 0 l 5 5 R-square=0.90 s t a t i s t i c s E l a s t i c i t y at means 3.4 1 5 . 8 10 .4 4 . 5 0 . 6 6 0 .12 . 004 Total: 0 . 7 8 4 Tests for constancy of parameters over time F ( 2 , 6 6 ) * 0 .2 2 .7 8 .4 0 .5 / £ ( 8 ) = 1 5 . 3 4 * * * On HO: parameters are constant over time. ** A l i k e l i h o o d r a t i o test on HO: A l l the parameters are constant over time. These results seem good. The f i t i s good. The estimated parameters seem well determined and the n u l l hypothesis of the constancy of parameters over time i s generally not rejected. (4) Multi-process Plants. Number of plants: 4 1 . Number of years: 3. Total number of observations: 1 2 3 . The form of the estimated cost function i s : 174 c/w=b(0)+b(1 )x(D+b(2 )x(2 )+b(3 )x(3 )+b(4 )x(4 )+b(5 )x(5) +b(6)x(6)+b(7)x(7)+u, where a l l the variables and c o e f f i c i e n t s are as previously defined. The most important s t a t i s t i c a l results are l i s t e d in table 5.5 below. Table 5.5 Multi-process Plants: Estimation Results. Tests for constancy Est imated t E l a s t i c i t y over time c o e f f i c i e n t s s t a t i s t i c s at means F(2,99)* b(0)=3.5 1 .4 ' 0.0 b(1)=.0178 23.4 0. 57 1 . 1 b(2)=.0625 4.8 0.06 0.6 b(3)=.0325 8.0 0.05 0.3 b(4)=.0213 5.3 0.02 4.2 b(5)=.0l61 10.7 0.16 14.7 b(6)=.0314 8.1 0.04 1 .8 b(7)=.0444 5.5 0.03 0.1 Total : 0.93 X\16)=54.9** R-square=0.95 * On HO: parameters are constant over time. ** A l i k e l i h o o d r a t i o test on HO: A l l the parameters are constant over time. Again these results seem sa t i s f a c t o r y . The f i t and significance of the estimated c o e f f i c i e n t s are good for panel data. There are indications, however, of nonconstancy of some of the parameters over time, especially the marginal cost of s a l t f i s h production. 175 (5) Frozen Fish, S a l t f i s h and Stockfish Production. We report here on the results of estimating one aggregate cost function for a l l the processing firms operating one or more of these processes. Number of plants in the sample: 85. Number of years: 3. Total number of observations: 255. The form of the estimated cost function i s : c/w=b(0)+b( 1)x(1)+b(2)x(2)+b(3)x(3)+b(4)x(4)+.b(5)x(5) + +b(6)x(6)+b(7)x(7)+u, where a l l the variables and c o e f f i c i e n t s are as defined above. The most important s t a t i s t i c a l results were as follows: 176 Table 5.6 Aggregat ive Processing Cost Funct ion: Estimation Results. Tests for constancy of parameters Est imated t E l a s t i c i t y over time coef f ic ients s t a t i s t i c s at means F(2,231)* b(0)=3.31 1 .7 0.2 b( 1 ) = . 0183 36.0 0.62 3.6 b(2)=.0463 4.2 0.04 1 .8 b(3)=.0271 8.2 0.04 12.6 b(4)=.0233 4.9 0.02 0.9 b(5)=.0l42 12.6 0.17 1 .2 b(6)=.0l30 4.5 0.02 1 .4 b(7)=.0242 4.0 0.01 0.1 Tota l : 0.92 £ l(16)=88.8** R-square=0.94 * On HO: parameters,are constant over time. ** A l i k e l i h o o d r a t i o test on HO: A l l the parameters are constant over time. The s t a t i s t i c a l properties of these estimates are generally good. The estimated equation has a reasonably high explanatory power and the estimated c o e f f i c i e n t s are well determined. The n u l l hypothesis of constancy of parameters over time are generally not rejected. Only the constancy of the marginal cost of freezing pelagic inputs is rejected on the 1% significance l e v e l . This finding, however, may be explained by technological changes that took place in capelin freezing during the sample period. Since the aggregated cost function i s , for reasons of s i m p l i c i t y , a t t r a c t i v e from the point of view of model construction, i t i s of considerable interest to check whether the implied r e s t r i c t i o n s on the parameters of the disaggregated 1 77 cost functions, i . e . (2), (3) and (4) above, are rejected by the data. The relevant test s t a t i s t i c were: F(17,231 ) =4 . 11 and X2(17)=65.3 <40>. Hence the aggregation r e s t r i c t i o n s are rejected. 