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 . 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 the number of electronic instruments were included but their significance was low. Of the fis h i n g e f f o r t c o e f f i c i e n t s , that of handline fi s h i n g could not be properly estimated due to few observations (and the consequent m u l t i c o l l i n e a r i t y problem). Hence, th i s c o e f f i c i e n t was r e s t r i c t e d to a value suggested by extraneous information. As mentioned above the hypothesis of l i n e a r i t y had to be rejected. The Box-Tidwell transformation parameters could, on the other hand, be r e s t r i c t e d to only two values. 198 Table 5.10 Deep-Sea Trawlers; Harvesting Cost Functions. Number of Observations: 96 Estimation Technique: Nonlinear Least Squares, Explanatory variables: Estimated coef fs t stats Box- Cost Tidwell elas-Transf. t i c i t y factors at q(i) means Capital variables k(h): Constant Size of vessel (tons) Engine power of vessel Age of vessel Location of vessel E f f o r t variables , e*(s) : Days at sea with trawl 25.4 3.3 1.0 .0403 4.5 1.0 -.0157 4.0 1.0 -.545 1.7 1.0 -1.6 2.6 1.0 .332 7.5 2.5 Catch variables, y(j) : Cod: .164** 13.2 0.7 0.32 Haddock: .164** 13.2 0.7 0.06 Saithe: 4.95 7.7 0.3 0.30 Redfish: .0037 4.0 1.1 0.05 Other species: .164** 13.2 0.7 0.03 Other variables: Landings in Europe <76> : 1.40 2.1 1.0 0.01. ** Restricted to be equal. F(3,84)=2.01 Other s t a t i s t i c s : Standard error of the estimate: 6.3% R**2: 0.94 Durbin-Watson s t a t i s t i c : 1.93 Test for normality of residuals: ;£*(3)=8.6 Test for homoscedasticity <75>: F(3,93)=2.13 Tests of functional form: Function i s l i n e a r : q(i)=1.0, a l l i : ?£(4)=79.1 Function is log- l i n e a r ; q(i)=0, a l l i : X L(5) = 431.2 199 As for the multi-purpose vessels these results seem satisfactory from a s t a t i s t i c a l point of view. In p a r t i c u l a r , the Durbin-Watson, homoscedasticity and normality of residuals tests did not suggest that the stochastic assumptions in equation (26) should be rejected. Hence some f a i t h in the other s t a t i s t i c s may be j u s t i f i e d . The o v e r a l l f i t is moreover good and the estimated parameters seem generally well determined. Although the parameter estimates of the deep-sea trawler harvesting cost function, on the one hand, and the multi-purpose fishing vessels, on the other, seem quite d i f f e r e n t at f i r s t glance, the respective cost e l a s t i c i t i e s evaluated at sample means are, in fact, numerically similar as indicated in table 5.11: Table 5.11 Harvesting Cost Functions: Estimates of Total E l a s t i c i t i e s . Total e l a s t i c i t i e s * Variables. Mult i-purpose vessels Deep-sea trawlers Fishing e f f o r t : Catch : 0.17 0.66 0.15 0.76 * Defined as the sum of the respective e l a s t i c i t e s l i s t e d in tables 5.9 and 5.10. 200 Appendix 5.3-A On Competitive U t i l i z a t i o n of a Common Resource. The following outlines b r i e f l y , and within a simple framework, the concept of competitiveness w.r.t. a resource <77>. Let the resource growth function be defined by: (A.1) dx/dt=G(x)-y, a l l x,y>0. Where x represents the resource stock and y the t o t a l harvesting rate at time t. G(x), the growth function of the resource, is assumed to be smooth and concave with G(x1)=G(x2)=0, for some unequal but nonnegative x1 and x2. Let there be N i d e n t i c a l firms (e.g. f i s h i n g vessels) in the industry and describe the production p o s s i b i l i t i e s of each of them by the following smooth and concave production function: (A.2) y(n)=Y(e(n),x), n=1,2,..,N, where e(n) represents firm's n use of economic inputs. Moreover, l e t the costs experienced by firm n be described by the smooth and convex cost function: (A.3) c(n)=C(e(n)), n=1,2,..,N. F i n a l l y l e t r represent the common rate of time discount. Assume now that each firm believes that the others are technologically i d e n t i c a l to i t s e l f and that they have the same objective, i . e . the maximization of discounted p r o f i t s . It then follows that each firm r a t i o n a l l y expects a l l the other firms to choose exactly the same input levels as i t s e l f . Hence, each firm solves the following maximization problem: (A.4) Max /"(y(e(n),x)-C(e(n))exp(-rt)dt, e(n) Subject to: (i) dx/dt=G(x)-NY(e(n),x), ( i i ) e(n),x^O. Where i t i s understood that a l l the variables depend on time, t. The corresponding current value Hamiltonian for firm n i s : (A.5) H(n)=Y(e(n),x)-C(e(n))+q(n)(G(x)-NY(e(n),x)), where q(n) i s the shadow price of the resource stock to firm n. 201 Now, an i n t e r i o r solution to problem (A.4) includes the equation: (A. 6) dq(n)/dt-rq(n)=-Y^(e(n) ,x)+q(n) (NYJC(e(n,x)-G^-(x) ) . In equilibrium, dq(n)/dt=0. Hence: (A.7) q(n)=Y x(e(n),x)/(NY^(e(n),x)+r-G^(x)), a l l n. It follows immediately from (A.7) that q(n), the imputed equilibrium shadow price of the resource, converges asymptotically to zero as N, i . e . the number of firms, increases. One implication of a p r a c t i c a l nature i s that, i f N i s large, the imputed shadow price of the resource is immaterial from the point of view of a description of the firms' behaviour. Appendix 5.3-B Decisions on A c t i v i t i e s and the Consumption of Inputs <78>. Consider a certain f i e l d of production, e.g. f i s h e r i e s . Firms desirious of entering t h i s f i e l d can generally choose a par t i c u l a r production technology from a c o l l e c t i o n of available technologies <79>. A pa r t i c u l a r technology i s adopted via investment in the corresponding c a p i t a l units. Once c a p i t a l investment has taken place, operating decisions are broadly limited to how to employ the c a p i t a l . In f i s h e r i e s , for instance, the main operating decisions concern which fishery to pursue, with what gear and for what duration. Moreover, once c a p i t a l i s i n s t a l l e d , input consumption for a given operating a c t i v i t y i s r e l a t i v e l y i n f l e x i b l e . What input s u b s t i t u t a b i l i t y the available technology offers is primarily of an ex ante, or pre-investment, nature. In the extreme case the e l a s t i c i t y of input substitution ex post i s zero. Proceeding on the assumuption of zero e l a s t i c i t y of substitution ex post, the essentials of the discussion may be formalized as follows: Technologies: I n i t i a l l y the following set of technologies i s avai l a b l e : (B.1) T=(T(1),T(2),...T(L)), where each technology i s embodied in a certain units of c a p i t a l and defines the following production function: (B.2) Y(k(l),act(i),e)=min[a(1,i)e(1), ,a(N,i)e(N)], 202 where k(l) denotes the c a p i t a l corresponding to technology 1, a c t ( i ) refers to a c t i v i t y i and the (1xN) vector e represents inputs. The a(n,i)'s are simply the input-output c o e f f i c i e n t s corresponding to a c t i v i t y i . (B.2), of course, holds for each a c t i v i t y , i . Choice of Technology. A p a r t i c u l a r technology is respecitve c a p i t a l . In other problem i s to: (B.3) Choose k(l) where T(l)eT. adopted by investing in the words, the firms' investment Post-investment use of inputs. Now, given T(l) and k(l) and assuming positive input prices, the p r o f i t maximizing use of inputs corresponding to a given a c t i v i t y , a c t ( i ) , i s defined by: (B.4) Y(k(l),act(i),e)=a(1,i)e(1)=a(2,i)e(2)= So, given certain regularity assumptions on Y(.,.,.), the inputs are unique functions of the a c t i v i t i e s as follows: (B.5) e(1)=F(1,act(i),k(l)) e(2)=F(2,act(i),k(l) ) Thus, to the extent that t h i s description of the production r e a l i t i e s is true, consumption of economic inputs can be inferred from knowledge of operating a c t i v i t i e s and the i n s t a l l e d c a p i t a l . 203 5.4 The Revenue Function. The t o t a l revenue of the harvesting process at time t is defined as <80>: H (28) R(t)= Z p(h,t)q(h,t). Where q(h,t) denotes the produced quantity and p(h,t) the price of f i n a l product h at time t. H is the t o t a l number of f i n a l products. Hence, in order to calculate R(t), estimates of p(h,t) and q(h,t) must be obtained. This is the subject of this section. Because of the potential implications for competitive behaviour in the fishing sector, a separate revenue function for this sector i s also of some in t e r e s t . Write this function as: i (29) R1(t) = Z p ( i , t ) y ( i , t ) . Where y ( i , t ) and p ( i , t) denote the landed volume and the corresponding unit price of species i at time t respectively. I is the t o t a l number of species <81>. Of these two sets of variables, i . e . the quantity and price variables that make up the revenue function, the former are c l e a r l y endogenous in the model while the l a t t e r are taken to be exogenous <82>. Consequently, the following discussion w i l l mainly be concerned with the functional determinants of the quantity variables given the volume of catch. In what follows, moreover, the indices, e s p e c i a l l y the time indices, w i l l frequently be omitted in the interest notational convenience. 204 5.4.1 The Quantity of F i n a l Products. The f i s h i n g a c t i v i t y generates a t o t a l catch of each of the three species; cod, haddock and saithe. Write t h i s as the vector: (30) y'=(y(1),y(2),y(3)), where y ( l ) , y(2) and y(3) denote the t o t a l catch of cod, haddock and saithe respectively and the superscript ' denotes a transpose. The catch of each species can be allocated to one or more of four basic processes; freezing, s a l t f i s h , stockfish and f i s h meal production <83>. These processes we refer to by the numbers 1,2,3 and 3, respectively. Thus, l e t t i n g q ( i , j ) denote the f i n a l output of species i from process j we represent the f i n a l products by the 12x1 vector: (31) q=(q(i,j)), i=1,2,3 and j=1,2,3,4. The r e l a t i o n between the catch vector and the f i n a l products vector, at a given time, may be written as <84>: (32) q=T-y, ~s where T i s a 12x3 matrix that transforms the catch into the vector of f i n a l products. Thus, taking y as given, the problem of determining the vector q i s equivalent to the problem of 205 estimating the elements of the transformation matrix T. Si m i l a r l y , the rel a t i o n between the the catch vector and the landed catch may be written as: (33) y=A-y. ~ ~> ~s Where y'=(y(1),y(2),y(3)) is the transposed vector of landed catch of cod, haddock and saithe, respectively and A is a 3x3 diagonal matrix that transforms the volume of catch of each species into landed catch. We now turn to the task of examining the structure of these transformation matrices, A and, in p a r t i c u l a r , T. The t o t a l catch of species i , y ( i ) , may be decomposed into the volume of landings, y ( i ) , and the volume loss at sea, 11(i) <85>, i . e . y(i)= y ( i ) + l 1 ( i ) . Let this relationship be determined by the "catch transformation parameter" or a ( i ) , say, as follows: (34) y ( i ) = a ( i ) y ( i ) , 1>a(i)>0, i=1,2,3. Hence, the volume loss at sea is given by: 11( i ) = ( 1 - a ( i ) ) y ( i ) , i=1,2,3. 206 Equations (7) define the volume of landed catch of each species in terms of the catch and the respective catch transformation parameters, a ( i ) ' s . The l a t t e r , of course, constitute the diagonal elements of the matrix A in equation (33). Obviously the attainable value of each a ( i ) i s bounded above by the available preservation technology. Within that l i m i t , however, economic considerations may influence the value of a(i) <86>. According to the available empirical measurements, on the other hand, the a ( i ) ' s are very stable and seemingly independent of catch and price fluctuations <87>. Hence, for the purposes of t h i s study, the a ( i ) ' s may be regarded as technical parameters. Landed catch i s allocated to one or more of the four production processes mentioned above. Thus, l e t t i n g y ( i , j ) denote the t o t a l quantity of species i allocated to process j , define the "process a l l o c a t i o n parameters", b ( i , j ) by: (35) y ( i , j ) = b ( i , j ) y ( i ) , 1>b(i,j)>0, a l l i , j . Since a l l of the landed catch must be allocated to a process we also have: Z b ( i , j ) = 1 , a l l i . j = i Within these r e s t r i c t i o n s , the values of the b ( i , j ) ' s can in general be chosen by the f i s h processors. Remember that in the Icelandic demersal fishing industry many processing plants operate two or more of the basic processes and, in addition, 207 there are resale markets for landed catch in some areas. Consequently, one expects the b ( i , j ) ' s to be, to some extent, responsive to economic signals, especially f i n a l output prices. The quantity allocated to each process, y ( i , j ) , may be decomposed into the volume of f i n a l output, q ( i , j ) , loss of volume in production, 12(i,j) and a transfer of volume to a dif f e r e n t process, z ( i , j ) . This last variable consists almost exclusively of transfers to the f i s h meal process <88>. In other words: y ( i , j ) = q ( i , j ) + z ( i , j ) + 1 2 ( i , j ) , a l l i , j . Where z ( i , j ) denotes transfer of species i i n i t i a l l y allocated to j t h process to the f i s h meal process. This transfer i s dictated by the " f i s h meal transfer parameter", d ( i , j ) , as follows: z ( i , j ) = d ( i , j ) y ( i , j ) , 1>d(i,j)>0, a l l i , j . Without loss of generality we impose d(i,4)=0, a l l i . On empirical arguments p a r a l l e l to those used for a(i) we take i t that the d ( i , j ) ' s are predominantly technologically determined <89>. Given the process a l l o c a t i o n s , y ( i , j ) ' s , the volume of f i n a l products are determined • by the " f i s h processing transformation parameter", c ( i , j ) as follows <90>: 208 q ( i , j ) = c ( i , j ) y ( i , j ) , j =1 ,2,3, a l l i . (36) q(i,4)=c(i,4)(y(i,4)+ L d(i,j)y(i,j)), a l l i . It follows that volume loss in processing is given by: 12(i, j) = ( 1 - c ( i , j ) - d ( i , j ) ) y ( i , j ) , a l l i , j = 1,2,3. I2(i,4)=(1-c(i,4))(y(i,4)+ I d ( i , j ) y ( i , j ) ) , a l l i , j . On arguments similar to those used to claim that the a ( i ) ' s and d ( i , j ) ' s are primarily technological parameters, we claim the same property for the c ( i , j ) ' s . Using equations (34), (35) and (36), we f i n a l l y obtain the algebraic equivalent to (32): q ( i , j ) = c ( i , j ) b ( i , j ) a ( i ) y ( i ) , a l l i , j = 1,2,3. (37) q(i,4)=c(i,4)(b(i,4)+ L d ( i , j ) b ( i , j ) ) a ( i ) y ( i ) , a l l i . The equations in (37) describe the d e t a i l s of the matrix equation (32). In par t i c u l a r (37) defines the matrix T in terms of 39 technical and economic c o e f f i c i e n t s . A summary of these c o e f f i c i e n t s i s contained in the following table: 209 Table 5.12 Transformation c o e f f i c i e n t s . Number of Total number of unknown Coef f i c i e n t s i j r e s t r i c t i o n s c o e f f i c i e n t s a ( i ) ' s 3 0 3 b ( i , j ) ' s 3 4 3 9 d ( i , j) 's 3 4 3 9 c ( i , j ) ' s 3 4 12 The matrix T may be v i s u a l i z e d as the multiple of 3 matrices, A,B,C, containing the catch transformation parameters, the process a l l o c a t i o n and f i s h meal transfer parameters and the process transformation parameters respectively. I.e. (38) T=C- B- A. w Where C is a 12x12 diagonal matrix of the process transformation parameters. B i s a 12x3 column diagonal matrix of process a l l o c a t i o n and f i s h meal transfer parameters as follows: B= b( 1 , 1 ) ,0,0 bd,2) ,0,0 b(1,3) ,0,0 b(1,4)+D(1),0,0 0,b(2,l) ,0 0,b(2,2) ,0 0,b(2,3) ,0 0,b(2,4)+D(1),0 0,0,b(3,1 ) 0,0,b(3,2) 0,0,b(3,3) 0,0,b(3,4)+D(3) 3 where D(i)= Z d ( i , j ) b ( i , j ) , i=1,2,3. 210 And A is a 3x3 diagonal matrix of catch transformation parameters. Some estimates of the elements of A, B, and C as well as the d ( i , j ) ' s during 1976-80 are recorded by the respective fishery s t a t i s t i c s i n s t i t u t e s <91>. These estimates, however, are not routinely c o l l e c t e d and are consequently rather sketchy. Nevertheless they c l e a r l y show that the c o e f f i c i e n t s that have been c l a s s i f i e d as predominantly technologically determined above, i . e . the elements of A, C and the d ( i , j ) ' s , are r e l a t i v e l y invariant over time although some show a small upward trend presumably r e f l e c t i n g the effects of technological advances. The b ( i , j ) ' s , on the other hand, fluctuate rather widely supposedly in response to changes in the economic envi ronment. Therefore, for the purposes of this study, i t seems plausible to adopt fixed values of the technological c o e f f i c i e n t s based on their h i s t o r i c a l values and trend. Since, on the other hand, the effects of exogenous s h i f t s in the economic environment, e.g. output prices, on fishery p o l i c i e s are of interest in t h i s study, the b ( i , j ) ' s must be related to their economic determinants. This task is undertaken in the next section. The following table, table 5.13, contains information on the history of the technical c o e f f i c i e n t s operative in the transformation of the volume of catch into f i n a l products as well as the values that w i l l be adopted in t h i s study. The data on which these estimates are based are l i s t e d in Appendix 1:Data (section 5.4). 21 1 Table 5.13 Technical Transformation C o e f f i c i e n t s . H i s t o r i c a l values Variance Adopted Coeff's Mean Trend % of mean values a(1 ) .8 No * 0.8 a(2) .8 No * 0.8 a(3) .8 No * 0.8 d(1 , 1 ) .46 No * 0.46 d d ,2) .22 No * 0.22 d(1,3) . 1 24 No * 0. 1 24 d(2, 1 ) .476 No * 0.476 d(2,2) NA NA NA NA d(2,3) . 1.24 No * 0.124 d(3, 1 ) .424 No * 0.424 d(3,2) .316 No * 0.316 d(3,3) . 1 24 No * 0. 1 24 c( 1 , 1 ) .39 Yes 7.0 0.43 c(1,2) .429 Yes 4.0 0.44 C(1,3) .17 No • 0.0 0.17 C(1,4) .251 No 2.0 0.25 c(2, 1 ) .371 Yes 8.0 0.41 c(2,2) .44 No * 0.44 c(2,3) . 1 7 No 0.0 0.17 c(2,4) .251 No 2.0 0.25 c(3, 1 ) .447 Yes 8.0 0.48 c(3,2) .450 No 0.0 0.450 c(3,3) .17 No 0.0 0.17 c(3,4) .251 No 2.0 0.25 * One observation only. NA= Not avai l a b l e . Sources: The Icelandic Fisheries Association; the a ( i ) ' s and d ( i , j ) ' s . The Marine Research Institute; the a ( i ) ' s . The National Economic Inst i t u t e ; the d ( i , j ) ' s and c ( i , j ) ' s . 212 5.4.2 The Process Alloca t i o n Parameters. We now turn to the determinants of the elements of B. For expositionary purposes we begin with a s i m p l i f i e d theoretical framework. Write the s t a t i c p r o f i t maximization problem of the f i s h processors as (39) Max H= Z Z p ( i , j ) q ( i , j ) - C ( q ( i , j ) , i , j ) . 