178 Appendix 5.2-A The Error Components Approach to Regression. Although, given correct factor demand functions, the cost function defined by (7) above, i s r e a l l y an ide n t i t y , there are several reasons why this expression w i l l not f i t any given data precisely. More to the point, i t i s of considerable s t a t i s t i c a l importance, as well as economic interest, to ident i f y the factors that might be responsible for these deviations or disturbances as they are often c a l l e d . Assume, as is in fact the case here, that the available data are panel data. Given th i s and comparing firms in the same time period, or cross-sectionally, the following causes of disturbances may be suggested: (i) The quantity of noncontrollable inputs, e.g. c a p i t a l on which we do not have data, may differ.among the firms. ( i i ) The factor demand functions may d i f f e r among the firms due to technological differences not embodied in physical c a p i t a l . ( i i i ) Prices may d i f f e r between firms, e.g. according to location <41>. To the e f f e c t that these factors are operative they give r i s e to firm-specific disturbances in cost functions f i t t e d to the data. C a l l this disturbance term e(s), s=1,2,...,S, where S is the t o t a l number of firms in the sample. Over time disturbances may be created i . a . by: (i) Changes in the technology of the firms. ( i i ) Changes in unobserved r e s t r i c t e d inputs e.g. c a p i t a l . Denote these disturbances by v ( t ) , t=1,2,..T. F i n a l l y , we may assume that there are some disturbances, whose causes we do not attempt to i d e n t i f y , that vary both over time and between firms. Denote these by z ( s , t ) , a l l t and s. This discussion brings us to the following stochastic s p e c i f i c a t i o n of processing cost functions: (A.1) c(s,t)=C(p,w,x(1);s,t)+u(s,t), a l l s,t. Where s i s , as before, an index for the firms in the sample, t is a time index, x(1) refers to the vector of observed inputs and p and w represent the price indices of inputs and outputs respectively. F i n a l l y , u(s,t) i s a stochastic disturbance term defined by: (A.2) u(s,t)=e(s)+v(t)+z(s,t), a l l s,t, where e(s), v(t) and z(s,t) represent cross-sectional, time related and mixed disturbances as discussed above. 1 79 The formulation defined by (A.1) and (A.2) takes us into the error components or random c o e f f i c i e n t s framework in the econometric l i t e r a t u r e <42>. In terms of random c o e f f i c i e n t s (A.2) corresponds exactly to the case of a constant term that varies both over time and across firms <43>. Arranging the data on the firms in column vectors for each year and stacking these vectors, we may write a t y p i c a l cost function for a l l available observations as: (A.3) c=C(w,p,X)+u. Where X i s a matrix of observations on the input variables and the ~ disturbance vector, u, i s defined by u'=(u(1)',u(2)',...,u(T)'), where ' denotes a transpose and each subvector Is defined" by u (t) ' = (u (1 , t) , u ( 2, t) , .,u(S,t). A l l the other vectors are correspondingly dimensioned. We assume that the d i s t r i b u t i o n of u i s given by: (A.4) u~N(0,U), where the variance-covariance matrix U i s defined by: (A.5) U= E u( 1 )• u( 1 ) ' u(2)-u(1 ) ' u(3).u(1 ) ' r X-u(1) u(2) ' u(2)•u(2)' u(3) u(2) ' u( 1 )• u(3) ' u(2). u(3) ' u(3)•u(3)' Where E denotes the expectation operator and ' denotes a transpose as before. In order to make the number of parameters in this TSxTS variance-covariance matrix manageable, we make the following simplifying asumptions <44>: (i) E ( u ( s , t ) u ( s , t ) ) = o u ( t ) , a l l s, t. I.e. diagonal disturbance covariances are constant in every .year but allowed to vary between years, thus, perhaps, r e f l e c t i n g convergence or divergence in the costs of the firms over time r e l a t i v e to those "predicted" by the estimated cost functions. ( i i ) E ( u ( s , t ) u ( r , t ) ) = o y ( t ) , s not equal to r, a l l t. This covariance measures the extent to which deviations in observed costs of firms from those "predicted" by the estimated cost function are common across firms thus, perhaps, r e f l e c t i n g the effects of "good" or "bad" operating years. Again t h i s period-specific covariance i s assumed to be constant in each time period but allowed to vary over time. ( i i i ) E(u(s,t)u(s,1)=o e(t,t-1), a l l s, 1 not equal to t. This covariance, presumably r e f l e c t i n g f i r m - s p e c i f i c disturbances that per s i s t over time, e.g. their r e l a t i v e 180 e f f i c i e n c y , i s thus assumed to depend on time, t, and the time difference, t-1, but to remain constant otherwise. (iv) E(u(s,t)u(j,r))=0, a l l s,j and a l l t,r not equal to each other. This means, in other words, that the covariances between the disturbances associated with d i f f e r e n t firms in d i f f e r e n t years are zero. According to these assumptions, the complete variance-covariance matrix of the u(s,t)'s for T=3 i s : (A.6) U= Q(D , a e d , D l , o e ( 2 , 2 ) I a eri,1)I , Q(2) , a e ( 2 , 1 ) I a e ( 2 , 2 ) l , o e T 2 , l ) l , Q(3) Where Q_(t) i s a (SxS) symmetrix matrix with diagonal terms equal to Oo.(t) and off-diagonal terms equal to o v ( t ) . J. is the (SxS) identity matrix. U, thus, contains 9 unknown parameters, namely o^O), 0^(2), a~(3), a v ( l ) , a v(2) , a v ( 3 ) , a e( 1 , 1 ) , a e ( 2 , 1 ) and a e ( 2 , 2 ) . Now, for reasonably large sample sizes, these covariance parameters can be estimated j o i n t l y with the functional parameters in order to improve the e f f i c i e n c y of the l a t t e r . As an example consider the following linear cost function: (A.7) c=Xb+u, where X i s the observation matrix, b the vector of c o e f f i c i e n t s to be~estimated and u^a disturbance vector defined by (A.4) and (A.6) above. Assuming that other c l a s s i c a l regression assumptions hold <45>, the maximum li k e l i h o o d estimator for b is given by the well known GLS estimator <46>. ~ Appendix 5.2-B. Test S t a t i s t i c s . Let the following equation represent observations on I explanatory variables over indices s and t. The s and t indices could e.g. refer to cross-sectional and time series observations respectively. A l t e r n a t i v e l y s could refer to the observations on firms of type s and t to di f f e r e n t firm types. (B.1) y(t)=X(t)b(t)+u(t), a l l t, ~ ~> ~ where y(t) i s the (Sxl) vector of observations on the endogenous variable, X,(t) i s the (Sxl) matrix of observations on the explanatory~variables, b(t) is the (1x1 ) vector of unknown c o e f f i c i e n t s and u(tT is the (Sx1) vector of stochastic disturbance terms. ~ 181 Assume that the index t goes from 1 to T and write a l l the ST observations as follows: (B.2) y=Xb+u, where y'=(y(1)',y(2)',...,y(T)'), where ' denotes a transpose, and the~cther matrices are stacked correspondingly. Now, tests of lin e a r r e s t r i c t i o n s on the b'=(b(1)',b(2)',..,b(T)') vector can be carried out within the c l a s s i c a l framework <47>. In f i n i t e samples, the resulting test s t a t i s t i c i s di s t r i b u t e d as F(r,ST-IT), where r i s the number of linear r e s t r i c t i o n s . An a l t e r n a t i v e large sample test i s the l i k e l i h o o d r a t i o test: X 2(r)=2(L(b(unr)-L(b(res))) , where L(.) denote the maximum values of the logrithm of the relevant l i k e l i h o o d function for unrestricted and r e s t r i c t e d estimates of the b vector respectively <48>. Clearly, i f s and t refer to cross-sectional and time series observations respectively, the above s t a t i s t i c s can be used to test whether the b(t) vectors are i d e n t i c a l , i . e . for constancy of the parameters over time. Al t e r n a t i v e l y , i f these indices refer to firms and firm types respectively, the s t a t i s t i c s can be used to test for aggregation over firm types. In that case too the n u l l hypothesis i s : H0:b(1)=b(2)=..,=b(T). Appendix 5.2-C Box-Tidwell Transformation. Consider the following transformation of an arbitr a r y variable x: (C.1) x ( q ) = ( x 9 - 1 ) / q . Clearly as q approaches zero x(q) converges to ln(x ) , and X(1)=X-1. Other values of q correspond to other less standard transformations of x. In recognizion of the o r i g i n a l contribution by Box, Tidwell and Cox <49>, the parameter, q, i s often referred to as the Box-Tidwell or Box-Cox transformation parameter. Now consider the regression equation: (C.2) y=Y(x(1),x(2), ,x(l))+u. where y and the x ( i ) ' s represent observations on the dependent and independent variables respectively and u represents a 182 stochastic disturbance term. Now, (C.2) may be approximated by: 1-1 (C.3) y=a(0)+ La(i)x(i,q(i))+u. i This representation of (C.2) is c a l l e d the Box-Tidwell transformation. It i s f l e x i b l e in that i t provides a good approximation to the function Y( ) in the neighbourhood of a