1 i q ( i , j) Where q ( i , j ) denotes the produced quantity and p ( i , j ) the price of product i , j . C(.,.,.) is the cost function of producing q( i , j ) • The maximization of H i s subject to the following constraints: (i) q ( i , j ) = c ( i , j ) y ( i , j ) , j = 1,2,3, a l l i . q(i,4)=c(i,4)(y(i,4)+ Z d ( i , j ) y ( i , j ) ) , a l l i . j *f * ( i i ) Z y ( i , j ) = y ( i ) , a l l i . •j -1 ( i i i ) y(i,j)>0, a l l i , j . Where, i t w i l l be recalled, y ( i ) denotes the landed volume A of species i and y ( i , j ) the volume of landings of species i allocated to process j . The c ( i , j ) ' s and d ( i , j ) ' s are technical transformation c o e f f i c i e n t s . The f i r s t set of constraints merely 213 repeats the information contained in equation (36) above. Constraints ( i i ) state that the sum of the process all o c a t i o n s must equal the t o t a l quantity of landings and constraints ( i i i ) simply require the all o c a t i o n s to each process to be nonnegat ive. Substituting (i) into (39) and rearranging we can rewrite the p r o f i t maximization problem in terms of process a l l o c a t i o n s , y ( i , j ) , and corresponding unit values, p ( i , j ) , as <92>: (40) Max H= Z Z p ( i , j ) y ( i , j ) - C ( y ( i ) , i , j ) . y ( i , j ) Subject to ( i i ) and ( i i i ) . Where the all o c a t i o n prices are given by: p ( i , j ) = p ( i , j ) c ( i , j ) + p ( i , 4 ) d ( i , j ) c ( i , 4 ) , a l l i , j , And the a l l o c a t i o n vector, y ( i ) , i s defined by: y ( i ) = ( y ( i , 1 ) , y ( i , 2 ) , y ( i , 3 ) , y ( i , 4 ) ) , a l l i . On regularity assumptions discussed in appendix 5.4-A, the solution to th i s problem i s given by the functions: (41) y ( i , j ) = Y ( p ( i ) , y ( i ) ) , a l l i , j . 214 Where the f i r s t argument i s the vector p(i)=(p(i,1),...,p(i,4)) and y ( i ) denotes the landed volume of species i . Process allocations depend, in other words, only on the vector of f i n a l product prices and the landed quantity of the species in question. The optimal a l l o c a t i o n parameters are accordingly given by: (42) b ( i , j ) = y ( i , j ) / y ( i ) = B ( p ( i ) , y ( i ) ) , a l l i , j . Thus we have derived i m p l i c i t functions for each of the 12 b ( i , j ) ' s of the model. Given observations on the b ( i , j ) ' s , p ( i , j ) ' s and y ( i ) ' s the structure and parameters of (42) may now, in p r i n c i p l e , be estimated. The e f f i c i e n c y of the estimates i s , however, increased by taking due notice of the r e s t r i c t i o n s on the functional form of (42) implied by the structure of the underlying maximization problem. A set of r e s t r i c t i o n s follows from the nature of the b ( i , j ) ' s as shares. That obviously implies: (i) 1>b(i,j)>0, a l l i , j . ( i i ) 1 b(i,j)=1, a l l i . We refer to r e s t r i c t i o n s (i) as the nonnegativity r e s t r i c t i o n s and r e s t r i c t i o n s ( i i ) as the adding-up r e s t r i c t i o n s <93>. It i s worth observing that these r e s t r i c t i o n s are d e f i n i t i o n a l in character and thus must not be violated even when applied to 215 empirical data. In addition to these " d e f i n i t i o n a l " r e s t r i c t i o n s i t is ea s i l y seen from (39) and (40) that i f the cost functions are homogeneous of degree 1 in the p ( i , j ) ' s then the B(.,.) functions w i l l be homogeneous of degree zero in the the same prices. Now, while output prices are not e x p l i c i t l y arguments in the cost functions, i t is well known that these functions may also be written as homogeneous functions of degree one in input prices. Thus, i f r e l a t i v e input and output prices remained constant, the cost functions would in fact be homogeneous of degree one in output price s . However, since the p ( i , j ) ' s are es s e n t i a l l y foreign market prices, t h i s condition i s rather unikely to hold empirically. Now, having derived the process a l l o c a t i o n functions, (42), the next step i s to obtain estimates of these functions. F i r s t , observations on a l l the variables contained in (42) during 1974-82 are available <94>. Second, apart from the nonnegativity and adding-up constraints, there i s very l i t t l e a p r i o r i information on the functional form of the functions in (42). A natural f i r s t step of the estimation procedure, therefore, i s to attempt to infer t h i s from the data. Since the cross equation r e s t r i c t i o n s implied by the adding-up constraints render a standard f l e x i b l e functional form approach to thi s task unattractive <95>, we elected to adopt the more ad hoc method described below. Several nonlinear versions of (42) were compared. These included smooth nonlinear functions with the nonnegativity constraints imposed through the functional forms of the 216 equations. This nonlinear approach, however, produced rather poor r e s u l t s . It turned out to be numerically d i f f i c u l t to obtain sa t i s f a c t o r y estimates of nonlinear equation systems s a t i s f y i n g both the adding-up cross equation constraints and having functional forms f l e x i b l e enough to f i t the data well <96>. For these reasons we resorted to a linear s p e c i f i c a t i o n of the functions in (42). This incidently exhibited a good f i t to the data. A major drawback of thi s functional form, however, is that "predicted" b ( i , j ) ' s are not automatically constrained to the i n t e r v a l [0,1]. This, however, was not a problem in the sample period <97>. A general linear s p e c i f i c a t i o n of (42) i s : (43) b ( i , j ) = a ( i , j ) + I d ( i , j , k ) p ( i , k ) + e ( i , j ) y ( i ) , a l l i , j , h where the a ( i , j ) ' s , d ( i , j , k ) ' s and e ( i , j ) ' s are parameters. K i s the the t o t a l number of output prices in each equation generally equalling the number of processes <98>. Given that there are 3 species and 4 processes (43) consists of 12 equations and 72 parameters. The actual system estimated was considerably simpler, however. F i r s t l y , the b(i,4)'s, i . e . the direct a l l o c a t i o n s to the f i s h meal process, were in a l l instances n e g l i g i b l e <99>. Thus, these equations were dropped from the system to be estimated. Secondly, the al l o c a t i o n of species 2, haddock, to process 2, s a l t f i s h production, was also n e g l i g i b l e . Moreover, perhaps, for that very reason, observations on the corresponding prices were nonavailable. Consequently, th i s equation and the corresponding 217 price term, p(2,2), was also dropped from the system. This leaves a system of 8 a l l o c a t i o n equations and 46 parameters to be estimated. The following r e s t r i c t i o n s on the parameters were tested: I. Adding-up r e s t r i c t i o n s , i . e . I b(i,j)=1. L a ( i , j ) = 1 , a l l i , Le(i,j)=0, a l l i , j Ld(i,j,k)=0, a l l i,k. j B. Homogeneity r e s t r i c t i o n s , i.e zero homogeneity in prices. Ld(i,j,k)=0, a l l i , j . h Since, as discussed in section 5.2, the p ( i , j ) ' s are es s e n t i a l l y foreign prices and the y ( i ) ' s largely outside the control of the processing firms i t seems safe to assume that the explanatory variables in (43) are s t a t i s t i c a l l y exogenous. However, to impose the cross equation adding-up constraints in a consistent manner the process a l l o c a t i o n equations must be estimated simultaneously for each species in turn. The estimation results are l i s t e d in the following table: 218 Table 5.14 Estimation of the share equations. Estimation technique: Zellner's Seemingly Unrelated Regression Equations technique with cross equation adding-up constraints imposed <100>. Tests of r e s t r i c t -C o e f f i c i e n t s S t a t i s t i c s ions a d(1) d(2) d(3) d(4) e R DW * I II System 1 Cod Eq. 1 . .681 .086 -.089 -.004 (19.) (6.7) (9.9) (0.7) Eq. 2 .612 -.033 .059 -.008 (30. ) (4.6) (12.) (2.4) Eq. 3 -.293 -.053 .030 .012 (9.6) (5.0) (4.0) (2.5) System 2 Haddock Eq. 1 .977 .012 -.009 (32. ) (3.6) (3.4) Eq. 3 .023 -.012 .009 System 3 Saithe Eq. 1 .576 .031 .016 -.024 (20. ) (4.6) (7.7) (8.1 ) Eq. 2 .281 .000 -.020 .008 (9.8) (0.1 ) (9.0) (2.8) Eq. 3 . 1 44 -.031 .003 .016 (37.) (34.) (11 .) (39.) NR NR -.007 -6-10^' .96 2.6 7.1 (res) (4.7) -.018 -6 -10^ .97 2.2 6.6 (res) (8.9) .011 1.2-1 0"^ .94 2.8 4.8 (res) (12.0) -.055 1.410"* .97 2.7 7.4 (6.9) (1.2) .055 - 1 . 4 - 1 0"3 .97 2.7 7.4 -.066 3.7 10° . 54 2.0 20. (7.0) (7.3) -.006 -.6 10"4 .76 2.1 18. (0.6) (1.2) .072 -•3.1 -10"° .89 2.0 20. ** RJ NR RJ (55.) (45.) Numbers in brackets are t - s t a t i s t i c s . NR= not rejected on 5% significance l e v e l . RJ= rejected on 5% significance l e v e l . *= Chi-square(3) test of normality of residuals. **= automatically s a t i s f i e d . These estimates seem reasonably s a t i s f a c t o r y . The s t a t i s t i c a l tests do not indicate a strong rejection of the s t a t i s t i c a l assumptions underlying the estimation process. The 219 f i t of the equations to the data is f a i r . The sign of the estimated c o e f f i c i e n t s are generally as predicted by theory and their standard errors low. The d ( i , j , k ) ' s for saithe are something of an exception in t h i s respect, however. This p a r t i c u l a r l y holds for the second equation of that system. There is some evidence of a unique market opportunities in the late seventies only p a r t i a l l y r e f l e c t e d by the price data, that may explain t h i s anomalty. F i n a l l y , the o v e r a l l v a l i d i t y of these estimates i s supported by the adding-up r e s t r i c t i o n s , I, not being rejected by the data. 5.4.3 Output Prices. As previously mentioned, the f i n a l output prices as well as the intermediary f i s h prices are considered to be exogenous in th i s model. Hence, l i t t l e further needs to be said about these prices within the confines of the model. However, in order to calculate the revenue function corresponding to a certain harvesting program, estimates of these prices need to be obtained. In Appendix 1:Data (section 5.4), we present a l i s t of the actual f i n a l and intermediate output prices during 1974-82. The former were, in fact, used for estimating the process a l l o c a t i o n parameters in the previous section. 220 Appendix 5.4-A Analysis. In t h i s appendix the derivations of equations (40) and (41) in section 5.4.2 w i l l be explained. Without loss of generality we may, for notational s i m p l i c i t y , concentrate on one species only. Accordingly, dropping the price index, i , rewrite the maximization problem of the producers as: f (A.1) Max H= Z p ( j ) q ( j ) - C ( q(j),j) q(j) J =1 Subject to (i) q ( j ) = c ( j ) y ( j ) , j =1 ,2,3 q(4)=c(4)(y(4)+ I d ( j ) y ( j ) ) . ( i i ) ?y(j)=y. j ( i i i ) y(j)>0, a l l j . Substituting (i) into the objective function y i e l d s : H= Z p ( j ) c ( j ) y ( j ) - C ( c ( j ) y ( j ) , j ) + p ( 4 ) c ( 4 ) ( y ( 4 ) + I d ( j ) y ( j ) ) j -C(c(4)(y ( 4)+Zd(j)y(j)),4). Or, H= Z p ( j ) y ( j ) - C ( y ( i ) , j) , where p(j)=p(j)c(j)+p(4)c(4)d(j) , j = 1,2,..4 , C(y(i),j)=C(y(i) , j) , j=1,2,3 =C(y(l),y(2),y(3),y(4),4), j=4. Notice that p ( j ) , the unit values (or shadow values) of allocations to processes 1, 2 and 3 partly depend, as should be expected, on transfers to process 4. Notice also that the cost functions of processes 1,2,3 only depend on a l l o c a t i o n s to these processes while the cost function of process 4, due to the volume transfers, also depends on the allocations to the other processes. Thus the producers' maximization problem becomes: 221 (A.2) Max H= 2 Z p ( i , j ) y ( i , j ) - C ( y ( i ) , i , j ) , i=1,2,3. 1 j y ( i , j ) Subject to ( i i ) and ( i i i ) . (A.2) v e r i f i e s equation (40) in section 5.4.2. Now,^if the cost functions, C (y ( i ) , i , j ) 1 s , are continuous in the y ( i , j ) ' s then, by the Weierstrass theorem <101>, a solution to t h i s problem e x i s t s . If we are, moreover, ready to assume _ f i r s t an i n t e r i o r solution and second that the C(£(i),i,j)'s are twice continuously d i f f e r e n t i a b l e convex functions in the con t r o l l e r s then the solution to this problem is given by equation (41) in section 5.4.2. I.e. <102>: y ( i , j ) = Y ( p ( i ) , y ( i ) , i , j ) , a l l i , j . Where p ( i ) = ( p ( i , 1 ) , p ( i , 2 ) , p ( i , 3 ) , p ( i , 4 ) ) . 222 5.5 The Fishing Capital Function. The topic of this section i s the f i s h i n g c a p i t a l and i t s dynamics. By f i s h i n g c a p i t a l we mean the f i s h i n g f l e e t as a physical e n t i t y . As explained in chapter 3 we do not, in t h i s study, consider the dynamics of f i s h processing c a p i t a l . 5.5.1 Characterization of the Fishing C a p i t a l . The Icelandic demersal f i s h i n g f l e e t i s far from being a homogeneous e n t i t y . It consists of a great number of vessels of various types, sizes and vintages <103>. Since, however, our concern i s primarily with the broader dynamics of the fishing process our characterization of the f i s h i n g c a p i t a l w i l l be in much more aggregative terms. Following the aggregative structure in section 5.3.2 we consider 2 classes of f i s h i n g vessels; multipurpose boats and deep-sea trawlers. From an a n a l y t i c a l point of view these vessel classes constitute two.distinct f i s h i n g f l e e t s . Let us refer to them as f l e e t 1 and 2, respectively. We describe each of the two f i s h i n g f l e e t s in terms of 3 properties only; (i) the number , ( i i ) the average age and ( i i i ) the average technical a t t r i b u t e s of the vessels in each f l e e t . The l a s t property refers to such relevant variables as the tonnage of the vessels, their length, engine si z e , number of electronic f i s h i n g instruments etc. For the purposes of exposition l e t us c o l l e c t the average technical attributes of the vessels in the vector k ( i , t ) , where i refers to the f l e e t and t to time. S i m i l a r l y , l e t n ( i , t ) and 223 a ( i , t ) represent vessel number and mean age of fl e e t i at time t respectively. A complete description of the fis h i n g c a p i t a l i s then contained in the vector: i//(i,t) = ( n ( i , t ) , a ( i , t ) , k ( i , t ) ) , i = 1,2. t=1,2, 5.5.2 Fishing Fleet dynamics. Having described the c h a r a c t e r i s t i c s of the fis h i n g c a p i t a l we now turn to i t s dynamics. Of the three properties of the fish i n g c a p i t a l discussed above, the vector of vessel a t t r i b u t e s , k ( i , t ) , w i l l be taken to be exogenous to the fis h i n g c a p i t a l dynamics <104>. It follows that, from the point of view of the autonomous dynamics of the fi s h i n g c a p i t a l , we may take i to be a s u f f i c i e n t s t a t i s t i c for k ( i , t ) so that the properties of the f i s h i n g c a p i t a l are f u l l y described by the vector: : (44) An(t)=n(t+1)-n(t)=I(t+1)-DI(t+1)-D(t+1), I,DI,DeI^. Where I(t+1) represents the investment and Dl(t+1) the 224 disinvestment in f l e e t i during the interval [t,t+l] <106>. D(t+1) denotes the reduction in the number of vessels of f l e e t i through depreciation during the same i n t e r v a l . Il denotes the space of posit i v e integers. Given the i n i t i a l number of vessels, n(0), (44) describes the path of n(t) as a function of l ( t ) , DI(t) and D(t). Of these variables l ( t ) and Dl(t) are naturally exogeneous. In fact, as discussed in chapter 3, these variables belong to the set of control variables for this f i s h e r i e s model. D ( i , t ) , on the other hand, i s supposed to represent the reduction in the number of vessels with the passage of time due to the ef f e c t s of the physical environment. This, presumably, depends on the other major c h a r a c t e r i s t i c of the fis h i n g c a p i t a l , the mean age of the f l e e t , a ( i , t ) . Thus, c l e a r l y , D(i,t) is not exogeneous and the dynamics of the number of vessels is incomplete without a spe c i f i c a t i o n of this function. As there i s a considerable amount of empirical data that have bearing upon the spe c i f i c a t i o n of D(i,t) t h i s issue w i l l be pursued in a separate subsection. To emphasize the technical and physical aspect of D(i,t) we w i l l , henceforth refer to i t as the deterioration funct ion. Turning now to the dynamics of the mean age of the f l e e t s , a ( i , t ) , we make the following two simplifying assumptions: (1) A l l investment takes • place in new vessels only. (2) A l l disinvestment takes place in vessels of the mean age, a ( i , t ) <107>. Given assumptions (1) and (2), an equation describing the path of a ( i , t ) through time is readily derived. Consider the 225 fleet at time t+1. Decompose the f l e e t into the "old f l e e t " and the "new f l e e t " . Let the old f l e e t consist of the vessels remaining of the fle e t in the previous period. Let the new fl e e t consist of vessels added to the fl e e t since previous period, i.e. I(t+1). Without loss of generality, we may assume that the age of the new fl e e t at t+1 i s 1 <108>. Denote the old fl e e t at t+1 by no(t+l) and the new fl e e t by nn(t+l). By the d e f i n i t i o n of these two concepts the t o t a l f l e e t at t+1 i s : n(t+1)=no(t+1)+nn(t+1 ) . Where nn(t+1)=1(t+1) and, by the d e f i n i t i o n of the old f l e e t : no(t+1)=n(t)-DI(t+1)-D(t+1). Now, by d e f i n i t i o n , a(t)+1 i s the age of the old fl e e t at t+1 while the age of the new f l e e t i s 1. Therefore the mean age at time t+1 i s given by: a(t+1)=((a(t)+1)no(t+1)/n(t+1)+1•nn(t+1))/n(t+1). Or, more concisely: (45) a(t+1)=(a(t)+1)w(t)+1 (l-w(t)) where w(t) i s given by <109>: 226 (46) w(t)=(n(t)-DI(t+1)-D(t+1))/(n(t)-DI(t+1)-D(t+1)+I(t+1)). So the mean age of the fl e e t at time t i s simply the weighted average of the age of the old fl e e t and that of the new fle e t where the weights are given by equation (46). Equations (44), (45) and (46) describe the dynamics of the fishi n g f l e e t s . Given the i n i t i a l conditions, (n(0),a(0)), the values of the deterioration function, D(t), and the control variables, I(t) and DI(t), these equations completely describe the time path of the fish i n g f l e e t s . 5.5.3 Estimation of the Deterioration Function. We f i r s t assume that the way in which deterioration a f f e c t s the f l e e t can be described by the following expression: D(t)=INT(s(t)n(t)), where INT( ) i s the integer operator <110> and s(t) i s the proportional deterioration rate of the fle e t at time t. We, moreover, hypothesize that the deteroriation rate i s a function of the age of the f l e e t , i . e . : (47) s(t)=S(a(t)) . Where we expect that S'(a)>0 and there exists some positive age, a*, say, with the property that S(a*)=1. Such an a* may be 227 interpreted as a technical age maximum beyond which no vessels survive. Of course (47) i s only defined for a ( t ) ^ a * . Our aim, in thi s subsection, i s to obtain reasonable estimates of the deterioration rate function, S(a). The history of the Icelandic trawler f l e e t during the past few decades contains a great deal of information about actual deterioration rates during t h i s period. On the basis of th i s history the National Economic Development Institute has extracted data describing, for a given i n i t i a l f l e e t , the number of vessels remaining every year over a period of 40 years <111>. Since these data include only exits from the f l e e t due to technical wastage, sinkings etc. they, presumably, r e f l e c t f a i r l y accurately the pa r t i c u l a r reduction in the number of vessels that the deterioration function i s supposed to represent. To relate this information to s ( t ) , l e t N(a) denote the fl e e t remainder function, i . e . the number of vessels remaining of the i n i t i a l f l e e t when i t s age i s a. Then, the deterioration rate is defined as: (48) s(t)=S(a)=(N(a)-N(a+1))/N(a). Investigation of the functional form for N(a) suggests the following s p e c i f i c a t i o n <112>: (49) N(a) = 1-bac +u(a) , a=1,2, a*. 