given point. Moreover, assuming that the relevant l i k e l i h o o d function can be maximized with respect to the q ( i ) ' s , the data may be allowed to choose the maximum l i k e l i h o o d functional form. On the assumption that u is white noise, i . e : U~N(0,CTl), the joint maximum l i k e l i h o o d estimator of the a ( i ) ' s and the q ( i ) ' s in (C.3) is given by: z-i (C.4) Min l { y ( t ) - a ( 0 ) - 2 a ( i ) x ( i , q ( i ) , t ) } 2 , t <• where t i s an index for observation t. Frequently, i t i s of interest to impose r e s t r i c t i o n s on some or a l l of the transformation parameters. Tests on such r e s t r i c t i o n s may be constructed by ca l c u l a t i n g the corresponding l i k e l i h o o d r a t i o s <50>. 183 5.3 Harvesting Cost Functions. A general description of the fishing industry as a whole and the harvesting sector in p a r t i c u l a r was provided in section 5.1. In t h i s section we concentrate on the formulation and estimation of cost functions for the harvesting sector. The harvesting sector is composed of a great number of firms. The operating decisions of these firms generally involve both decisions concerning the level s of the firms' c a p i t a l and the employment of the existing c a p i t a l <51>. The f i r s t category of decisions, which may be referred to as long run decisions, includes i . a . investment in f i s h i n g vessels, instruments and gear <52>. The second category includes t y p i c a l short run decisions such as what fishery to pursue, with what gear, for what duration etc. These, as any other operating decisions, normally result in both costs and benefits to the firm. The benefits w i l l be considered in the next section. Here the focus is on the costs. More precisely, our objective, in this section, is to estimate functions r e l a t i n g t y p i c a l short run harvesting decisions and, as the case may be, other variables to t o t a l costs. 5.3.1 Harvesting Cost Functions; Theoretical Formulation. In c l a s s i c a l economic theory cost functions are formulated in terms of prices and, generally, either inputs or outputs <53>. About th i s type of cost functions and their r e l a t i o n with the underlying production technology, considerable knowledge has been accumulated <54>, some of which was u t i l i z e d in our 184 examination of the processing cost functions above. In f i s h e r i e s and, in fact, other natural resource extraction industries, the existence of a scarce non-producible input, i . e . the resource stock, requires modification of the c l a s s i c a l theory of cost functions. We w i l l now b r i e f l y outline the derivation of appropriately s p e c i f i e d cost functions for the Icelandic demersal harvesting sector <55>. Given that the number of harvesting firms in the Icelandic demersal f i s h e r i e s , exploiting a handful of common property f i s h stocks, exceeds 200, i t seems safe to assume that t h i s fishery is highly competitive with respect to the resource base <56>. Hence, the short run behaviour of the fishing vessels can be regarded as the solution to the following problem <57>: (13) Max H= Lp(j)Y(j,e(j),k,x(j))-rk'-wZe(j)'. j ~ ~ ~ ~ ~ ~-a l l e(j) Subject to: e=Le(j). Where Y(j,e,k,x) i s a production function for species j . e(j) i s a (1x1) vector of variable inputs directed at species j . The sum of e(j) equals e, the t o t a l use of variable inputs. This i s expressed as a constraint in (13) because, in general, some of the inputs, i . a . fish i n g time, may be bounded above, k i s a (1XH) vector of fixed inputs e.g. c a p i t a l . - . x(j) represents the stock l e v e l of species j as i t appears to the vessel. This variable w i l l usually be referred to as the vessel s p e c i f i c f i s h stock below. p(j) i s the market price of a unit of 185 catch of species j and r and w are vectors of fixed and variable input prices, respectively. ' denotes a transpose. The l e v e l of f i s h stocks i s , in many respects, the central variable of thi s formulation. Its presence in the production function r e f l e c t s the fundamental assumption that harvesting depends, c e t e r i s paribus, upon the quantity of a nonproducible input, i . e . the resource stock <58>. Notice that, in thi s micro-formulation, x(j) denotes