228 Where b and c are parameters and u(a) is a stochastic error term. We assume that u~N(0,aI), u=(u( 1 ) ,u(2) , u(a*)) Maximizing the l i k e l i h o o d function for (49) under these assumptions yielded the following s t a t i s t i c a l r e s u l t s : Table 5.15 Estimation of equation ( 4 9 ) . Estimation technique: Nonlinear least squares Number of observations: 40 Parameters Estimates t-values b 0.00119 15.3 c 1.869 18.9 R2": 0.96 DW: 0.32 t Test for normality of residuals: /C(2)=2.8 These results seem reasonable. The f i t i s good and the c o e f f i c i e n t s seem well determined. In the l i g h t of the Durbin-Watson s t a t i s t i c , however, this may be overoptimistic. If the DW value r e f l e c t s a misspecification of the N(a) function the estimated c o e f f i c i e n t s w i l l be inconsistent. However, since other functional forms did not produce better results we proceed on the basis of these r e s u l t s . According to (48) the deterioration rate function i s given by: 229 S(a)=(b((a+1 ) C - a C ) / ( l - b a c ). The estimated N(a) and S(a) are depicted in figure 5.2 below. Figure 5.2 The-estimated f l e e t remainder function and deterioration rates. FLEET REMAINDER DETERIORATION RATES 0 10 20 30 40 AGE OF FLEET As indicated in figure 5.2, the estimated f l e e t maximum age, a*, i s 36-37 years. No comparable data for the deterioration of the multipurpose f i s h i n g f l e e t are available. What information there is <113>, however, does not suggest s i g n i f i c a n t l y d i f f e r e n t deterioration rates for the multipurpose f l e e t . Therefore, the above estimates of deterioration rates w i l l be adopted for the 230 t h i s f l e e t also. 5.5.4 Investment and Disinvestment in the Fishing Fleet. In addition to physical deterioration, the dynamics of the fish i n g f l e e t are determined by investment and disinvestment in the f l e e t . These, as discussed in section 3.3, are natural control variables in most f i s h e r i e s . Empirically speaking, however, these controls can generally not be applied instantaneously and c o s t l e s s l y . In the technical jargon, f i s h i n g c a p i t a l i s generally not perfectly malleable <114>. As i s well known from the work of Clark et a l . <115>, the degree of c a p i t a l m a l l e a b i l i t y is generally an important determinant of the nature of e f f i c i e n t harvesting paths. In the case of the Icelandic demersal f i s h e r i e s , the essentials of the fi s h i n g f l e e t m a l l e a b i l i t y situation seem to be the following <116>: As a l l types of fi s h i n g vessels can be purchased from foreign shipyards at a short notice and their capacity, r e l a t i v e to Iceland's needs, is huge, i t seems plausible to assume that investment can take place at any relevant rate at fixed prices. However, a time lag of 1-1.5 years between order and delivery must be allowed for. According to the available s t a t i s t i c s , the investment price of the two types of fi s h i n g vessels on which th i s study focuses, are given in the following table: 231 Table 5.16 Fishing Capital Investment Prices. Vessel classes Unit price (M.Ikr. 1974 prices) Multi-purpose vessels: Trawlers: 54.0 167.0 Source: National Economic Inst i t u t e ; (Unpublished material). Investment in a fis h i n g vessel does not mean that costs of the magnitude l i s t e d in table 5.16 are incurred. Investment is simply a transformation of one asset into another. The costs associated with a pa r t i c u l a r piece of investment are ref l e c t e d in the reduction in i t s resale value with the passage of time. To some extent at least t h i s is ref l e c t e d in depreciation rates that are included in the estimated harvesting cost functions above. Disinvest takes place through resale. For the demersal fl e e t as a whole th i s means export of used f i s h i n g vessels. What information there exists on foreign resale markets suggests that very considerable transaction costs are involved. They include a very s i g n i f i c a n t loss of operating time, agent fees, high price discounts r e l a t i v e to the depreciated investment value of the vessels etc. Given these costs, however, i t seems that a great number of vessels may be unloaded with a lead time of, say, 0.5-1.5 years. This disinvestment situation thus resembles that of imperfect resale markets as discussed by Clark et a l . <117>. Let the resale costs as a fraction of the depreciated investment price, p, say, be represented by q. The c a p i t a l 232 adjustment costs are thus defined by the equation: (50) C3(t)=qpDI(t), where DI(t), i t w i l l be recalled, represents disinvestment, q may be regarded as a measure of c a p i t a l m a l l e a b i l i t y , with q=0 suggesting perfect c a p i t a l m a l l e a b i l i t y . More conveniently, define a m a l l e a b i l i t y parameter, m, say, as follows: m=1-q. Hence m=1 suggests perfect m a l l e a b i l i t y . 233 Appendix 5.5-A. Fishing Capital Equilibrium. According to the previous section, the internal dynamics of the f i s h i n g c a p i t a l are described by the following equations: (A.1) n(t+1)=n(t)+I(t+1)-DI(t+1)-D(t+1), (A.2) D(t+1)=S(a(t))n(t), (A.3) a(t+l)=(a(t)+l)w(t)+l(l-w(t)), (A.4) w(t)=(n(t)-DI(t+1)-D(t+1))/n(t+1). In f i s h i n g c a p i t a l equilibrium: (A.5) n(t)=n(t+1)=n*, say, and (A.6) a(t)=a(t+1)=a*, say. Imposing (A.5) and (A.6) upon (A.1)-(A.4) and simplifying y i e l d s : (A.7) n*=(l(t)-DI(t))/S(a*), (A.8) a*=n*/l(t). Hence the equilibrium age and vessel number i s given by the equat ions: (A.9) a*S(a*)=1-DI(t)/I(t), (A.10) n*S(n*/l(t))=I(t)-DI(t). If DI(t)=0 for a l l t, (A.9) reduces to: (A.9') a*S(a*)=1. Therefore, in t h i s special case, there i s a unique autonomous equilibrium solution for a. In fact, according to the estimate of S(a) in section 5.5.3, a*=21.5 years. The corresponding equilibrium f l e e t size, on the other hand, may be any p o s i t i v e number, since the remaining equilibrium solution i s simply that t o t a l depreciation equals investment. I.e. by (A.10): (A.10') n*S(a*)=I(t). 234 5.6 The Rate of Discount. Given certain well known regularity conditions upon the structure of preferences and production p o s s i b i l i t i e s , there exi s t s , under an i n s t i t u t i o n a l arrangement c a l l e d market economy, a set of equilibrium prices <118>. Under some further conditions, these equilibrium prices, moreover, support a Pareto optimal a l l o c a t i o n of resources <119>. For our present purposes, i t is convenient to represent these prices by the NT-dimonsional row vector: p=(p(n,t)), n=1,2,..N,t=1,2,..T, where the n subscript refers to the commodity and the t subscript to the period in question. T, of course, does not have to be f i n i t e . Now, l e t N refer to the money commodity. Then, by convention <120>, the nominal discount factor i s defined by the one period r e l a t i v e money prices as follows: (51) d(t)=p(N,t+1)/p(N,t). The corresponding nominal rate of discount is defined by: (52) r(t)=p(N,t)/p(N,t+1)-1. In an i n f l a t i o n a r y environment, the price of money one period hence i s given by: 235 p(N,t + 1)=p*(N,t+1)/(1+s(t)) , where s(t) stands for the i n f l a t i o n rate between t and t+1 and p*(N,t+1) represents the undeflated money price at period t+1. Define the real rate of discount as: (53) r*(t)=p(N,t)/p*(N,t+1)-1. From this and (52) i t follows that the real rate of discount i s related to the nominal one by: (54) r * ( t ) = ( r ( t ) - s ( t ) ) / ( 1 + s ( t ) ) . It should be noticed, in this context, that, since the seperating hyperplane that defines the equilibrium price vector depends on the structure of preferences and production p o s s i b i l i t i e s so also w i l l the real rate of discount. Hence, in addition to being an e x p l i c i t function of time, the rate of discount w i l l depend on the development of tastes and production p o s s i b i l i t i e s . Actual market economies t y p i c a l l y generate a c o l l e c t i o n of market discount rates. As other actual prices, however, the market rates of discount may not accurately r e f l e c t the Pareto optimal ones <121>. In the case of Iceland, however, these thorny problems need not detain us. A large fraction of Icelandic investment i s financed by c a p i t a l raised in international f i n a n c i a l markets <122>. Since these borrowings 236 can apparently be r e l a t i v e l y freely expanded and contracted, the corresponding rate of interest defines the relevant cost of c a p i t a l to the Icelandic economy and thus the appropriate discount rate. According to o f f i c i a l s t a t i s t i c s on the nominal rate of interest paid by Iceland on foreign debt in recent years and the repective rates of i n f l a t i o n , the real rate of interest was as l i s t e d in table 5.17. Table 5.17 Iceland's Real Rate of Interest on Foreign Dept. Source: Iceland's Central Bank (partly unpublished material). For the present value calculations in part III of this study, i t seems reasonable to use the most recent of these rates, i . e . 3.63%, as an estimate of the appropriate discount rate. This number p a r t i a l l y r e f l e c t s recent developments in international f i n a n c i a l markets as well as being f a i r l y close to the h i s t o r i c a l real interest rate. However, in addition to the approximate nature of th i s p a r t i c u l a r number, the reader i s reminded of the i m p l a u s i b i l i t y of employing a fixed rate to discount an income stream over possibly a very long period. Periods Average rate of interest % 1980-83 1975-83 1970-83 3.625 3.011 1 .543 237 Footnotes. 1. This function was christened V(A) in section 3.1 where A indicates a p a r t i c u l a r harvesting program. 2. A negli b i b l e f r a c t i o n , well below 1%, of the t o t a l demersal catch is consumed domestically. 3. Most of the quantitative data in this section i s obtained from F i s k i f e l a g Islands, 1978a and b, and pertains, unless otherwise s p e c i f i e d to 1977. Since then structural changes have been n e g l i g i b l e . 4. In other periods they are either i d l e or engaged in pelagic f i s h e r i e s . 5. For further d e t a i l s see Jonsson (1978). 6. It is interesting to compare th i s figure with figures 4.3 and 4.4 in section 4.2. 7. With population t y p i c a l l y ranging from about 500 to 5000 individuals. 8. This information was primarily extracted' from F i s k i f e l a g Islands, 1978a. 9. The pattern of landings with respect to the r e l a t i v e productivity of the fishing grounds, the location of the ports and their capacity and possible crowding raises several interesting e f f i c i e n c y issues some of which are discussed in Jensson, 1975. 10. These contracts are usually made for one season at a time. 11. There are deviations but they are i n s i g n i f i c a n t . 12. The q u a l i t y of the catch refers here to both i t s freshness and size d i s t r i b u t i o n . 13. I.e. i t s species composition and q u a l i t y . 14. During 1974-80, for instance, the simple c o r r e l a t i o n c o e f f i c i e n t between annual percentage changes in the price of a unit catch of cod and i t s export price was only 0.29. 15. This example i s , of course, a greatly s i m p l i f i e d version of the f i s h e r i e s model of this work. It , nevertheless, manages to convey adequately the main message of the current argument. 16. Within these processes there are, of course, several subprocesses, each resulting in a particular type of f i n a l product. 17. This i s doubtlessly due to some economics of running these 238 processes concurrently. 18. For more d e t a i l s , see Appendix 1: Data (section 5.2). 19. In 1981 over 33 marine species were commercially harvested in Icelandic waters on a s i g n i f i c a n t scale. 20. Thus, from a s t a t i s t i c a l point of view, these variables are exogenous or at least predetermined. 21. Of course, I=I(1)+I(2). 22. To see t h i s define new price vectors as p'=ap and w'=aw, where a is a nonnegative real scalar. From (6) itT i s ~ c l e a r ~ that the p r o f i t function corresponding to these prices is simply H(p',w',x)=aH(p,w,x). Therefore the p r o f i t maximizing factor demand functions,~ x*(2), are independent of a and from (7) we immediately deduce: ~ C(p',w',x(1))=aC(p,w,x(1)). 23. The quantitative data are l i s t e d in Appendix 1:Data (section 5.2). 24. In s t a t i s t i c a l currency, the value of the disaggregated approach may be inferred from the results of a test on the v a l i d i t y of aggregation over firm types to be presented below. 25. I f , however, the aggregation r e s t r i c t i o n s are rejected by the data this cost i s largely i r r e l e v a n t . 26. The reason the estimation is carried out in two stages i s to reduce the computational costs. 27. For d e t a i l s on the data and i t s origins see appendix: data (sect ion 5.2). 28. See I n t r i l i g a t o r , 1978, pp. 64, on t h i s . 29. The structure of the variance-covariance matrix of the u(s,t)'s i s important from the point of view of e f f i c i e n t estimation of the parameters of this function. This issue i s examined in appendix 5.2-A. 30. This power transformation, suggested by Box and Tidwell, 1962, provides a f l e x i b l e functional form for (8), i . e . i t yiel d s good l o c a l approximation to any continuous functional form. For more d e t a i l s see appendix 5.2-C. 31. This suggests, moreover, that the underlying production technology i s semi-linear or simpler. For the cost function to be linear in r e s t r i c t e d inputs, the factor demand functions must be linear in the same inputs. This requires, however, the f i r s t derivatives of the production function to be linear or constant. If t h i s i s the case, we say 239 that the production function i s semi-linear. 32. See e.g. Wallace and Hussain, 1969, on t h i s . 33. This implies, as previously mentioned, that the hypothesis that the underlying production function i s semi-linear cannot be rejected. 34. See e.g. Schmidt, 1976, pp.67. 35. The relevant l i k e l i h o o d function i s given by: L(y,X;b,z)=aDet(U)",texp(-(y-Xb) ' U"'(y-Xb)/2 ) , where z refers to the unknown parameters of U and a is a constant given the sample. ~ The actual numerical procedure used, however, was an i t e r a t i v e one, which, subject to the usual provisions, yields results a r b i t r a r i l y close to the maximum l i k e l i h o o d ones. 36. For d e t a i l s on the data on thi s class of firms as well as the others discussed below, see appendix: data. 37. For more information on t h i s test consult appendix 5.2-B. 38. For a brief discussion about th i s test see appendix 5.2-B. 39. Notice, however, that the l i k e l i h o o d r a t i o test i s designed as a large sample test and i s thus should not be taken e n t i r e l y at face value in this case. 40. On the construction of these tests, see appendix 5.2-B. 41. As pointed out in section 5.1, thi s i s , however, unlikely to be of significance in thi s p a r t i c u l a r case. 42. Good references are contained in Swamy, 1974, and Judge et a l . , 1980. 43. See e.g. Wallace and Hussain, 1969, Swamy, 1974, and Judge et a l . , 1980. 44. Compare t h i s with the disturbance structure in (A.2) above. 45. See e.g. Schmidt, 1977, pp. 1. 46. See e.g. I n t r i l i g a t o r , 1978, pp. 165. 47. See e.g. Schmidt, 1957, pp. 22-31. 48. For more information, see e.g. Silvey, 1970, pp. 108. 49 See Box and Tidwell, 1962, and Box and Cox, 1964. 50. See e.g. White, 1972, and Judge et a l . , 1980, pp. 310. 240 51. In a world of perfect c a p i t a l markets, only one of these two classes of decisions poses an independent problem. In the real world, however, this well known result, see e.g. Clark et a l . , 1979, is largely irrelevant. 52. In general this investment may be either negative or p o s i t i v e . The c a p i t a l issue i s further examined in section 5.5 below. 53. See e.g. Malinvauld, 1972, pp. 64-68. 54. See e.g. Diewert, 1978 and the references therein. 55. Actually, most of the argument applies to other natural resource extraction as well. 56. For a more detailed discussion of t h i s p a r t i c u l a r concept of competitiveness see appendix 5.3-A. Here i t should s u f f i c e to state that a high degree of competitiveness w.r.t. the resource base corresponds to a low imputed shadow price of the resource by each firm. 57. This i s , of course, a highly s i m p l i f i e d version of the actual p r o f i t maximization problem solved by the firms. The essential argument i s not thought to be dependent upon t h i s s i m p l i f i c a t i o n , however. 58. This should actually be c a l l e d the f i r s t axiom of natural resource economics. If production, or u t i l i t y for that matter, were independent of the resource stock, problem (13) would be indistinguishable from a standard neoclassical p r o f i t maximization problem and there would be no need for a separate natural resource theory. The functional dependence of costs upon the resource stock does not, however, have to be monotonic or cont inuous. 59. For confirmation of t h i s well known fact, see e.g. Malinvauld, 1972, and Diewert, 1978. 60. Notice f i r s t , following the approach described in footnote 22 above, that the p r o f i t maximizing factor demand functions are homogeneous of degeee zero in g and w. The claimed result then follows immediately from equation (16). 61. Assuming that the dimensionality of e* at least equals that of x, the necessary and s u f f i c i e n t conditions for t h i s result are~ embodied in the i m p l i c i t function theorem, see e.g. Chiang, 1974, p. 218. 62. Such as on what fishery to pursue, what gear to use, how much time to spend on f i s h search etc. 63. In fact, in the Icelandic demersal f i s h e r i e s , i t seems that, given the vessel type and fi s h i n g gear, economic input combinations are largely fixed. This may, of course, hold for 241 many other types of production as well. 64. I.e. equation (17). 65. A detailed discussion, as well as l i s t i n g , of the data i s provided in appendix: data (section 5.3.2). However, as explained there, the data for 1975 are incomplete. 66. I.e. what species the f i s h i n g a c t i v i t y i s directed towards. 67. See Box and Tidwell, 1962. 68. As discussed in appendix 5.2-C, the Box-Tidwell transformation provides a l o c a l approximation to any continuous functional form. As such, i t includes various functions, notably the linear and log-linear forms as special cases. Other f l e x i b l e functional forms, i . a . the translog and generalized Leontief forms are not well suited to functions of many explanatory variables as i s the case here. 69. As an example, p(X) = ( (p( 1 ) V , )-1 )/X( 1 ) , (p(2)*°-1 )/X(2) , . . .) . 70. H=6, J=5 and 1=5, where N, J, I are the t o t a l number of c a p i t a l variables, species and f i s h i n g a c t i v i t i e s contained in the sample. 71. Assuming f i r s t that the systematic part of (23) may be reprsented as: c/w=Lk(h)+ZL(aj 1)p'+a(2)w+aj3)k'+a(4)e'+a(5)y'), =Zk(h)+IJ(a(2)w+Za(3,h)k(h)+Za(4,s)e(s)) + I ( Z a d , j)p( j)+Ia(5, j)y( j) ) . j J where A indicates Box-Tidwell transformations and ' a transpose. Assume, moreover, that the following approximation holds: L(k(h)+IJa(3,h)k(h))=b(0)+Lb(1,h)k(h). where the b( )'s are parameters. Thus, defining the b( ) c o e f f i c i e n t s appropriately, (24) follows. 72. This c l a s s i f i c a t i o n i s in conformance with F i s k i f e l a g Islands, 1978b. 73. E s s e n t i a l l y measured as days at sea. 74. Since, given the data, the concentrated l i k e l i h o o d function i s : L=a1-a21og(c-C(X;b,q))'(c-C(X;b,q)), where a1 and a2 are constants. 242 75. The test consisted of f i r s t running the following simple regression: Dvar(u)= Z a ( i ) D ( i ) , c-1 where Dvar(u) denotes the deviation of the squared residual from the sample residual variance, the a ( i ) ' s are c o e f f i c i e n t s and the D(i)'s are dummy variables equalling one in year i and zero otherwise. The subsequent F-test on HO: a(i)=0, a l l i , was taken to be a test of hereroscedasticity. This test i s similar to the so-called modified Glejser t e s t . See Goldfeld and Quandt, 1972, pp. 93. 76. These landings t r i p s to Europe were not uncommon during the sample period. 77. For further d e t a i l s , see Arnason, 1979. 78. The arguments of this appendix are largely derived from the putty-clay hypothesis in production theory. See e.g. Gapinski, 1982 pp. 300. 79. The concept of technology i s here used in a broad sense including the i n s t i t u t i o n a l arrangements that are usually associated with geographical locations. 8 0 . See equation (9) in chapter 3. 81. It may be noticed that equations (1) and (2) also define the revenue function for the f i s h processing industry as R2(t)=R(t)-R1 (t) . 82. A d e t a i l discussion of t h i s can be found in chapters 3, especially section 3.2.4. 83. Within these main categories there are of course several subproducts that are below the l e v e l of disaggregation in t h i s study. 84. Compare t h i s with equation (10) in chapter 3. 85. The volume loss at sea primarily stems from the effects of preliminary processing and storage aboard the vessels. 86. I.a. by choosing an appropriate preservation technology. 87. In fact, according to the o f f i c i a l s t a t i s t i c s , this parameter has remained constant for the past 10 years or more. While t h i s estimate is hardly e n t i r e l y accurate, i t nevertheless suggests a high l e v e l of s t a b i l i t y . 8 8 . It mainly consists of various f i s h remains such as bones etc. 89. In fact, according to the o f f i c i a l measurements, the 243 d ( i , j ) ' s have remained remarkably invariant over time. 90. It i s readily seen that c ( i , j ) , in fact, i s composed of d ( i , j ) and a more fundamental process transformation parameter, e ( i , j ) , say, where c ( i , j ) = e ( i , j ) ( 1 - d ( i , j ) ) . Here c ( i , j ) is used because i t f i t s the available data. For more d e t a i l s see appendix: data (section 5.4). 91. I.e. The Icelandic Fisheries Association, The Marine Research Institute and The National Economic I n s t i t u t e . 92. For d e t a i l s of this derivation see appendix 5.4-A: Analysis. 93. A similar kind of share equations and adding-up constraints have been extensively examined in consumer demand theory. See e.g. Deaton and Muellbauer, 1980. 94. For d e t a i l s see Appendix 1: Data (section 5.4). 95. By "standard f l e x i b l e functional form approach" we mean approaches such as the Box-Tidwell-Cox and translog transformations. These were considered "unattractive" in t h i s case because of the functional and consequently numerical complexity resulting from imposing these transformations on a system of equations with cross equation constraints. 96. This, in a l l l i k e l i h o o d , is not an insurmountable problem, however. 97. But suggests, on the other hand, the a d v i s a b i l i t y of proceeding with caution in simulations. 98. The functional r e s t r i c t i o n s on a cost function compatible with these linear process a l l o c a t i o n functions are stringent as indicated in appendix_ 5.4-A: Analysis. 99. See Appendix 1 :Data ( sect ion 5.4). 100. See e.g. Zellner, 1962. 101. For d e t a i l s on the Weierstrass theorem see e.g. Takayama, 1974, pp. 29-30. 102. A statement and discussion of the i m p l i c i t function theorem can e.g. be found in Chiang, 1967 pp. 222-223. 103. For d e t a i l s see chapter 5.1. 104. This, of course, does not exclude exogenous s h i f t s in k(i ,t) . 105. This w i l l be recognized to be a variant of the standard c a p i t a l growth identity in neoclassical economics. See e.g. Solow, 1956, and Hahn and Matthews, 1964. 244 106. Later we w i l l e x p l i c i t l y assume that investment and disinvestment takes place at the outset of each period, [ t , t + l ] , 107. More r e a l i s t i c a l l y , disinvestment might take place in vessels at any age. However, since the current approach only allows a single age variable at a point of time, namely a ( i , t ) , t his assumption is natural. 108. This i s equivalent to assuming that new investment takes place instantly at the beginning of each period, ( t , t + l ) . 109. This follows from equation (44) and the d e f i n i t i o n s of no(t+1) and nn(t+1). 110. INT(x) i s the integer closest to x including x i t s e l f . 111. See NEDI, 1977 pp. 37-48. 112. Notice that by normalization N(0) equals unity. 113. See NEDI, 1977, pp 37-48. 114. See e.g. Burmeister, 1980, p. 39. 1 V5. See Clark, Clarke and Munro, 1979. 116. See NEDI, 1977. 117. See Clark, Clarke and Munro, 1 979. 118. For d e t a i l s see e.g. Debreu, 1959, e s p e c i a l l y chapter 5 1 19. Again see Debreu, 1959, especially chapter 6. Also Takayama, 1974, pp. 185-201. 120. See e.g. Debreu, 1959, pp.33-34, and Burmeister, 1980, pp 9-25. 121. A very good discussion of the problems with actual market rates of interest can be found in Dasgupta, Marglin and Sen, 1972, chapter 13. 122. The Icelandic foreign debt service r a t i o in the last few years has hovered around 20% of t o t a l foreign earnings. 245 6. The Fishing Mortality Production Function. The basic structure of the empirical model of th i s essay was outlined in chapter 3 above. There i t was assumed that the b i o l o g i c a l and economic parts of the model could be linked by a special function mapping economic inputs into f i s h i n g m o r t a l i t i e s . Having many of the c h a r a c t e r i s t i c s of a production function t h i s function may be c a l l e d the f i s h i n g mortality production function. In this chapter we w i l l develop this idea in more d e t a i l s both a n a l y t i c a l l y and empirically. The main aim of th i s chapter i s to derive a t h e o r e t i c a l l y reasonable f i s h i n g mortality production function and estimate i t s form and parameters for the Icelandic demersal f i s h e r i e s . This is the subject of sections 6.2 and 6.3 below. F i r s t , however, in section 6.1, the conventional s p e c i f i c a t i o n s of the fis h i n g mortality function are reviewed and their l i m i t a t i o n s discussed. One of the general results of th i s chapter i s that for both a n a l y t i c a l and empirical reasons the widespread assumption of a linear r e l a t i o n s h i p between f i s h i n g mortality and economic inputs <1> can hardly be maintained. In p a r t i c u l a r , f i s h i n g mortality may, in addition to being a more complex function of economic f i s h i n g inputs, depend on the volume of the f i s h stocks. If t h i s i s indeed the case, i t w i l l be appreciated that fishery p o l icy recommendations, based on the usual simple l i n e a r i t y assumption, w i l l , in general, be unsound. The theoretical implications of th i s are b r i e f l y explored in the last section of th i s chapter. 246 6.1 A Brief Review of the L i t e r a t u r e . Constituting as i t were the link between the a c t i v i t y of the fishing f l e e t and the population dynamics of the f i s h stocks i t is unavoidable that a f i s h i n g mortality production function be included in any complete f i s h e r i e s model <2>. Moreover, as the p i v o t a l role of this function indicates, i t s s p e c i f i c a t i o n fundamentally determines important properties of the model, not the least i t s optimality properties <3>. As discussed at length in section 4.1 above, there are two basic approaches to f i s h population dynamics, the aggregated one as exemplified by the Schaefer harvesting function, and the disaggregated one of which the Beverton-Holt model is the prime example. Apart from differences in aggregation structure, however, these two f i s h e r i e s models contain the same basic fishing mortality production function. We w i l l now b r i e f l y demonstrate t h i s : For one cohort the Beverton-Holt instantaneous f i s h i n g mortality production function i s <4>: (1) f(t)=ae(t). Where f ( t ) i s the fishing mortality of the cohort and e(t) the so-called f i s h i n g e f f o r t , both at time t. a i s simply a constant factor of proportionality between the two variables. The Schaefer harvesting function is <5>: (2) y(t)=be(t)x(t). 247 Where y(t) denotes catch, e(t) fis h i n g e f f o r t and x(t) the volume of the f i s h stock, i . e . biomass, a l l at time t. b i s a positive constant, sometimes c a l l e d the c a t c h a b i l i t y c o e f f i c i e n t <6>. Now, to show the equivalence of (1) and (2), l e t n(t) denote the number and w(t) the weight of individual f i s h of the cohort of question. Then, by the d e f i n i t i o n s of fis h i n g mortality, instantaneous catch and biomass <7>: f(t)=-n(t)/n(t)=-w(t)n(t)/w(t)n(t)=y(t)/x(t) . Hence i t i s readily seen that the instantaneous Schaefer harvesting function, (2), i s equivalent to the Beverton-Holt fishing mortality production function, (1). It follows that the Schaefer harvesting function i m p l i c i t l y defines a fi s h i n g mortality production function of the Beverton-Holt kind. Moreover the c a t c h a b i l i t y c o e f f i c i e n t , b, and the f i s h i n g mortality factor of proportionality, a, are i d e n t i c a l . A somewhat more general version of the Schaefer function, (2), i s the recently popular type of harvesting functions <8>: y(t)=Y(x(t))e(t), which implies the fishing mortality function: (3) f ( t ) = e ( t ) Y ( x ( t ) ) / x ( t ) . 248 Clearly (1) and (3) are very r e s t r i c t i v e functional forms. Among other things they exhibit a constant, in fact unitary, e l a s t i c i t y of fishing mortality with respect to f i s h i n g e f f o r t . Although th i s c l e a r l y s i m p l i f i e s the arithmetic in the fishery planning o f f i c e , there is of course no assurance that t h i s benefit exceeds the cost of a potential misspecification. Nevertheless t h i s class of fi s h i n g mortality functions largely dominates the theoretical l i t e r a t u r e on fishery economics and ce r t a i n l y the applications of thi s theory <9>. A generalized version of the Schaefer harvesting function, is <10>: (4) y(t)=Y(e(t),x(t) ) , implying the fis h i n g mortality function: (5) f ( t ) = Y ( e ( t ) , x ( t ) ) / x ( t ) , which includes (3) as a special case. (5) does not impose a p r i o r i r e s t r i c t i o n s on the e l a s t i c i t y of fi s h i n g mortality with respect to fis h i n g e f f o r t . Given e l a s t i c i t i e s of fishing mortality may be i d e n t i f i e d with certain fishery s i t u a t i o n s . An e l a s t i c i t y below unity may for instance be attributed to congestion on the fishing grounds. An e l a s t i c i t y exceeding unity may' on the other hand be interpreted as economies of scale in fishing due for instance to cooperation in locating f i s h schools. Of course, the same fis h i n g mortality production function may ex i b i t variable e l a s t i c i t y of fi s h i n g mortality 249 over i t s domain <11>. A key fault with equation (5), and consequently the other less general f i s h i n g mortality functions, i . e . (1) and (3), i s that these formulations imply harvesting functions in which e(t) and x(t) are mathematically separable in a certain sense to be made more precise in section 6.4 below. Here i t may be stated in very simple terms that according to equation (5), economic inputs are d i r e c t l y translated into catch without the intermediation of the stock size. This, of course, may not be empirically accurate. In fact the results below indicate that at least for the Icelandic demersal f i s h e r i e s the opposite i s true. A more general fi s h i n g mortality production function in th i s respect i s <12>: (6) f(t)=F(e(t),x(t)) . As (5), th i s s p e c i f i c a t i o n does not r e s t r i c t the e l a s t i c i t y of fish i n g mortality with respect to fis h i n g e f f o r t at a l l . Moreover, i t permits the volume of the f i s h stocks to influence the effectiveness of economic inputs in generating f i s h i n g m o r t a l i t i e s . This function, of course, contains (1) and (3) as special cases. 250 6.2 Analysis. In t h i s section we w i l l examine the elements of the fishing a c t i v i t y . By fis h i n g a c t i v i t y we mean the process of applying economic inputs for the purpose of generating output in terms of catch. Since the focus i s on the elements of fi s h i n g a c t i v i t y that are common to most, i f not a l l , f i s h e r i e s the conclusions of t h i s section should be widely applicable. As already mentioned our a n a l y t i c a l aim is to obtain reasonable mathematical formulation of the fis h i n g mortality production function. The analysis suggests the s p e c i f i c a t i o n of two separate functions; a fi s h i n g mortality function and a fish i n g time function. 6.2.1 The Fishing Mortality Function. Let there be 1 economic inputs in the harvesting process. Each fi s h i n g e n t i t y , a firm, say, selects a vector of these inputs according to i t s objectives and constraints. Denote these input vectors by e=(e(1),e(2),...,e(l)). Assuming that only nonegative e(.)'s can be chosen, eeR+ . Let there moreover be m species of f i s h and n cohorts for each species <13>. Denote the fi s h i n g mortality of the i t h cohort of the j t h species by f ( i , j ) . The vector of fishing m ortalities then is f=(f(1,1),f(2,1)...,f(n,m)). Also by the constraints of nature feR. . F i n a l l y , l e t the environmental conditions, including the number of f i s h in each cohort, be represented by the (1xk) vector, z. 251 The f i s h i n g mortality generated by each firm can now be formally defined as the following mapping: F: (e,z)—>f, eeR+ ,zeRk ,feR","> . <•>- ~ ^ — ~ ~ This mapping i s formally analogous to a joint production function in standard economic theory <14>. For thi s reason i t s name, f i s h i n g mortality production function, i s not inappropriate. Having set out this general case, there are some rather obvious benefits from r e l a t i n g i t to standard fishery theory the c r u c i a l variable of which i s fi s h i n g e f f o r t . Hence define instantaneous fishing e f f o r t , e ( t ) , as the result of applying economic inputs to f i s h stocks: (7) e(t)=E(e)5. Where the s h i f t variable, 6, equals 1 i f actual fi s h i n g i s being conducted and 0 otherwise. The s p e c i f i c a t i o n of (7) represents an attempt to capture what are, in our b e l i e f , important empirical features of the fis h i n g a c t i v i t y . Hence a few additional comments may be useful. Notice, f i r s t , that (7) i s not a behavioural assumption. It merely defines a new variable c a l l e d f i s h i n g e f f o r t . According to (7) f i s h i n g e f f o r t is generated only when actual fi s h i n g is taking place and not otherwise. This is in accordance with standard fishery biology where fis h i n g e f f o r t i s i d e n t i f i e d with 252 the act of f i s h i n g . In fact, what fishery b i o l o g i s t s usually refer to as f i s h i n g e f f o r t are measures such as trawl-hours, time f i s h i n g etc. <15>. Secondly, any fishery consists of a number of s u b a c t i v i t i e s some of which are not and cannot be c a r r i e d out simultaneously. These s u b a c t i v i t i e s include i . a . actual f i s h i n g , preliminary f i s h processing, f i s h search, s a i l i n g between the fishing grounds and the f i s h i n g port, landing the catch etc. The number of economic inputs is correspondingly large, and their use in the various s u b a c t i v i t i e s uneven. Only one of the f i s h i n g s u b a c t i v i t i e s , actual f i s h i n g , i s d i r e c t l y related to the generation of f i s h i n g m o r t a l i t i e s . It follows that only a subset of the economic inputs used in a t y p i c a l fishery are d i r e c t l y engaged in producing f i s h i n g m o r t a l i t i e s . Rather than redefining the vector e to r e f l e c t t h i s empirical proposition we here delegate the task of c o r r e c t l y mapping the complete vector of economic inputs into an appropriate measure of f i s h i n g e f f o r t , e ( t ) , to a special function, E ( . ) . F i n a l l y , in most f i s h e r i e s many important economic inputs exhibit the c h a r a c t e r i s t i c s of stocks rather than flows. These include the vessel and i t s a t t r i b u t e s , the f i s h i n g gear, the crew etc. The flow inputs, l i k e fuel for instance, are for technical reasons generally used at a f a i r l y i n f l e x i b l e rate, given the p a r t i c u l a r f i s h i n g a c t i v i t y . Once the f i s h i n g t r i p , or even the f i s h i n g season, has commenced the stock variables and the technology are given and fixed. The economic decisions that remain thus primarily concern the choice between the various possible f i s h i n g a c t i v i t i e s . Hence, once the f i s h i n g t r i p i s 253 under way, the f i s h i n g e f f o r t exerted depends upon (i) the available economic input stocks and ( i i ) whether and for how long the vessel is engaged in fishing a c t i v i t y . This l a t t e r factor i s represented by the s h i f t variable 6. Referring back to the t r a d i t i o n a l l i t e r a t u r e on f i s h i n g mortality production functions, in particular the Beverton-Holt function, (1), in the previous section, we immediately derive with the help of (7): (8) f(t)=aE(e ) 6 . "V/ A l t e r n a t i v e l y , on the basis of the general f i s h i n g mortality function (6) in section 6.1 and generalizing i t s t i l l by substituting the vector z for x(t) we obtain: f(t)=F(E(e )6,z) . Now, i t seems axiomatic that no fishing e f f o r t produces no fi s h i n g mortality, i . e . F(0,.)=0. It follows that the l a s t equation can equivalently be written as: (9) f(t)=F(E(e),z ) 6 . Now, fishery data are usually not available in continuous form. Therefore, in order to apply these fishery mortality functions, i t i s necessary to obtain their time integrals. More precisely we seek the i n t e r v a l f i s h i n g mortality: F = / i f (t)dt = / i F(E(e) ,z)odt. 254 Where F represents the fishing mortality over the time in t e r v a l [0,1]. The integration exercise is complicated by the vectors e and z being dependent on time. Thus, in order to obtain tractable r e s u l t s , we resort to some simplifying assumptions. F i r s t l y , we take i t to be a general empirical rule, that during each f i s h i n g t r i p , say, a fis h i n g entity only intermittently engages in actual f i s h i n g and then only for a certain duration of time. C a l l these inte r v a l s fishing periods. Between fi s h i n g periods the entity performs other f i s h i n g a c t i v i t i e s . Secondly, we assume that the f i s h i n g periods are r e l a t i v e l y short, l a s t i n g perhaps a few hours or less, so that we may take i t that the vectors e and z are constant during each f i s h i n g period. Since, as discussed above, the elements of e are mostly stocks that are normally not altered during each f i s h i n g t r i p this assumption does not seem unduly r e s t r i c t i v e with respect to e. The constancy of z, on the other hand, depends heavily on the length of each fis h i n g period. What i s c r u c i a l here i s of course the length of the fis h i n g period r e l a t i v e to the rate of change of z. In any case i t is clear that t h i s constancy assumption is only approximate. Now, on the basis of the above assumptions the integration problem is t r i v i a l . Refer to separate f i s h i n g periods by the index k, k=1,2,..K during [0,1], We then f i n d : F=f f (t)dt= Z F(k)t(k) . 