the resource stock as i t appears to a given vessel with certain operating c h a r a c t e r i s t i c s in a particular location. Thus x(j) i s not, in this case, the l e v e l of the resource as a whole, although probably closely correlated with i t . It should also be noticed that (13) involves the important assumption that d i f f e r e n t f i s h e r i e s can be pursued separately. This, of course, may not be true in any pa r t i c u l a r case. If fi s h e r i e s are not completely separable, i.e the catch of one species depends on the fish i n g a c t i v i t y directed at another, the problem becomes one of joint production which i s considerably more complex, although by no means intractable. In the case of the Icelandic demersal f i s h e r i e s , however, the available q u a l i t a t i v e information indicates that the cod, haddock and saithe f i s h e r i e s are, in fact, largely separable. The solution to (13), i f i t ex i s t s , includes the factor demand functions: (14) e(j)*=E(j,p,w,k,x), a l l j , r*~> *«v ^ #"W ^ where x is now a vector of the x( j ) ' s and p of the p ( j ) ' s . 1 86 Al t e r n a t i v e l y and equivalently <59>: (15) e(j)*=E(j,w,k,x,y), a l l j , where y i s the (1XJ) vector of catch volumes from the d i f f e r e n t species and E( ) i s , of course, not the same function as E( ) in (14). From (14) we otain the harvesting cost function: (16) c= Ir(h)k(h)+Zw(i)ZE(i,j;p,w,k,x)=C(p,r,w,k,x). h i j ~ ~ ~ ~ ~ ~ ~ ~ ~ This function d i f f e r s from c l a s s i c a l cost functions in only one important respect. It includes a nonproduced factor, the resource stock, among i t s arguments. Notice that this harvesting cost function is additive in f i s h e r i e s . Moreover, as i s easy to show, i t i s homogeneous of degree 1 in a l l the prices, i . e . r, w and p <60>. The factor demand functions can, as usual, be derived from knowledge of the production technology and vice versa. However, due to the presence of the resource stock in the production function, this duality r e l a t i o n is l i k e l y to be somewhat more complicated than in c l a s s i c a l production theory. The cost function, (16), applies to a single vessel. One d i f f i c u l t y in applying t h i s expression to empirical work i s that, while estimates of the t o t a l resource stock are often available, data on v e s s e l - s p e c i f i c resource stocks, i . e . the vector x, are generally not so. However, on seemingly un r e s t r i c t i v e assumptions <61>, at least from an empirical point 187 of view, (15) i m p l i c i t l y defines the following estimator for x: (17) x=X(w,k,y,e*). Where e* is now a vector of the e ( j ) * ' s and the elements of x, y^, x(s,j) say, represent the resource stock of species j as i t appears to a certain fishing vessel employing fishing input e ( s , j ) , i . e . f i s h i n g input s directed at species j . As discussed in more d e t a i l s in appendix 5.3-B, operating decisions in various l i n e s of production may be conceived of as consisting of (i) choice of a c t i v i t i e s (including their levels) and ( i i ) choice of combinations of inputs for each a c t i v i t y . Given a choice of the f i r s t kind, the l a t t e r i s frequently severely r e s t r i c t e d by the technology and i n s t i t u t i o n a l environment of the firm. While, in the case of the Icelandic demersal harvesting sector, decisions on harvesting a c t i v i t i e s <62> are taken very frequently, the combination of inputs corresponding to each a c t i v i t y i s , due to technological and i n s t i t u t i o n a l constraints, rather i n f l e x i b l e <63>. To thi s extent, the a c t i v i t y decisions imply a certain consumption of inputs, at least in the short run. To formalize these ideas, write the vector e as: (18) e=(e(1),e(2)) . Where e(l) refers to the operating a c t i v i t i e s and e(2) to the economic inputs. Moreover, assuming that a c t i v i t i e s , in fact, determine inputs: 188 (19) e(2)=F(e(1),k), where the technological and i n s t i t u t i o n a l constraints are refle c t e d in the form of the functions F( ) and the c a p i t a l variable k. Thus, on these arguments, the factor demand functions and, consequently, the estimator of the v e s s e l - s p e c i f i c resource stock <64>, may be written in terms of a c t i v i t i e s and not inputs. Thus, proceeding on thi s basis and substituting (17) into (16) we arrive at a harvesting cost function in observed variables only: (20) c= Lr(h)k(h)+Iw(i)ZE(i,j;p,w,k,X(w,k,y,e(1)*,F(e(1)*,k)) ) =C(p,r,w,k,y,e(1)*). Where, as usual, ~ denotes appropriately dimensioned vectors. 5.3.2 Harvesting Cost Functions: Empirical Estimation. The empirical problem i s to obtain "good" estimates of the harvesting cost function, (20), with the help of the available data. The available data consist of a combination of cross section and annual time series observations on the the fish i n g vessels during 1974-77 <65>. The data cover observations on input and output prices as well as a number of c a p i t a l variables, catch l e v e l s and the 189 yearly duration of a few f i s h i n g a c t i v i t i e s for each fi s h i n g vessel in the sample. The price data consist of six Laspeyre's price indices; (a) one for the inputs and (b) one landings price index for each of 5 categories of demersal catch, i . e . cod, haddock, saithe, redfish and other species. The c a p i t a l data are composed of six physical c a p i t a l measures on the vessels, namely their type (2 v a r i a b l e s ) , size, engine power, age and number of electronic instruments. In addition there are observations on the location of the home port of the vessels which, from a formal point of view, may also be regarded as a c a p i t a l variable. The catch data i s disaggregated into the yearly catch of cod, haddock, saithe, redfish and other demersal species. The f i s h i n g a c t i v i t y observations are the t o t a l yearly operating time employing one or more of fiv e d i f f e r e n t types of gear, i . e . trawl, g i l l n e t s , longline, hand l i n e and purse seine. In conformance with t r a d i t i o n a l terminology in f i s h e r i e s economics, we w i l l sometimes find i t convenient to refer to these operating variables as f i s h i n g e f f o r t below. Other important f i s h i n g a c t i v i t i e s , in the context of the Icelandic demersal f i s h e r i e s , concern i . a . the actual fishery pursued <66>, the fi s h i n g grounds chosen etc. Unfortunately, data on these variables are not available. With regard to the appropriate s p e c i f i c a t i o n of ( 2 0 ) , however, t h i s may not be as damaging as i t may seem. In the Icelandic demersal f i s h e r i e s , the p a r t i c u l a r fishery pursued i s strongly correlated with the vessel type and the fishing gear employed. S i m i l a r i l y , the choice of f i s h i n g grounds is to a s i g n i f i c a n t extent dictated by the physical c h a r a c t e r i s t i c s of the vessel and the location of 190 i t s home port on which there are observations. F i n a l l y , there is l i t t l e a p r i o r i reason to expect t o t a l costs to be p a r t i c u l a r l y sensitive to which of the three demersal species considered i s being pursued. Thus, in terms of the available data, an empirical version of (20) may be written as: (21) c=C(p,w,k,y,e)+u. Where c denotes harvesting costs as before, w represents the input price index and p the (1x5) vector of the output price indices contained in the data, k i s the (1x6) vector of c a p i t a l c h a r a c t e r i s t i c s and y the (1x3) vector of catch volumes as spe c i f i e d above. e i s the (1x5) vector of the fi s h i n g e f f o r t variables also as sp e c i f i e d above. The c a p i t a l price vector, r, on which there are no separate observations, has been dropped from the equation. The elements of r, however, are contained in w and, in estimating (21), we w i l l have to assume that w i s a s u f f i c i e n t s t a t i s t i c for r. F i n a l l y u i s a stochastic disturbance term. According to (20) and subject to the r e s t r i c t i o n s of the data, the functional s p e c i f i c a t i o n of (21) i s : (22) c=wZk(h)+wZ LE(i,j;p,w,k,X(w,k,y,e,F(e,k)))+u. h i j ~ ~ ~ -> ~> S ~ ~ The factor demand functions in (22) may be highly nonlinear. Hence an estimation technique that permits some f l e x i b i l i t y in 191 this respect is preferable. Of the various f l e x i b l e functional forms available the one proposed by Box and Tidwell <67> was selected