255 Where t(k) denotes the length of the k-th f i s h i n g period. Moreover, (10) F= 2 F(k)t(k)= I {F(E(e(k)),z(k))t(k)/tf}tf=F(E(e),z)tf. Where e and z are the appropriate weighted average vectors of economic inputs and environmental conditions respectively and tf=Lt(k), i . e . the t o t a l f i s h i n g time during [0,1]. S i m i l a r l y , for the simple case defined by equation (8) we der ive: (11) F=F(E(e))tf. It is perhaps helpful to pause here to compare equations (8) and (9) as well as their integrated counterparts (10) and (11) with the standard f i s h i n g mortality functions of the l i t e r a t u r e defined by equations (1) and (3) in section 6.1 above. F i r s t , what is c a l l e d f i s h i n g e f f o r t in the t r a d i t i o n a l approach of section 6.1 appears here as a multiple of t o t a l f i s h i n g time and a possibly complex function of economic inputs and environmental variables. Thus, from this point of view, the concept of f i s h i n g e f f o r t appears as an attempt to represent a multidimensional phenomenon by a univariate measure. Al t e r n a t i v e l y , taking the input stocks as predetermined, tf in equations (10) and (11) may be regarded as a convenient measure of fi s h i n g e f f o r t . This, in fact, seems to be the intention of 256 the p r i n c i p a l authors in the t r a d i t i o n a l f i s h i n g mortality l i t e r a t u r e <16>. If this is the point of view taken, an important contribution of equations (10) and (11) i s to demonstrate the potential complexity of the so-called c a t c h a b i l i t y c o e f f i c i e n t in the t r a d i t i o n a l approach <17>. According to the present analysis this c o e f f i c i e n t depends on the input stocks and very probably also on a series of environmental conditions <18>. On the whole, however, our results do not contradict the t r a d i t i o n a l theory on fishery mortality functions. This is of course not surprising, since through equations (1), (6) and (7) we i n t e n t i o n a l l y adopted the t r a d i t i o n a l framework. The present analysis rather amounts to an elaboration of the t r a d i t i o n a l theory. It uncovers the assumptions necessary to obtain the usual simple versions of f i s h i n g mortality functions and e x p l i c i t l y states their potential complexity. A more fundamental contribution of the above analysis, however, i s to make i t clear that there is no reason to expect a simple re l a t i o n between fi s h i n g mortality and economic costs. Taking F(E(e),z)tf in (10) as proportional to f i s h i n g e f f o r t i t i s clear that this variable is not necessarily closely related to the use of economic inputs and thus costs. This suggests a d i s t i n c t i o n between what may be c a l l e d b i o l o g i c a l fishing e f f o r t and economic fishing e f f o r t . The former concept being closely related to f i s h i n g mortalities and the l a t t e r to the use of economic inputs. Fishery b i o l o g i s t s have spent much time ref i n i n g their measures of b i o l o g i c a l f i s h i n g e f f o r t . Economists, on the other hand, tend to proceed in terms of 257 economic f i s h i n g e f f o r t . Both sides largely ignore the d i s t i n c t i o n between the two concepts <19>. In the remainder of this chapter the term fishing e f f o r t w i l l refer to b i o l o g i c a l f i s h i n g e f f o r t unless otherwise e x p l i c i t l y stated. Turning our attention b r i e f l y to the harvesting functions corresponding to the fi s h i n g mortality functions we notice f i r s t that fishing e f f o r t i s generally not applied to the f i s h stocks as a whole. Rather i t is applied to the f i s h concentrations that have been located or found by the vessel in question. C a l l t h i s quantity nf. nf i s of course a result of f i s h search. The optimal search process including the decisions whether and for how long to search for f i s h is of course a complicated issue that i s not central to this essay. However, i t may be pointed out to the interested reader that the job search l i t e r a t u r e is to a considerable extent concerned with analogous situations and offers results that are seemingly applicable to f i s h search <20>. It seems reasonable that nf depends on the vector of economic fishery inputs, e, environmental conditions, z, including the t o t a l size of the cohort and the accumulated search time, t s . Thus write the f i s h concentrations located as: (12) nf=NF(e,z,ts). It also seems reasonable that NF(.,.,.) i s nondecreasing in e and ts <21>. Now, for a single vessel, we have by d e f i n i t i o n : 258 y ( t ) = n f ( t ) f ( t ) . Hence combining (12) with (8) and (9) yiel d s the instantaneous harvesting functions: y(t)=aE(e)NF(e,z,ts)6, y(t)=F(E(e),z)NF(e,z,ts)5. We now assume that f i s h search and actual f i s h i n g can not be car r i e d out simultaneously <22>. It follows that at the beginning of each f i s h i n g period nf is already given. Moreover, i f the catch is small r e l a t i v e to nf, which may not be true, nf may be regarded as constant during the fis h i n g period. On these assumptions and those used to derive (10) and (11) integration i s again very simple and we obtain the corresponding harvesting equat ions: (13) Y=NF(e,z,ts)F(E(e),z)tf, (14) Y=NF(e,z,ts)F(E(e))tf. i Where Y represents the accumulated catch during the interval [0,1] and * denotes the appropriate weighted averages of the respective variables. For notational convenience the averaging symbol w i l l be dropped below. 259 6.3.2 The Fishing Time Function. Turning our attention now to the fi s h i n g time, t f , we notice that t h i s variable can not, in general, be chosen f r e e l y . The attainable f i s h i n g time i s c l e a r l y bounded above by the t o t a l time period in question as well as the technical conditions of the part i c u l a r fishery. More importantly, perhaps, at least from the point of view of obtaining e f f i c i e n c y in the fishery, actual fishing i s only one of several s u b a c t i v i t i e s that constitute the fish i n g a c t i v i t y . Therefore, as suggested above, fis h i n g time or more sophisticated measures of b i o l o g i c a l f i s h i n g e f f o r t , may not be good indicators of the t o t a l application of economic inputs in the fish i n g a c t i v i t y . In fact, the empirical results of section 5.3.2 support t h i s conjecture <23>. Hence, r e c a l l i n g our d e f i n i t i o n of the fi s h i n g mortality production function as the li n k between the economic and b i o l o g i c a l parts of a fi s h e r i e s model <24>, any analysis of thi s function i s incomplete unless i t e x p l i c i t l y relates f i s h i n g time to the t o t a l application of economic inputs in the fish i n g a c t i v i t y . We w i l l now attempt th i s task. Our approach i s to decompose the f i s h i n g a c t i v i t y into s u b a c t i v i t i e s that cannot be pursued simultaneously. Each of these a c t i v i t i e s implies s p e c i f i c use of economic inputs. Due to the r e l a t i v e novelty of t h i s approach we begin by an informal discussion. During a prespecified period, a year, say, the t o t a l time available for actual fishing i s limited by the length of thi s period. Moreover, as has been indicated, some time has, in general, to be allocated, to other other a c t i v i t i e s that are 260 incompatible with actual f i s h i n g . F i r s t l y , there are the t r i p s from the port of departure to the fi s h i n g grounds and back for landing the catch. The time required for these t r i p s back and forth, the " s a i l i n g time", i s , in many instances, very considerable <25>. More importantly, the t o t a l " s a i l i n g time" varies with the frequency of t r i p s , which in turn tends to increase with the catch l e v e l . Secondly, landing the catch also takes time. Obviously this "landings time" tends to increase with the catch l e v e l . Thirdly, there i s normally a good deal of time required to operate the fi s h i n g gear during which actual f i s h i n g can not take place. This includes time for hauling in the gear, emptying i t of catches, perhaps repairing i t and eventually submerging i t again for more catch. The time required for these a c t i v i t i e s , the "gear operation time", tends to increase with the le v e l of the catches. The main reason i s that increased catches require an increased frequency of hauling in the gear in order to avoid gear saturation and possible gear damage <26>. Fourthly, fi s h i n g vessels generally also operate as f i s h processing plants. After a l l the catch has to be disposed of in some way. The f i r s t steps in processing the catch are normally taken on board the fish i n g vessels. When the catch rate i s high the processing capacity tends to become a bottleneck in the fishi n g process with the result that actual f i s h i n g has to be temporarily halted. This time, the " f i s h processing time", also reduces the time available for actual f i s h i n g . As higher catch levels per vessel are, c e t e r i s paribus, synonymous with a higher l e v e l of f i s h stocks, the existence of 261 the above four f i s h i n g s u b a c t i v i t i e s thus suggests a negative relationship between fis h i n g time and the size of the f i s h stocks. A f i f t h f i s h i n g a c t i v i t y that might possibly counteract thi s negative effect of the volume of the f i s h stocks on fishing time i s the search a c t i v i t y . As f i s h stocks r i s e i t i s perfectly conceivable that i t w i l l prove optimal to reduce search time, thus freeing more time for the actual fishing a c t i v i t y . However, as w i l l be argued below, there i s no a p r i o r i reason to expect r i s i n g f i s h stocks to af f e c t the optimal search time in any par t i c u l a r d i r e c t i o n . We w i l l now pursue these ideas in a more formal manner. Assume that there e x i s t s , for each vessel, a function which may be c a l l e d the fi s h i n g time function. This function maps the to t a l available f i s h i n g time, appropriately defined, through the f i l t e r of the technical, environmental and economic conditions into actual f i s h i n g time. Our objective is to derive reasonable a p r i o r i hypotheses or propositions concerning the structure of this function. Define the operating time as the t o t a l time during which the vessel in question i s engaged in fishing a c t i v i t y . Denote the operating time by to. Clearly to. In other words the following ident i t y i s assumed to hold: 262 (15) to=tf+ts+tsa+tl+th+tp. Where tf denotes fishing time, ts search time, tsa s a i l i n g time, t l landings time, th gear operation time and tp processing time. Let t = ( t f , t s , t s a , t l , t h , t p ) be the time a l l o c a t i o n vector. Assume that at the outset of each fi s h i n g t r i p there exists a p r o f i t function for the vessel in question. Without any loss of generality we may write this function as: h=H(t,e,z,p). /•s/ ^ —' Where p represents the vector of relevant pr ices. e and z are the vectors of economic inputs and environmental variables as already defined. The captain's problem, in this context, i s to choose the time a l l o c a t i o n vector, t, so as to maximize th i s p r o f i t function subject to (15). Formally his problem i s : Maximize H(t;e,z,p) t S.t. to=tf+ts+tsa+tl+th+tp. The solution to this problem, i f i t exists, is the p r o f i t maximizing time a l l o c a t i o n vector: (16) t*=T(to,e,z,p) 263 The important result expressed in equation (16) i s that fis h i n g time i s , in general, a function of environmental conditions as well as other exogenous variables. In pa r t i c u l a r f i s h i n g time depends on the volume of the f i s h stocks being harvested. We have thus arrived at our f i r s t proposition concerning the fi s h i n g time function: Proposition 6.1 The f i s h i n g time function must in general be written as tf=TF(to,e,z,p). As a general statement proposition 6.1 seems so eminently plausible that i t borders on being t r i v i a l . Nevertheless i t has fundamental implications concerning the s p e c i f i c a t i o n of f i s h e r i e s models. These implications are formalized in the following c o r o l l a r i e s : Corollary 6.1 Fishing e f f o r t depends on the state of the f i s h stocks. More precisely: ef=EF(to,x;e,z,p). Where the variable ef represents f i s h i n g e f f o r t , x represents the size of the p a r t i c u l a r f i s h stock or cohort in question and 264 z is now the vector of environmental variables excluding x. Proof: Fishing e f f o r t i s defined in equation (10) as F(E ( e ) , z ) t f . But as stated in proposition 6.1 (17) tf=Tf(to,x;e,z,p). Q.e.d. Obviously, the result in corollary 6.1 has an important implication for the formulation of an optimal fishery policy. A generalized version of the very popular Schaefer harvesting function is equation (4) in section 6.1, i.e. (4) y=Y(e,x). Where e i s supposed to be f i s h i n g e f f o r t and x i s the aggregate volume of the f i s h stocks. Comparing t h i s function with harvesting functions (13) and (14) uncovers some of i t s i m p l i c i t assumptions. Among those i t may be pointed out that i t presumes the constancy of the exogenous variables z, p. Accepting, however, for the sake of the argument, the si m p l i f i e d framework of the generalized Schaefer harvesting function, we immediately 265 derive the following second c o r o l l a r y to proposition 6.1: Corollary 6.2 Harvesting functions of the Schaefer type must, in general, be s p e c i f i e d as: y=Y(E(e,x),x). Where e denotes the vector of economic inputs. Proof: This result immediately follows from equation (4) and coro l l a r y 6.1. Al t e r n a t i v e l y substitute the equation in proposition 6.1 into the harvesting functions (13) and (14) and make the Schaefer simplifying assumptions. Q.e.d. Corollary 6.2 states, in other words, that economic inputs and the volume of the f i s h stocks j o i n t l y determine the fis h i n g e f f o r t . To repeat, these two variables are not separable in the harvesting function. Proposition 6.1 and i t s c o r o l l a r i e s concern the arguments of the f i s h i n g time function. The nature of the relationship, in parti c u l a r the sign of the p a r t i a l derivative of t f with respect to x, is also of primary i n t e r e s t . However, in order to deduce a p r i o r i information on th i s derivative i t i s necessary to make some simplifying assumptions and thus reduce somewhat the generality of our approach. 266 As discussed above, i t seems plausible that tsa, t l , th and tp are nondecreasing functions of the catch l e v e l . The situation is p e rfectly analogous with standard cost theory, with the time al l o c a t i o n s playing the role of inputs and catch that of output. Hence i t seems legitimate to assume that the p r o f i t maximizing l e v e l of these variables, given the values of t f and ts, can be j o i n t l y written as: tsa+tl+th+tp=S(y;z,e,p) *V ^/ Where y i s the harvest defined by functions (13) and (14). For convenience of notation we now drop the exogenous variables and write t h i s equation as: (18) tsa+tl+th+tp=S(Y(tf,ts,x)) Where Y(.,.,.) stands for the harvesting functions defined by (13) and (14) with the exogenous variables notationally suppressed. We moreover make the following assumptions about the function S(y): S'(.)>0, S''(. )>0, for a l l y>0, where 1 denotes a derivative. 267 Lim S(y)=a>0. Lim S(y)=«. y—>CD The general shape of S(y) according to these assumptions i s i l l u s t r a t e d in figure 6.1. Figure 6.1 Shape of the S(y) Function. > y Furthermore make the very plausible assumption that the harvesting function Y(.,.,.) i s increasing in i t s arguments. Hence we may write (17) as: S(Y(tf,ts,x))=S(tf,ts,x), where S(.,.,.) has the same general shape as S(Y(.,.,.)). The time constraint now i s : 2 6 8 ( 1 9 ) t o = t f + t s + S ( t f , t s , x ) . H e n c e we i m m e d i a t e l y d e r i v e t h e f o l l o w i n g p r o p o s i t i o n : P r o p o s i t i o n 6 . 2 F o r a g i v e n o p e r a t i n g t i m e , t o , 3 t f / 3 x < 0 i f f 3 t s / 3 x > - S 3 / ( 1 + S 2 ) , w h e r e S 2 a n d S 3 d e n o t e t h e f i r s t d e r i v a t i v e s o f S w . r . t . i t s 2 n d a n d 3 r d a r g u m e n t s , r e s p e c t i v e l y . P r o o f : T h e p r o p o s i t i o n f o l l o w s i m m e d i a t l y f r o m d i f f e r e n t i a t i n g ( 1 9 ) w i t h r e s p e c t t o x k e e p i n g t o a n d t h e e x o g e n o u s v a r i a b l e s f i x e d . Q . e . d . T h e a r g u m e n t s d e v e l o p e d i n a p p e n d i x 6 . 2 - A s u g g e s t s t h a t t h e s i g n o f 3 t s / 3 x i s , i n g e n e r a l , i n d e t e r m i n a t e . T h e r e f o r e t h e s i g n o f t h e p a r t i a l d e r i v a t i v e o f t f w i t h r e s p e c t t o x c a n n o t b e d e t e r m i n e d b e f o r e h a n d . H o w e v e r , s i n c e f o r t h e o p p o s i t e r e s u l t 3 t s / 3 x h a s t o b e r e l a t i v e l y h i g h n e g a t i v e , we h y p o t h e s i z e t h a t 3 t f / 3 x i n m o s t c a s e s n e g a t i v e . M o r e o v e r , s i n c e S(y)—*°° a s x—»°=, 3 t f / 3 x m u s t u l t i m a t e l y b e c o m e n e g a t i v e w i t h t f a s y m p t o t i c a l l y a p p r o a c h i n g z e r o . T h e s e c o n c l u s i o n s a r e i l l u s t r a t e d i n f i g u r e 6 . 2 . 