for this purpose <68>. Thus, adopting the Box-Tidwell approach, approximate the factor demand function, E ( i , j ; . , . , . , . ) in (22), by: (23) E(i,j;p,w,k,y,e,F(e,k))=a(l)p(q (D)'+a(2)w(q(2)) +a^(3)k(q(3) ) '+aU)e(q(4)) '+aj5)y(q(5)) ' . Where a ( l ) , a(2),...,a(5) are the appropriately dimensioned vectors of c o e f f i c i e n t s and p( ), w( ), k( ), e( ) and y( ) the corresponding vectors of the explanatory variables in (22) transformed according to the Box-Tidwell parameter vectors q ( l ) , q(2),...,q(5) <69>. ' denotes a transpose of the respective vector. Now, substituting (23) into (22) yi e l d s a f l e x i b l e functional form for the harvesting cost function. The resulting equation, however, includes a huge number of parameters <70> and is consequently not very a t t r a c t i v e for empirical work. However, on further simplifying assumptions <71>, we may be j u s t i f i e d in rewriting (22) as: (24) c/w=b(0)+Zb(1,h)k(h)+b(2)w+Zb(3,j)p(j)+Lb(4,s)e(s) h j s +Lb(5,j)y(j) . j where ~ on top of variables indicates a Box-Tidwell transformat ion. 1 92 Now, having specified an estimable form for the f i s h e r i e s cost function, our estimation strategy is to search for the simplest variant of t h i s function that is not rejected by the data. More s p e c i f i c a l l y , the procedure w i l l be as follows: Within each class of vessels we w i l l consider a series of increasingly simple functional forms that are nested in (24). Having obtained the simplest functional form compatible with the data the next step is to consider aggregating these cost functions over types of vessels. F i n a l l y , having reached a conclusion regarding the aggregation l e v e l , the parameters of the corresponding harvesting cost function w i l l be estimated. The data, as mentioned above, consist of cross section observations on a number of fi s h i n g vessels during 1974-7. This might suggest an error components approach to the estimation along the l i n e s employed in section 5.2.2. In t h i s case, however, t h i s i s not e n t i r e l y appropriate. F i r s t l y , the number of cross section observations varies a great deal from one year to another and very few vessels are represented in the sample in every year. Secondly, observations on a very important class of vessels, deep-sea trawlers, are not available in 1975. The available observations, thus, do not constitute panel data and the interpretation of disturbance covariances over time would be problematic. Thirdly, as w i l l become apparent below, the hypothesis of harvesting cost functions linear in parameters i s rejected by the data. A GLS estimation procedure on these nonlinear equations would be computationally burdensome. F i n a l l y , but perhaps most importantly, the test s t a t i s t i c s on 1 93 the d i s t r i b u t i o n of the disturbance terms that were calculated, although perhaps not the most powerful ones, did not suggest the need for a GLS routine. According to the estimation strategy described above, we w i l l now seek as simple functional form for each vessel type as permitted by the data. Following the s t a t i s t i c a l conventions regarding the Icelandic fi s h i n g f l e e t we w i l l consider 5 classes of vessels <72>; 3 size categories of multi-purpose fi s h i n g vessels, (i) 20-50, ( i i ) 51-110 and ( i i i ) 111-500 tons, and 2 size categories of deep-sea trawlers, (iv) under 500 and (v) over 500 tons. By means of a rather tedious t r i a l and error process several possible s i m p l i f i c a t i o n s were tested. In addition to the exclusion r e s t r i c t i o n s on the parameters l i s t e d below, the hypothesis that the fi s h i n g e f f o r t variables, e(s) <73>, and the engine power of the vessel, k(H), say, could be combined in a single set of variables defined by e*(s)=k(H)e(s), a l l s, was not contradicted by the data. Refer to thi s as hypothesis I. The test results, in terms of l i k e l i h o o d r a t i o s , are l i s t e d in table 5.7: Table 5.7 Harvesting Cost Functions: Tests of Simplifying Restrictions, Vessel Classes Restrictions (i) ( i i ) ( i i i ) (iv) (v) b(2)=0; Zl(]) b(3,j)=0, a l l j ; Xl(5) b(2)=b(3,j)=0, a l l j ; &H6) Hypothesis I; %.l(2) b(2)=b(3,j)=0, a l l j and Hypothesis I; /C(8): 4.05 3.81 