269 Figure 6.2 Fishing Time as a Function of Vessel Specific Fish Stocks 270 Appendix 6.2-A Fish Search. Our primary aim in t h i s appendix is to argue that the sign of the p a r t i a l derivative, 3ts/9x, in proposition 6.2 i s , in general, indeterminate. Without much extra cost, however, this argument can be pursued j o i n t l y with an elementary discussion of f i s h search and i t s a f f i n i t y with the standard theories of consumer and job search <28>. Consider a fishing e n t i t y , a vessel, say. Assume there i s a number of f i s h i n g locations or fi s h i n g grounds on which the vessel may operate. At any point of time there i s a certain concentration of f i s h on each of these fi s h i n g grounds. C a l l t h i s concentration "ground density". Ex ante the captain of the vessel i s ignorant of the ground densities. This he can discover only by engaging in f i s h search. The nature of the f i s h search is l e f t unspecified apart from s t i p u l a t i n g that i t i s (a) time consuming and (b) can not be performed simultaneously with actual f i s h i n g . It follows that f i s h search i s costly i f only because i t takes time from f i s h i n g . Therefore, assuming that the objective of the fishing entity i s to maximize the net value of the f i s h i n g t r i p , the captain must try to s t r i k e a balance between the marginal y i e l d and cost of f i s h search. Depending on the captain's subjective pr o b a b i l i t y density function of discovering a higher concentration of f i s h , additional search has an expected value as well as cost to him. Calculating the net expected value of additional search the captain reaches a conclusion as to whether to search or f i s h . The correspondence between th i s very general description of the f i s h search problem and the economic l i t e r a t u r e on consumer search and job search should be obvious. The discovered ground densities correspond to price or wage offers in consumer and job search theories and the captain of the vessel corresponds to the consumer or worker. It follows that the results of these theories should also be applicable to f i s h search. Denoting the expectations operator by E(.) we can begin by formulating the following s t a t i c version of the captain's search problem: Max E(H(NF(ts,x,u))-C(ts)=P(ts,x,u) ts Where NF(.,.,.), the findings function, denotes the number of f i s h located by the search <29>. ts denotes search time, x stands for the volume of the f i s h stocks and u is a random variable representing the stochastic nature of the search. H(.) is a function that transforms NF(.,.,.) into value terms. C(.) is a cost function. P(.,.,.) thus i s the expected p r o f i t function of search. A necessary condition for an ex ante solution to t h i s 271 problem i s given by the ts s a t i s f y i n g the equation: P,(ts,x,u)=0. Hence the expected p r o f i t maximizing r e l a t i o n between ts and x i s : 9ts/9x=-P, 2*/Pi1*, where * indicates that the derivatives are evaluated at the maximal value of P ( . , . , . ) . Now, since P l 1*<0 for a maximum, the sign of 9ts/9x is determined by the sign of P 1 2*. More pr e c i s e l y : sign(9ts/9x)=sign(P 1 2*) . If , in other words, an increase in the volume of the f i s h stocks increases the expected marginal productivity of search, more time w i l l be allocated to f i s h search. However, there does not seem to be any a p r i o r i reason why t h i s condition should hold in general. Intuition suggests that the sign of P 1 2 * depends on the d i s t r i b u t i o n of an additional volume of f i s h over the fis h i n g grounds. If the absolute difference in ground densities is reduced, less rewards may be expected from f i s h search and reverse. As an i l l u s t r a t i o n , consider the following NF(.,.,.): NF=A(x)+B(ts)C(x,u). Here A(x) denotes the findings without any search, i . e . the f i s h concentration on the f i r s t f i s h i n g ground. C(x,u) is a random factor representing the variance of ground densities over the fis h i n g grounds. This term depends on the stock siz2, x, and a pure random e f f e c t , u. Clearly, given t h i s expected findings function, P 1 2 * depends on the function C(x,u). In order to make further progress l e t us now adopt an approach developed in job search and consumer search theory. Assume that the captain forms a certain reservation ground density l e v e l . If he, during the search process, encounters a f i s h concentration above th i s level he terminates the search and begins to f i s h . Otherwise he continues to ' search. The reservation l e v e l i s of course a subjective variable, based upon the captain's perceived f i s h d i s t r i b u t i o n function and may be modified during the course of the search. As an example of how this reservation l e v e l may be formed and modified consider the following model: Let there be a fixed time horizon, T. Let the fi s h i n g a c t i v i t y consist of a single f i s h search period followed by a fish i n g period so that the t o t a l duration of f i s h search and fishing periods equals T. Let the search process occur in f i n i t e i n t e r v a l s , representing e.g. the time required to move from one fis h i n g ground to another. Assume, moreover, that f i s h movements 272 are rapid r e l a t i v e to the travel time between fi s h i n g grounds eliminating any p o s s i b i l i t y of accumulating knowledge <30>. Consider a sequential search process. After each search interval the captain decides whether to undertake an additional search or to commence f i s h i n g . Let V ( x , t ) denote the expected net value of the fishing t r i p when a ground density x has been discovered and t periods (out of T) have elapsed. At t the expected value of searching one more period is (A.1) S(t,f (x,h) ,c)= f°°V(x,t + 1 )f (x,h)dx - c. 0 Where f(x,h) is the captain's subjective probability density function for the ground densities, h is a s h i f t variable for this p r o b a b i l i t y density function, representing i . e . new information. c>0 i s the unit cost of one additional search. Notice that since T is the time horizon V(X,T+1)=0 and hence S(T,.,.)=-c. The value of beginning to f i s h at t when x has been located is on the other hand defined by the p r o f i t function: P(x,t)=P(x,T-t), where we assume that P^_(.,.)>0. Now, assuming risk neutrality (or that an appropriate risk premium is included in c ) , the expected value of the fishing t r i p at t i s : V(x,t)=max[P(x,T-t),S(t,f(x,h),c)]. The reservation ground density l e v e l at t i s thus given by the x*(t) s a t i s f y i n g : (A.2) P(x*(t),T-t)=S(t,f(x,h),c). On the basis of (A.1) and (A.2) we immediately derive: 9x*(t)/9c<0. (A.3) 9x*(t)/9f(x,h)>0. Where we interpret 9f(x,h) as an increased subjective probability of locating ground density above x. Thus higher search cost reduces and higher subjective probability density of finding good f i s h concentrations increases the reservation ground density l e v e l . This, of course, is i n t u i t i v e l y appealing. 273 Having outlined a theory of the formation and modification of the reservation ground density, x*, we are now in a better position to consider the sign of 3ts/3x. Let F(x*) be the true probability d i s t r i b u t i o n function given x*. F(x*) i s , in other words, the pr o b a b i l i t y of encountering a ground density below x* in any given search. Therefore, p=1-F(x*) i s the corresponding probability of ground density equal to or above x*. Given x*, the probability of searching for t* periods before finding a ground density above x* is defined by <31>: Pr (t*=t) = ( 1-pf"'p. The expected number of search periods needed to find a ground density equal to or above x* i s : (A.4) E(t*)+1=(1-p)/p+1=l/p=l/(1-F(x*)). . It follows that the expected number of search periods increases, c e t e r i s paribus, with x*. Hence making the innoccuous assumption that t o t a l search time i s an increasing function of search periods, we deduce with the help of (A.3) and (A.4): 3ts/9c<0 (A.5) 9ts/9f(x,h)>0. 3ts/3x=-(3p/3x)/p 2, where x denotes the t o t a l stock s i z e . The last result i l l u s t r a t e s our i n i t i a l contention that the sign of 9ts/9x i s indeterminate. The question is how x* is modified when the t o t a l stock size increases. I f , as seems l i k e l y , x* does not adjust immediately, 9ts/9x<0 i n i t i a l l y . I f , in the longer run, x* adjusts so as to maintain a constant p, 3ts/3x=0. What actually happens depends on the s p e c i f i c s of equation (A.2) in particular the expectations formation or learning process implied by the subjective p r o b a b i l i t y density function, f(x,h). 274 6.3 Empirical Estimation. Now, we propose to use the available data to estimate a fi s h i n g mortality production function for the Icelandic demersal f i s h e r i e s . On the basis of the results of the previous section we represent t h i s function by two subfunctions, a fishing mortality and a f i s h i n g time function. These two functions constitute a simultaneous equation system. Due to the shortcomings of the data, however, we are,' in the case of multipurpose fi s h i n g vessels, forced to estimate a single fi s h i n g mortality production function. We begin by describing the data available for the estimation process and continue to discuss the estimation of each function for the trawler f l e e t , f i r s t separately and then j o i n t l y . In the last subsection we discuss the estimation of fi s h i n g mortality production functions for the multipurpose fis h i n g f l e e t . 6.3.1 The Data. The analysis in section 6.2 led to the following f i s h i n g mortality and f i s h i n g time functions: (10) F=F(E(e),z)tf (17) tf=TF(to,x;e,z,p) Where F represents the average fis h i n g mortality of a given cohort of a p a r t i c u l a r species during a certain period of time. 275 e and z are vectors of average economic fis h i n g inputs and environmental variables respectively, tf represents the fi s h i n g time, i . e . the t o t a l application time of the fi s h i n g gear, and to i s the t o t a l operating time of the vessel during the same period. F i n a l l y , x i s the so-called vessel s p e c i f i c stock size and the vector p represents the prices relevant to the time a l l o c a t i o n vector, t, <32>. We have obtained data on the fishing a c t i v i t y of the Icelandic trawler f l e e t that p a r t i a l l y s a t i s f y the informational requirements of equations (10) and (17). The data describe the fi s h i n g behaviour of the Icelandic trawler f l e e t on vessel basis during the 5 year period 1976-80 <33>. Let us refer to vessels by the index i and years by the index j . The key variables covered by the data are: (i) Total demersal catch by each vessel, y ( i , j ) . ( i i ) The size or tonnage of each vessel, k ( i , j ) . ( i i i ) The location of the home port of each vessel, 1 ( i , j ) . (iv) The t o t a l fishing time of each vessel, t f ( i , j ) . (v) The t o t a l o v e r a l l catch for each species, y(j,h), where the index h refers to species, (vi) The size of the f i s h stocks both in number and volume terms, n(j) and x(j) respectively, ( v i i ) The average fi s h i n g mortality of each species, F(j,h), where the index h refers to the species, ( v i i i ) The t o t a l operating time of each vessel, t o ( i , j ) . Although these data are, by the usual standards of fishery 276 s t a t i s t i c s , f a i r l y extensive, they are severely limited r e l a t i v e to the informational requirement of equations (10) and (17). F i r s t l y , as already mentioned, only the f i s h i n g a c t i v i t y of the trawler f l e e t i s included in the data. Comparable data on the fi s h i n g a c t i v i t y of the multipurpose vessels are, as w i l l be further discussed below, not available <34>. Secondly, only one f i s h i n g c a p i t a l variable, k ( i , j ) is included in the data. Thirdly, the vector z represents, in p r i n c i p l e , the average environmental conditions under which vessel i operates in each year. No d i r e c t measurements on these conditions are available. On the other hand some proxies for the state of the environment may be employed. The location of the home port of each vessel, l ( i , j ) , roughly defines i t s geographical scope of operations and thus represents environmental conditions to the extent they coincide with a given geographical area. Another indirect indication of environmental conditions is provided by the t o t a l volume of the f i s h stocks, x ( j ) , in each year. Fourthly, f i s h i n g mortality due to the a c t i v i t y of each vessel, F ( i , j ) , i s unfortunately not recorded. However, as argued in appendix 6.3.1-A below, th i s variable may be estimated by the equation: F ( i , j ) = F ( j ) y ( i , j ) / y ( j ) , a l l i , j . Where the variables and indices are those already defined. F i f t h l y , while equation (10) i s derived and formulated in terms of each cohort, the data are much more aggregated. The 277 catch of each vessel for instance is not d i f f e r e n t i a t e d according to cohorts. Moreover, the fi s h i n g time, although c a r e f u l l y recorded for each vessel, i s not specified according to the species, l e t alone the cohorts, to which th i s e f f o r t may be directed. Hence, although the f i s h i n g mortality function was derived in terms of each cohort, the data only allow us to estimate a f i s h i n g mortality function aggregated over both cohorts and species. This gives r i s e to a formidable aggregation problem. If , in fact, the f i s h i n g mortality functions d i f f e r according to cohorts and species, attempts to represent these functions with one aggregative function may lead to serious errors < 3 5 > . On the other hand, there are conditions under which this aggregation is exact. This i s further discussed in appendix 6 . 3 . 1 - B below. Sixthly, there are no d i r e c t observations on the time al l o c a t i o n costs, p. However, i t seems plausible that the r e l a t i v e p ( i , j ) ' s are largely invariant over the indices i and j . In that case this variable may be omitted from the estimation without serious loss. F i n a l l y , x ( i , j ) i s the stock l e v e l on which vessel i operates in year j . As i s perhaps to be expected there are no direct observations on t h i s variable. A reasonable estimator on x ( i , j ) , however, is given by x ( i , j ) = y ( i , j ) / t f ( i , j ) , where variables and indices are again as previously defined. This expression i s , in fact, the so-called "catch-effort" r a t i o 278 that is s t i l l widely used in f i s h population studies as an estimator of stock abundance <36>. Notice, however, that in this case i t i s the vessel s p e c i f i c and not the t o t a l abundance that is being estimated. The data on the fi s h i n g a c t i v i t y of the multipurpose vessels are much more limited. In p a r t i c u l a r , there are no observations on the f i s h i n g time of the multipurpose vessels. The available data are e s s e n t i a l l y those used to estimate the harvesting cost functions for the multipurpose vessels in section 5.3.2 above. These data cover, i t w i l l be recalled, the fis h i n g a c t i v i t y of a number of multipurpose vessels on a yearly basis during 1974-77. Apart from the nonobserved f i s h i n g time, the observed variables correspond, in most respects, with those for the trawler f l e e t described above. There are observations on catch, several fishing c a p i t a l variables including vessel and engine size, indirect environmental variables such as the geographical location of the home port of the vessel, the t o t a l volume of f i s h stocks as well as the t o t a l operating time of each vessel. It should be realized, however, that this last variable, i . e . operating time, i s somewhat d i f f e r e n t from the corresponding one for the trawler f l e e t . While, for the trawler f l e e t , recorded t o t a l operating time coincides almost exactly with the theoret i c a l d e f i n i t i o n in section 6.2.2, t h i s i s not so for the multipurpose fi s h i n g f l e e t . Extended periods in port, due e.g. to holidays, bad weather or the necessary set-up time for switching from one fishery to another, are counted as operating time in the recorded s t a t i s t i c s . As these periods of "nonfishing a c t i v i t y " t y p i c a l l y l a s t for 10-12 weeks each year, 279 i t is clear that the measured operating time exceeds, by far, the t h e o r e t i c a l concept. Even more seriously from a s t a t i s t i c a l point of view, the difference between the the o r e t i c a l and the measured operating time generally varies a great deal from one vessel to another. Thus, as we have seen, the available data are severely limited r e l a t i v e to the informational requirements of equations (10) and (17). In fact, some key variables in these functions are not d i r e c t l y observed and have to be estimated. These estimates w i l l of course contain errors giving r i s e to an errors in variables problem in the empirical estimation of equations (10) and (17). In the case of the multipurpose vessels we may not even hope to obtain consistent estimates of the f i s h i n g mortality production function. This problem i s fundamental. The available information i s simply not extensive enough to permit a complete estimation of the t h e o r e t i c a l structure. This, of course, is commonplace in empirical work. 280 Appendix 6.3.1-A Fishing Mortality by Individual Vessels. In section 6.3.1 (A.1) F ( i , j ) = F ( j ) y ( i , j ) / y ( j ) was proposed as a reasonable estimator of F ( i , j ) . We now examine the properties of this estimator. As demonstrated in section 4.1.2 the t o t a l catch in numbers from species j during a certain period can be written as: (A.2) c ( j ) = n ( j , t - 1 ) F ( j ) ( 1 - e x p ( - Z ( j ) ) ) / Z ( j ) . In which expression c ( j ) denotes the t o t a l catch, n ( j , t - l ) the to t a l number of individual f i s h at the beginning of the period, F(j) the average fishing mortality and Z(j) the t o t a l mortality during the period. S i m i l a r l y , the catch in numbers due to the fishing a c t i v i t y of vessel i can be written in an obvious notation as: (A.3) c ( i , j ) = n ( j , t - 1 ) F ( i , j ) ( l - e x p ( - Z ( j ) ) ) / Z ( j ) . From (A.2) (And (A.3) i t follows that (A.4) F ( i , j ) = F ( j ) c ( i , j ) / c ( j ) . Now, l e t the number of cohorts be L and let w(j,l) represent average individual weight of cohort 1. Clearly L L (A.5) y ( i , j ) = Zw ( j , 1 )c ( i , j , 1) =c ( i , j )Zw ( j , 1 )c ( i , j , 1 )/c ( i , j ) * * u = c ( i , j ) Z w ( j , l ) A ( i , j , l ) . JL And (A.6) y(j)= Z w ( j , l ) c ( j , l ) = c ( j ) Z w ( j , l ) c ( j , l ) / c ( j ) * 6 L = c ( j ) Z w ( j , l ) B ( j , l ) . A ( i , j , 1 ) = c ( i , j , 1 ) / c ( i , j ) is the share of cohort 1 in vessel's i catch of species j . B(j,1)=c(j,1)/c(j), s i m i l a r l y , i s the share of cohort 1 in the t o t a l catch of species j . Substituting (A.5) and (A.6) in for c ( i , j ) and c ( j ) in (A.4) y i e l d s : (A.7) F ( i , j ) = F ( j ) y ( i , j ) / y ( j ) ( Z w ( j , l ) B ( j , l ) / Z w ( j , l ) A ( i , j , l ) ) . Comparison of (A.7) with (A.1) uncovers the conditions under which the l a t t e r expression provides an exact estimator of F ( i , j ) . These are summarized in the following lemmas: 281 Lemma 1. (Necessary and s u f f i c i e n t conditions). F ( i , j ) = F ( j ) y ( i , j ) / y ( j ) i f f Zw(j,1)B(j,1)= Zw(j,1)A(i,j,1). Lemma 2. (S u f f i c i e n t condition). B ( j , l ) = A ( i , j , l ) , a l l 1, i f F ( i , j ) = F ( j ) y ( i , j ) / y ( j ) . Lemma 2 i s f a i r l y i n t u i t i v e . It simply states that i f the cohorts feature equally in the catch of a l l the vessels aggregation i s feasibl e . However, in general, t h i s requirement is too strong. If the number of cohorts i s one, L=1, the right hand condition of lemma 1 i s automatically s a t i s f i e d . For L=2 lemma 2 is also a necessary condition. For L>2 the conditions in lemma 2 are not necessary. Appendix 6.3.1-B Aggregation of Fishing Mortality Functions. Consider a multispecies fishery. For the purposes of thi s appendix i t i s convenient to disregard the d i s t i n c t i o n between species and think of the f i s h stocks as simply a c o l l e c t i o n of dif f e r e n t cohorts. The same yearclasses of d i f f e r e n t species are counted as d i f f e r e n t cohorts. Let the t o t a l number of cohorts in the fishery be L. In accordance with equation (10), write the set of fi s h i n g mortality functions as: (B.1) F ( l ) = F ( E ( e ) , z , l ) t f ( 1 ) , 1=1,2,...L. **** «N> Notice that we have made F ( l ) e x p l i c i t l y dependent upon the parti c u l a r cohort in question while for convenience of notation the vessel and year indices as well as other symbols irrelevant to the present problem have been suppressed. Now, restating equation (A.2) of the previous appendix s l i g h t l y , we may write the Beverton-Holt catch function for the fishery as a whole as: c(j)=F(j)n(j) , Where n(j)=n(j,t-1)(1-exp(-Z(j)))/Z(j ) , i.e the appropriately defined average stock size of our aggregative f i s h stock, j . Simi l a r l y , for a given cohort, 1, in the fishery: c ( j , l ) = F ( j , l ) n ( j , l ) . By d e f i n i t i o n : 282 c(j ) = L c ( j , l ) Thus, dropping the irrelevant j index, we obtain the following expression for the aggregated f i s h i n g mortality: (B.2) F=SF(l)n(l)/n. JL Substituting (B.1) into (B.2) yiel d s the fundamental aggregation function: (B.3) F = Z F ( E ( e ) , z , l ) t f ( l ) n ( l ) / n . The aggregative fi s h i n g mortality function i s , in other words, simply the weighted sum of the f i s h i n g mortality functions for each cohort with the weights for each cohort being the r a t i o of the number of individuals in that cohort to the t o t a l number of individuals. The aggregation problem is whether (B.3) can be written as: (B.4) F=F(E(e),z)tf, where tf i s the t o t a l f i s h i n g time. The disaggregation problem is whether the disaggregative functions, (B.1), can be inferred from knowledge of the aggregative function, (B.4). The structure of (B.3) makes i t clear that the aggregative function does not in general e x i s t . Thus, imposing (B.4) on empirical data is in general a case of misspecification. However, there are conditions under which th i s aggregation i s in fact exact. We now b r i e f l y consider such instances. Case 1. Selective f i s h i n g impossible. This case may be described by the expression: (B.5) f=aF(L). Where £ is the (1xL) vector of f i s h i n g m o r t a l i t i e s and a=(a(lT,a(2),....,a(L-1),1) is a vector of constants. F(L) i s an arbitr a r y base f i s h i n g mortality, or numeraire. The c r u c i a l point of (B.5) i s that a depends only on the arbit r a r y numeraire fishing mortality. ~ From an empirical point of view this case arises e.g. when the cohorts are not geographically d i s t i n c t and the fi s h i n g technology does not permit variable cohort selection from a given concentration of f i s h . In t h i s case, c l e a r l y , f i s h i n g time can not be di f f e r e n t i a t e d according to cohorts. Hence tf(1)=tf(s)=tf, a l l s , l , and (B.3) may be rewritten as: 283 F=F(E(e),z,L)tfZa(l)n(l)/n=F(E(e),z)tf, since the vector z contains a l l the information under the summation sign. ~ Moreover, the cohort s p e c i f i c fishing mortality functions may c l e a r l y be estimated i f F ( l ) i s known since t f ( l ) i s . Case 2. Non-differentiated fi s h i n g time. This case i s defined by: t f ( l ) = t f ( s ) = t f , a l l l , s . Empirically this case may arise because the sp a t i a l d i s t r i b u t i o n s of the cohorts are id e n t i c a l with the result that i t i s impossible to concentrate fishing on some and not others. Notice that this does not imply that the cohorts are subject to a fixed r e l a t i v e fishing mortality. This case d i f f e r s from case 1 in that cohort selection may s t i l l be affected through varying the economic inputs in F ( . , . , . ) . In this case again (B.4) exists and i s given by: F=tfZF(E(e),z,l)n(l)/n. x ~ ~ The cohort s p e c i f i c f i s h i n g mortality functions can on the other hand not be inferred from knowledge of the aggregative function and the b i o l o g i c a l parameters in t h i s case. Case 3. Equation (B.3) implies that i f (B.6) F ( . , . , l ) n ( l ) = F ( . , . , s ) n ( s ) , a l l l , s then the aggregative function, (B.4), again e x i s t s . (B.6), however, seems very r e s t r i c t i v e . To elucidate the conditions under which (B.6) might hold consider the following example: Assume that the captain of the vessel choses t f ( l ) so as to maximize the following p r o f i t function: ZF(. , . , l ) n ( l ) t f ( D p ( l ) - C ( t f (1) ) , where p(l) i s the output price of an individual f i s h of cohort 1 and C ( t f ( D ) represents the cost of t f ( l ) . Solution to th i s problem i s : F ( . , . , l ) n ( l ) p ( l ) = C , ( t f ( D ) , 1=1,2,...L, where C'(.)=9C(.)/3tf. Thus, i f C ( t f ( 1 ) ) / p ( l ) = C ( t f ( s ) ) / p ( s ) , a l l l , s , 284 the condition in (B.3) w i l l hold. In the case of the Icelandic demersal f i s h e r i e s we may take i t that the marginal costs of the various cohort s p e c i f i c f i s h i n g times are very s i m i l a r . In that case p(l)=p(s), a l l l , s is required for the v a l i d i t y of (B.6). In the Icelandic demersal f i s h e r i e s cohort selective f i s h i n g i s possible but only to a certain extent. S i m i l a r i l y f i s h i n g time may be d i f f e r e n t i a t e d according to cohorts but again only to a limited extent. Thirdly, p(l) is not, in general, equal to p(s). However, when comparing the same yearclasses of di f f e r e n t species the difference is r e l a t i v e l y small. The conclusion of the above discussion i s that for the Icelandic demersal f i s h e r i e s an aggregative f i s h i n g mortality function of the (B.4) kind probably misrepresents the true s i t u a t i o n . As pointed out, however, there i s some reason to believe that the misspecification error may not be too great. In any case, due to the absence of disaggregated data, there i s hardly any alternative to estimating an aggregative f i s h i n g mortality function. 285 6.3.2 Estimation of the Fishing Mortality Function. According to the results in section 6.2.1, the fish i n g mortality function i s specified as: (10) F=F(E(e),z)tf. A natural f i r s t stage of the estimation of this function i s to investigate i t s functional form. A general power transformation of the variables in (10) provides a f l e x i b l e s p e c i f i c a t i o n of the function for use in empirical work. More precisely, following the suggestion of Box and Tidwell <37>, define the following transformation of an arbitrary variable x. x(q) = (x ?-1 )/q. Because of the limi t a t i o n s of the data and numerical d i f f i c u l t i e s <38> we are forced to confine our s p e c i f i c a t i o n tests to the following expression: (20) F(r1)=a+bk(r2)+dtf(r3)+u, Where k represents the tonnage of the vessel in question and r1, r2 and r3 are Box-Tidwell transformation parameters as defined above, u i s a stochastic disturbance term assumed to have the d i s t r i b u t i o n : 286 u(i) ~NIID(o,a) , a l l i . The s p e c i f i c a t i o n test adopted here consists of maximizing the l i k e l i h o o d function for t h i s equation with and without r e s t r i c t i o n s on the Box-Tidwell transformation parameters. The res u l t i n g values of the l o g - l i k e l i h o o d function form a basis for l i k e l i h o o d r a t i o tests on the r e s t r i c t i o n s <39>. The results are l i s t e d in table 6.1 Table 6.1 Testing Restrictions on the Fishing Mortality-Function, (20). According to the l i k e l i h o o d r a t i o tests in table 6.1 the r e s t r i c t i o n s under II, III and IV are a l l rejected on the 1% significance l e v e l . However, there are arguments against adopting the unrestricted Box-Tidwell version. In the f i r s t place, i t implies that zero f i s h i n g time generates a nonzero f i s h i n g mortality. This, of course, i s unacceptable. Secondly, the estimation of equation (20) is subject to considerable s t a t i s t i c a l d i f f i c u l t i e s . The data i s , as has already been discussed at length, severely r e s t r i c t i v e giving r i s e to aggregation and errors in variables problems. In addition, tf is endogenous rendering the Box-Tidwell estimation procedure Log-1i kelihood values Likelihood r a t i o values against I. %*(3) I. The r's unrestricted I I . r1=r2=r3=0 II I . r1=r2=r3=1 IV. r1=r3=0, r2=1 -1338.6 -1350.0 -1346.9 -1347.6 2 2 . 8 16.6 1 8 . 0 287 inconsistent. These s t a t i s t i c a l problems c e r t a i n l y reduce the a p p l i c a b i l i t y of the li k e l i h o o d r a t i o tests above <40>. Hence, comparing those with unadjusted significance tables may be misleading. Therefore i t does not seem j u s t i f i e d to s a c r i f i c e functional s i m p l i c i t y on the basis of the above tests. On the other hand the results of the s p e c i f i c a t i o n tests c e r t a i n l y caution that the fi s h i n g mortality function, i t s theoretical or empirical s p e c i f i c a t i o n or both, should be regarded as suspect. On the basis of the above discussion we adopt the following functional form for the fish i n g mortality function: (21) F=aexp(bk)tf 9. As mentioned above there are no direc t observations on the environmental conditions under which each vessel operates. We, however, try to capture t h i s i n d i r e c t l y . F i r s t , measurements on the aggregate stock size of the demersal species each year. Secondly, we use the location of the home port of the vessels as further measures of their environmental conditions. Four such geographical locations are defined; the Southwest, Northwest, North and East areas. These categories correspond to a p r i o r i information about possible environmental differences. Thirdly we try to account for environmental fluctuations by including dummy sh i f t variables for each year in the sample. The estimation equation thus i s : 288 (22) F(i,j)=a+bk(i,j)+cz(j)+Zh(j)d(j)+Zl(m)d(m) j m + g t f ( i , j ) + u ( i , j ) , a l l i , j . Where we r e c a l l that F ( i , j ) i s the estimated average fi s h i n g mortality generated by vessel i in year j . k ( i , j) i s the tonnage of vessel i in year j . z ( j ) i s the volume of the demersal f i s h stocks in year j . t f ( i , j ) i s the to t a l f i s h i n g time of vessel i in year j . d(j) is a dummy variable equalling 1 in year j and 0 otherwise. d(m) i s a dummy variable for location equalling 1 when the location i s m and 0 otherwise. d(l) refers to the Southwest region etc. u ( i , j ) is a stochastic disturbance term. Apart from the c a p i t a l and the dummies a l l the variables are in logarithmic form. This i s indicated by the symbol on top of the respective variables, a, b, c, g, the h(j)' s and l(m)'s are parameters. Since out of the 10 intercepts ( 9 dummies and 1 regular) only 8 can be i d e n t i f i e d we a r b i t r a r i l y drop the f i r s t time and the f i r s t location dummy from equation (22) prior to estimating i t . We moreover assume pr o v i s i o n a l l y that the disturbances s a t i s f y the c l a s s i c a l assumptions, i . e . u~N(0,aI ) , where u denotes the vector of disturbance terms for the IxJ observations. Now, the fi s h i n g mortality function and the fish i n g time functions constitute a simultaneous equation system. Hence i t can hardly be maintained that t f ( i , j ) i s s t a t i s t i c a l l y exogenous 289 in equation (22). We therefore resort to the instrumental variables estimation technique known as 2SLS <41>. Moreover we are interested in obtaining the simplest possible form of (22). In p a r t i c u l a r we w i l l test the hypotheses: HI: c=0 H2: g=1. The estimation results are given in table 6.2: 290 Table 6.2 Estimates of the Fishing Mortality Function. Estimation technique: 2SLS The set of instruments: t o ( i , j ) , k ( i , j ) , d ( j ) , d ( m ) and t r ( i , j ) , where the last variable denotes the t o t a l number of fi s h i n g t r i p s undertaken by vessel i in year j . Number of observations: 199 Parameters Unrestricted Est imates Restricted estimates Est imates t-values Estimates t-values a -1.9 -5.3 -1.4 -31.3 b .00051 9.1 .00052 9.5 c . 1 2 .00 .00 (restricted) h(2) .09 2.5 .08 2.3 h(3) -.08 -2.4 -.09 -2.5 h(4) -.07 -1.9 -.08 -2.2 h(5) .13 3.6 .12 3.4 1(2) .10 1 .5 .12 1.7 1(3) -.08 -2.9 -.07 -2.6 1(4) -.25 -7.9 -.24 -7.7 g 1 .07 23.8 1.00 (restricted) RS 0.83 0.83 SE: 0.16 0.16 DW: 1 . 64 1 .66 Test for normality of residuals: 2T(3)=13.1 £14) = 19.2 Tests: H1: F(1,189)=.01 Not rejected. H2: F(1,189)=1.3 Not rejected. These estimation results seem f a i r . The explanatory power of the equation i s reasonably high and the estimated parameters have the expected signs and magnitudes. Of the environmental proxies, z ( j ) , d(j) and d(m), i t turns out that z ( j ) , the t o t a l volume of the demersal f i s h stocks each year, i s not s i g n i f i c a n t at a l l . The yearly dummy, on the other hand, i s s i g n i f i c a n t perhaps s i g n a l l i n g that environmental conditions other than the volume of the f i s h stocks, i . e . the 291 weather, migration patterns etc., have a more marked effect on fish i n g mortality than the aggregate volume of the f i s h stocks. The location dummies are s i g n i f i c a n t indicating that the respective locations in fact represent d i f f e r e n t environmental conditions. As expected the t o t a l f i s h i n g time, t f , is found to be a si g n i f i c a n t determinant of fis h i n g mortality. The theoretical hypothesis that the e l a s t i c i t y of fis h i n g mortality with respect to f i s h i n g time equals 1 i s not rejected. 6.3.3 Estimation of the Fishing Time Function. In section 6.2.2 we derived the following fishing time function for each vessel: (17) tf=TF(to,x;e,z,p), Where we moreover, on theoreti c a l grounds, hypothesized that TF X(.,.;.)<0, TF X X(.,.?.)>0, a l l x above a certain minimum. Lim TF(.,.;.)=0 x—>°> In t h i s section we propose to use data on the fishing behaviour of the Icelandic demersal fi s h i n g f l e e t , outlined in section 6.3.1, to obtain estimates of this function and i t s 292 parameters. A form roughly s a t i s f y i n g our a p r i o r i s p e c i f i c a t i o n s of the f i s h i n g time function i s (23) tf=ato c k fexp( L b(h)d(h)x+gx 2) • u. Where the index h refers to the geographical location of the home port of the p a r t i c u l a r vessel as discussed above. d(h) is a dummy variable equalling 1 whenever a vessel belongs to the h-th location and the zero otherwise, a, c, f, g and the b(h)'s are constants, u i s a stochastic error term. Equation (23), of course, implies certain functional r e s t r i c t i o n s that may not be j u s t i f i e d . One way to test for the appropriateness of this p a r t i c u l a r s p e c i f i c a t i o n i s to estimate a more general functional form and check whether the r e s t r i c t i o n s in (23) are rejected by the data. For this purpose a general Box-Tidwell power transform of the variables in (23) was also estimated and compared with the results of the r e s t r i c t e d s p e c i f i c a t i o n . More precisely the following Box-Tidwell version of (23) was estimated: tf(r1)=a+cto(r2)+fk(r3)+Lb(h)d(h)x(r4)+gx 2(r5)+u. h under various constraints on the r's. The main results of this exercise are l i s t e d in the following table: 293 Table 6.3 Testing for the Sp e c i f i c a t i o n of the Fishing Time Function, (23). Log-likelihood values I. Unrestricted estimate -1339.5 II. 5 r e s t r i c t i o n s . r1=r2=r3=0, r4=r5=1 -1345.9 Likelihood r a t i o test on the r e s t r i c t i o n s in (23): TC (5)=12.6. The results are not very conclusive. The n u l l hypothesis, that the r e s t r i c t i o n s are true, i s rejected on the 5% l e v e l but not on the 2.5% l e v e l . Hovever, following our rule of adopting s i m p l i f i c a t i o n s that are not strongly rejected by the data, we continue on the basis of equation (23). We now proceed to estimate the parameters of equation (23). As discussed in section 6.3.1 on the data, x ( i ) , i.e. vessel's i s p e c i f i c f i s h stock, i s estimated by x ( i ) = y ( i ) / t f ( i ) . x(i) in other words depends on t f ( i ) . Moreover the harvesting functions (13) and (14) demonstrate that the catch, y ( i ) , also depends on the f i s h i n g time. For these reasons x in equation (23) can hardly be regarded as exogenous although the true vessel s p e c i f i c f i s h stock may be. Furthermore, even i f x were exogenous, i t nevertheless contains an estimation error in general, thus giving r i s e to a c l a s s i c a l errors in variables problem <42>. Both of these d i f f i c u l t i e s , the simultaneity problem and the errors in variables problem, can, at least in theory, be overcome in an e f f i c i e n t way by resorting to the instrumentals variables estimation technique known as two stage least squares, 2SLS. That method, according to the theory, 294 y i e l d s , c e t e r i s paribus of course, consistent and e f f i c i e n t estimates of the parameters in (23) <43>. In addition to obtaining s t a t i s t i c a l l y "good" estimates of the parameters of (23) we are also interested in the most simple version of the f i s h i n g time equation that i s not contradicted by the data. In p a r t i c u l a r we would l i k e to test the following three hypotheses: H1 : c=1, H2: b(1)=b(2)=b(3)=b(4) . H3: g=0. Making the c l a s s i c a l stochastic assumption that u~N(0,aI) . and maximizing the instrumental r e s u l t s : the relevant l i k e l i h o o d function of variables constraint yielded the (23) under following 295 Table 6.4 ' Estimates of the Fishing Time Equation, (23). Estimation technique: 2SLS The set of instruments: t o ( i , j ) , k ( i , j ) , b(h), t r ( i , j ) , t ( j ) , where t r ( i , j ) refers to the t o t a l numb-er of t r i p s by vessel i and t ( j ) repre-sents a dummy variable for year j . Number of observations: 199 Unrestricted Estimates Restricted Estimates Parameters Estimates t-values Estimates t-values a -1 .27 -7.7 -1.19 -14.9 c 1.01 59.1 1.0 (restricted) f .092 6.1 .093 6.2 b( 1 ) -.111 -3.4 -.112 -3.5 b(2) -.045 -1.1 -.045 -1.1 b(3) -.100 -2.7 -.100 -2.8 b(4) -.033 -0.8 -.034 -0.8 g .0 (restricted) .0 (restricted) R*=0.96 Ri = 0.96 SE=0.06 SE=0.06 DW=2.14 DW=2.14 Test for normality of residuals: £*(6)=7.2 £17)=7.1 Tests of hypotheses: HI: F(1,192)=0.33. Not rejected. H2: F(2,192)=6.9. Rejected. H3: F(1,192)=1.8. Not rejected. These estimates seem s a t i s f a c t o r y . The explanatory power of the equation is high and the standard error of estimate correspondingly low. As far as can be judged on the basis of the Durbin-Watson s t a t i s t i c and the chi-square test for normality of residuals, our s t a t i s t i c a l assumption concerning the di s t r i b u t i o n of the disturbance term does not seem to be contradicted by the data. Consequently we may have f a i t h in the t-values for the significance of the estimated c o e f f i c i e n t s . The signs and magnitudes of the estimated c o e f f i c i e n t s hold 296 no surprises. It is interesting to note that, according to the sp e c i f i c a t i o n tests above, the e l a s t i c i t y of t f w.r.t. to i s independent of to and not s i g n i f i c a n t from 1. This means that the search time, t s , as a proportion of t o t a l operating time, to, is not influenced by to. In other words the vessels with high operating time spend just as much of the time searching for f i s h as vessels with low operating time. This result l i s of course open to many interpretations. One possible inference i s that the information gained by search is not accumulative with respect to the vessel, perhaps because i t soon becomes obsolete due to environmental v a r i a b i l i t y . The tonnage of the vessel i s estimated to have a small positive effect on the fis h i n g time. As predicted by the analysis in section 6.2 i t turns out that the vessel s p e c i f i c stock size has a s i g n i f i c a n t negative effect on fis h i n g time. Moreover, the general shape of 3tf/3x hypothesized in section 6.2 is not contradicted by the data. The p o s s i b i l i t y of a negative second derivative of tf w.r.t. x at low values of x is not supported by the data. The estimated relationship between t f and x calculated at the sample means of the other variables is depicted in f i g . 