5.02 5.70 8.15 0.17 0.11 0.21 0 .91 1.33 2.33 3.14 3.91 2.06 3.79 2.72 3.80 4.32 3.81 5.11 1.22 0.96 0.53 1.45 2.15 1 94 According to the results in table 5.7, we are not able to reject any of the r e s t r i c t i o n s considered. The following functional form, thus, seems to adequately explain the data: 6 ~ s s (25) c/w=b(0)+Zb(1,h)k(h)+Lb(4,s)e*(s)+Zb(5,j)y(j)+u. The second stage of the estimation procedure consisted of investigating to what extent the harvesting cost functions, (25), could be aggregated over vessel types. The pertinent res u l t s , in terms of F - s t a t i s t i c s on the parameter r e s t r i c t i o n s involved in the corresponding aggregations are l i s t e d in table 5.8: Table 5.8 Harvesting Cost Functions: Aggregation Tests. Aggregation over vessel types F - s t a t i s t i c s Types ( i ) , ( i i ) and ( i i i ) : F(34,172)=1.39 Types (iv) and (v): F(14,68) =1.85 Types ( ( i ) , ( i i ) , ( i i i ) ) and ( ( i v ) , ( v ) : F(14,291)=4.31 On the basis of these res u l t s , i t seems that we may r e s t r i c t our attention to two classes of vessels: (a) multi-purpose vessels and (b) deep-sea trawlers. We now proceed to estimate the parameters of harvesting cost functions for multi-purpose fi s h i n g vessels and deep-sea trawlers as defined by equation (25). A l l the observations on the estimation equation may be written as: 195 c=C(X;b,q)+u, • ^ ~> where c represents observations on t o t a l harvesting costs and X observations on the explanatory variables. b is a vector of unknown c o e f f i c i e n t s and q a vector of Box-Tidwell transformation parameters. u, as before, represents the disturbance vector. We assume the following stochastic s p e c i f i c a t i o n for the disturbance vector: (26) u ~N(o, al) , where is a constant and I i s an appropriately dimensioned identity matrix. Now, given (26), the maximum l i k e l i h o o d estimator for b and q is given by the functions s a t i s f y i n g <74>: -v- sis (27) Min (c-C(X;b,q))'(c-C(X;b,q), b,q where ' denotes a transpose. We refer to this rule as the nonlinear least squares estimator. Nonlinear least squares estimation of the harvesting cost functions carrying out further, r e l a t i v e l y i n s i g n i f i c a n t , s i m p l i f i c a t i o n s , seemingly supported by the data, yielded the results reported in tables 5.9 amd 5.10: 196 Table 5.9 Multi-purpose fi s h i n g vessels: Harvesting Cost Functions. Number of Observations: 223 Estimation Technique: Nonlinear Least Squares. Box- Cost Tidwell elas-Transf. t i c i t y Estimated t factors at Explanatory variables: coeffs stats q ( i) means Capital variables k(h): Constant: Dummy, Vessels 51-110 tons: 111-500 tons: Vessel age: Number of electron, inst.m: -1.97 1.5 1.0 1.12 2.1 1.0 3.95 5.4 1.0 .0099 1.8 .83 .247 0.6 .83 E f f o r t variables, e* (s): Days at sea with Tl t l trawl: g i l l n e t s : longline: handline: .868 2.896 2.049 .300 other gear: 1.590 2.0 5.6 3. 1 * 1 .8 .83 .83 .83 .83 .83 0.03 0.09 0.04 0.00 0.01 Catch variables, y(j) : Cod: .0533** 17.0 .83 0.43 Haddock: .0533** 17.0 .83 0.08 Saithe: . 041 4 9.8 .83 0.06 Redf i sh: .0920 6.4 .83 0.03 Other species: .0533** 17.0 .83 0.06 * Not estimated. ** Restricted to be equal. F(3,208)=0.53 Other s t a t i s t i c s : Standard error of the estimate: 6.9% R**2: 0.93 Durbin-Watson s t a t i s t i c : 1.86 Test for normality of residuals: £ l(6) = 16.2 Test for homoscedasticity <75>: F(4,219)=1.21 Tests of functional form: Function i s linear : q(i)=1.0, a l l i Function is log-linear; q(i)=0, a l l i q(s)=q(j), a l l s and j ( i . e . e f f o r t and catch) (1) = 1 4.8 XL(2)=322.9 ^C19)=7.2 1 97 From a. s t a t i s t i c a l point of view these results seem sa t i s f a c t o r y . Most importantly, according to the tests for homoscedasticity, autocorrelation and normality of residuals, we are unable to reject any of the stochastic assumptions embodied in (27). The explanatory power of the equation i s also reasonably good and most of the estimated c o e f f i c i e n t s are well determined. Several of the c a p i t a l variables included in the data were not found to be s t a t i s t i c a l l y s i g n i f i c a n t in the equation. The role of vessel size could be captured by two dummy variables for vessel classes. Vessel age and