6.3. This figure i s comparable with i t s theore t i c a l counterpart in figure 6.3. 297 Figure 6.3 The Estimated Fishing Time Stock Size Relationship. FISHING TIME; TRAWL-HOURS 4000 3000 2000-1000 — I I I I | I I I I | I I I I | I I I I | I I I I ] I I I I | I I I I | 5 10 15 20 25 30 35 VESSEL SPECIFIC STOCK: MULTIPLES OF SAMPLE X The e l a s t i c i t y of f i s h i n g time with respect to the volume of the f i s h stocks i s given by: E(tf,X)= -0.0947-X, where X denotes the average vessel s p e c i f i c stock. Clearly the numerical value of this e l a s t i c i t y increases with X. Calculated at the sample mean of X, E(tf,X)= -0.0836. In section 6.3.2 we found that the e l a s t i c i t y of f i s h i n g mortality w.r.t. fishing time was indistinguishable from unity, i . e . E(F,tf)=1. It follows that the e l a s t i c i t y of f i s h i n g mortality w.r.t. average vessel s p e c i f i c f i s h stocks evaluated at the sample mean i s E(F,X)=-0.0836. As an i l l u s t r a t i o n l e t us assume that a fishery policy for 298 a previously mature common property fishery prescribes halfing of the f i s h i n g f l e e t with the result of doubling the f i s h stocks. The vessel s p e c i f i c f i s h stock thus increases by a factor of 4. According to our estimates the e f f i c i e n c y of each vessel in generating fishing time would be reduced by some 22%. Clearly, t h i s i s not an i n s i g n i f i c a n t e f f e c t . We f i n a l l y notice that the p a r t i a l derivative of fishing time w.r.t. vessel s p e c i f i c stock is s i g n i f i c a n t l y d i f f e r e n t for the four general locations considered. In fact the numerical value of thi s derivative is markedly lower in the Northwest and East regions than in the Southwest and North. Given that the gear operation, processing and landings time are similar for a l l parts of the country, which seems reasonable, t h i s result suggests that the Northwest and East fishing ports enjoy an environmental advantage with respect to the distance to the fishing grounds. A glance at figure 5.1 in section 5.1 supports this hypothesis. 6.3.4 Joint Estimation of the Fishing Mortality and Fishing Time Equations. As already pointed out, the fish i n g mortality and fi s h i n g time functions constitute a simultaneous equation system. Since both equations are ov e r i d e n t i f i e d according to the order condition on i d e n t i f i a b i 1 i t y a gain in e f f i c i e n c y may be attained by exploiting the f u l l system information in the estimation process <44>. For t h i s reason equations (22) and (23) were reestimated using the 3SLS technique, which according to 299 the stochastic s p e c i f i c a t i o n of the equations is asymptotically equivalent to the f u l l information maximum li k e l i h o o d method <45>. The main results of thi s procedure were as follows. Parameter estimates were hardly affected at a l l . There was generally only a very s l i g h t increase in the calculated t-values for the parameters. The r e s t r i c t i o n s imposed in tables 6.2 and 6.4 above were not rejected. The reason for t h i s rather i n s i g n i f i c a n t improvement may be found in the var-covariance matrix of the system. It was estimated to be: 3.6 0.3 0.3 24.1 which i s f a i r l y close to being diagonal <46>. 6.3.5 Estimation of Fishing Mortality Functions for Multipurpose Fishing boats. The theory of fishing mortality production functions developed in chapter 6.2 above i s intended to apply to any fishing e n t i t y . In the case of the Icelandic multipurpose fishing f l e e t , howevever, lack of data severely r e s t r i c t our a b i l i t y to u t i l i z e this theory. 300 F i r s t l y , estimation of separate f i s h i n g mortality and fish i n g time functions is not practicable because, as discussed in section 6.3.1, observations on the f i s h i n g time of the vessels are not available. Therefore, in order to obtain any estimates of the fi s h i n g mortality function, the two subfunctions have to be combined. Secondly, a key step in the estimation of the fishing mortality production function for the trawler f l e e t was the argument that a reasonable estimator of the vessel s p e c i f i c stock, x ( i ) , was given by x ( i ) = y ( i ) / t f ( i ) . Since, for the multipurpose f i s h i n g boats, t f ( i ) , the fis h i n g time, i s not known, t h i s estimator i s , however, not fe a s i b l e . Thirdly, the multipurpose f i s h i n g f l e e t i s far more heterogeneous with respect to vessel types, f i s h e r i e s , f i s h i n g gear and seasonal variations than the trawler f l e e t . Hence the aggregation problems discussed in appendix 6.3.1-B above are even more serious in the case of the multi-purpose f l e e t . From t h i s i t is clear that the empirical basis for estimating a fishing mortality production function for the multipurpose fi s h i n g vessels is much weaker than for the trawler f l e e t . Nevertheless, with the appropriate modification of the estimation procedure, reasonable estimates of the parameters of this function may s t i l l be attainable. Combining equations (10) and (17) we obtain the "reduced form" equation: F=F(E(e),z)TF(to,x;e,z,p). 301 This we may write in a stochastic form as: (24) F=G(e,z,to,p,x,u), where u represents a random term. As mentioned above, observations on z, e and to are available. F may be estimated as described in section 6.3.1. There are no observations on p but, as argued in section 6.3.3, that is probably of minor consequence. The f i n a l variable, the vessel s p e c i f i c stock, x, i s not d i r e c t l y observed. The most plausible estimator of x available seems to be: (25) x ( i ) = y ( i ) / t o ( i ) , a l l i , where y ( i ) i s the catch of vessel i and t o ( i ) is i t s operating time. As discussed above, however, (25) is a poor estimator of x ( i ) . Since operating time always exceeds actual fi s h i n g time i t is clear that (25) is a negatively biased estimator of x ( i ) . Since the expected value of the error is nonzero the standard econometric techniques of dealing with an errors in variables problem are not applicable <47>. In p a r t i c u l a r , the instrumental variables technique w i l l generally neither y i e l d consistent estimates of the c o e f f i c i e n t s of (24) nor i t s s t a t i s t i c a l properties. Thus i t appears that since, in this case, consistent estimators of x are not available, feasible estimates of (24) w i l l generally be inconsistent. This suggests that less 302 confidence should be placed in these estimates than would otherwise be the case. In part i c u l a r i t now seems plausible to combine extraneous information or b e l i e f s with the sample information thus modifying the resulting estimates. One way of doing this i s to adopt a formal Bayesian estimation procedure <48>. Here we resort to the less elegant device of simply constraining parameters of (24) according to our a p r i o r i b e l i e f s . These are primarily based upon the analysis presented in section 6.2.1 and in part i c u l a r the estimates of the fi s h i n g time function for the trawler f l e e t in section 6.3.3 above. We now turn to the estimation of (24). Apart from the constrained estimation procedure employed at the very end, the structure of the estimation process is as in previous sections. Hence we w i l l proceed r e l a t i v e l y quickly. The f i r s t step of the estimation process is to consider the functional form of.(24). For this purpose consider the following Box-Tidwell s p e c i f i c a t i o n : (26) F(r1)=a+bk1(r2)+ck2(r3)+dx(r4)+ex 2(r5)+fto(r6)+u Where the r's are Box-Tidwell transformation parameters. k1 refers to the tonnage of vessel i and k2 to i t s number of electronic f i s h i n g instruments. Maximizing the l i k e l i h o o d function of (26) under the c l a s s i c a l stochastic assumptions <49> subject to various constraints on the r's yielded the following results: 303 Table 6.5 Testing for the Specification of (24). Log-likelihood values I. r1=r2=r3=r4=r5=r6: II . r1=r2=r3=r4=r5=r6=1: II I . r1=r2=r3=r6=0, r4=r5=1: 347.0 279.8 342.3 134.4 9.4 On the basis of these r e s u l t s , the questionable s t a t i s t i c a l properties of the l i k e l i h o o d r a t i o tests and our rule of adopting the simplest functional s p e c i f i c a t i o n not strongly rejected by the data, we continue on the basis of hypothesis I I I . Including environmental s h i f t variables in (24) and dropping highly i n s i g n i f i c a n t terms we arr i v e at the spec i f icat ion: (27) F= Z h(m)d(m)+biTl+cr2+dx+ex2 + fto+u, where, as before, h(m) represents environmental s h i f t variables in the form of yearly dummies and ~ indicates natural logs of the respective variables. It i s interesting to note that contrary to the estimation results for the trawler f l e e t , geographical location was not found to have a s i g n i f i c a n t e f f e c t on f i s h i n g mortality for the multipurpose f l e e t . A test of the n u l l hypothesis that the effects of a l l four geographical locations, South-west, West, North and East, on fis h i n g mortality are i d e n t i c a l produced the 304 test s t a t i s t i c F(3,211)=1.3. Consequently the n u l l hypothesis could not be rejected. We now turn to the estimation of the parameters of (27). The estimation method was 2SLS. F i r s t (27) was estimated on the basis of the sample information. This produced, as we w i l l see, seemingly implausible estimates of the e l a s t i c i t y of fis h i n g mortality with respect to f i s h stocks. Therefore, we imposed a linear constraint on the c o e f f i c i e n t s of (27) as follows: (28) E(F,X)=(d+2eX)X<0.25, where X denotes the sample mean of x. Given the sample, this constraint translates into the following linear equality constraint on d and e: d+4.8e=0.1 According to sections 6.3.2 and 6.3.3, the estimate for the e l a s t i c i t y of f i s h i n g mortality w.r.t. vessel s p e c i f i c stock size at the sample mean for x was E(F,X)=-0.08. As shown in table 6.7 below the corresponding estimate for the multipurpose fis h i n g f l e e t i s E(F,X)=1.07. Hence the constraint in (28) may be regarded as a compromise between the two results or an adjustment of the unrestricted estimate of E(F,X) towards that for the trawler f l e e t . The primary j u s t i f i c a t i o n for thi s procedure being that the l a t t e r estimate i s on a much sounder s t a t i s t i c a l basis than the former. 305 The estimation results are l i s t e d in table 6.6: Table 6.6 Estimation of the Fishing Mortality Function. Estimation technique: 2SLS The set of instruments: k1,k2,d(m),d(j) and the age, type and length of each vessel. Number of observations: 185 (Vessels with operating time less than 200 days excluded). Parameters Unconstrained Estimates Constrained Estimates Estimates t-values Est imates t-values h d ) -8.25 -10.2 -8.88 -36.5 h(2) -8.34 -10.2 -9.04 -37.3 h(3) -8.59 -10.5 -9.30 -39.2 h(4) -8.56 -10.5 -9.27 -37.8 b 0.0301 0.8 0.3549 6.7 c 0.0225 0.7 0.2914 • 2.4 d 0.7675 15.7 0.1871 (re s t r i c t ^ e -0.0669 -8.9 -0.0181 (rest r i c t ) f 1.012 7.7 1 .000 ( r e s t r i c t ) R* : 0.99 0.72 SE : 0.06 0.28 DW : 1 .73 1.81 Test for normality of residuals: t 7T(4)=90.8 /£(6) = 15.6 Tests of r e s t r i c t i o n s : I. h(1)=h(2)=h(3)=h(4): F(3,176 ) =230.1 Rejected II. d+4.8e=0.1: F(1,176)=139.4 Rejected III f=1.0: F(1,176)=0.008 Not rejected Apart from the e l a s t i c i t y of fi s h i n g mortality w.r.t. vessel s p e c i f i c stock, the unconstrained estimates are similar to those obtained for the trawler f l e e t . The fishing c a p i t a l variables, vessel size and electronic f i s h i n g equipment, have a weak positive effect upon fis h i n g mortality. Total operating time has a highly s i g n i f i c a n t positive effect on fi s h i n g 306 mortality with an estimated e l a s t i c i t y not distinguishable from unity. The yearly s h i f t variables are estimated to have a very s i g n i f i c a n t effect on f i s h i n g mortality. Before these variables have been interpreted to r e f l e c t the ef f e c t s of environmental conditions such as the migratory behaviour of f i s h , the weather and so on. Since the multipurpose boats are much smaller than the trawlers and thus more susceptible to v a r i a b i l i t y of this nature, this result is not surprising. According to table 6.6 the cost, in terms of the calculated l i k e l i h o o d values, of imposing the constraint (28) i s very high and the corresponding tests suggest a conclusive rejection of the constraint. Because of the inconsistency of the estimates, however, these results should not be taken at face value. It is quite possible that by imposing the constraint we come closer to maximizing the true l i k e l i h o o d function of (27). In fact t h i s is the reason for imposing the constraint. The improvement in the calculated chi-square s t a t i s t i c for the normality of the residuals in the constrained version may be regarded as support for this hypothesis. In the following table we present calculated values of E(F,X) based upon the above estimates: 307 Table 6.7 E l a s t i c i t y of Fishing Mortality w.r.t. Vessel Specific Stock, Vessel s p e c i f i c stock Multiples of x evaluated at the sample mean (i.e X) Calculated e l a s t i c i t y of Fishing mortality Unconstrained Version Constrained Version 0.5 1 .0 2.0 3.0 4.0 0.73 1 .07 0.60 •1 .41 •4.96 0.17 0.24 0.06 -0.53 -1 . 54 The relations between fis h i n g mortality and vessel s p e c i f i c stock at the sample means of the other variables for both the constrained and unconstrained versions are further i l l u s t r a t e d in figure 6.4. Figure 6.4 The estimated f i s h i n g mortality stock size relationship. FISHING MORTALITY: A MEASURE 0.8—, 0.6 — 0.4 — 0.2-0.0- I I I I | I I I I | I I I I | I I I I | I I I I | I I I I |TI I I | 1 2 3 4 5 6 7 VESSEL SPECIFIC STOCK: MULTIPLES OF SAMPLE X According to the above estimates, even the r e s t r i c t e d one, 308 E(F,X) i s s t r i c t l y positive in the neighbourhood of the sample mean of x. Since this is at variance with our a n a l y t i c a l propositions in section 6.2 as well as the empirical results for the trawler f l e e t in section 6.3.3 a few comments may be warranted. F i r s t , this may be an e n t i r e l y spurious result caused by the shortcomings of the data in which case there i s nothing further to explain. On the other hand i t is conceivable that the result in fact describes the true s i t u a t i o n . In that case, according to proposition 6.2, search time must f a l l r e l a t i v e l y rapidly with increases in the size of the f i s h stocks, i . e . the derivative 9ts/3x i s r e l a t i v e l y high negative. This may be due to i n e l a s t i c expectations by the captains concerning the ground densities, i . e . their slowness in adjusting the reservation ground density. After a l l the sample, i t w i l l be recalled, only covers 4 years. Another possible explanation i s s h i f t s in the sp a t i a l d i s t r i b u t i o n of the f i s h stocks with fishing grounds closer to the shore becoming r e l a t i v e l y richer than the others as the f i s h stocks increase. Simplifying somewhat in order to concentrate on the essentials, we may write the t o t a l yearly catch of a vessel as y=f x, where f and x here represent the appropriate yearly averages of vessel s p e c i f i c f i s h i n g mortality and f i s h stock respectively. Hence (29) E(y,x)=E(f,x)+1, 309 where E(y,x) stands for the e l a s t i c i t y of catch w.r.t. vessel s p e c i f i c stock and E(f,x) the corresponding e l a s t i c i t y of fishing mortality. Equation (29) in combination with our estimates of E(f,x) suggest an interesting feature of the fi s h i n g process. For a given f i s h i n g c a p i t a l and technology, there i s a f i s h stock l e v e l , x* say, beyond which additional stocks w i l l not increase the catch l e v e l . To see th i s one has only to r e c a l l that according to our estimates, E(f,x) i s continuous and E(f,x)->-» as x — < 5 0 > . Moreover, for x exceeding x*, i t i s equally clear that an increase in the lev e l of the f i s h stocks w i l l , c e t e r i s paribus, reduce the catch l e v e l . The empirical relevance of thi s observation i s probably minor although h i s t o r i c a l examples are easy to find <51>. F i n a l l y we notice that according to our estimates of E(f,x), the multipurpose f i s h i n g f l e e t i s , at low stock l e v e l s , able to make greater use of increases in the f i s h stocks than i s the trawler f l e e t . For higher stock levels t h i s situation i s reversed and the multipurpose f l e e t reaches i t s capacity at much lower stock l e v e l s , or x*, than the trawler f l e e t . The estimated value of x* for the multipurpose f l e e t i s approximately 3.5 times the average sample value of the vessel s p e c i f i c stock. The corresponding value for the trawler f l e e t i s 12. 310 6,4 Theoretical Implications. A theory of the f i s h i n g mortality production function has now been developed. The a n a l y t i c a l results have been subjected to empirical tests on the basis of the available data and we have managed to estimate seemingly reasonable f i s h i n g mortality and f i s h i n g time functions. An important result of t h i s chapter is the proposition that the l e v e l of f i s h stocks, or, in aggregative terms, the biomass l e v e l , influences in general the e f f i c i e n c y of economic inputs in generating f i s h i n g e f f o r t and subsequently f i s h i n g mortality. Moreover, there is evidence that t h i s stock-effort relationship i s , in most cases, negative. We w i l l now b r i e f l y discuss the theoretical implications of t h i s r e s u l t . Consider the following aggregative f i s h e r i e s model: A. Economy: H(t)=Y(e(t),x(t))-C(e(t)) , where e(t) represents economic inputs and x(t) the biomass l e v e l at time t. Y(.,.) is a harvesting function and C(.) a cost function. Consequently H(t) is the p r o f i t function. H(e,x) is taken to be j o i n t l y concave in both i t s arguments. There is a rate of time discount, r>0. 311 B. Biology: x(t)=G(x(t))-Y(e(t),x(t)), where x(t)=9x(t)/9t. G(.) i s the biomass growth function. It i s assumed that G"(x)<0, and G(a)=G(b)=0, b>a>0. The p r o f i t maximizing equilibrium solution for e(t) of thi s f i s h e r i e s model, i f i t exists, i s given by: (30) G x+C eY x/H e = r, where we have, for notational s i m p l i c i t y , dropped e x p l i c i t reference to the arguments of the functions involved. Equation (30) corresponds to the well known optimal equilibrium condition in fishery economics derived by Clark and Munro <52>. The second term in equation (30), C^Y^/H^, i s what Clark and Munro c a l l the marginal stock e f f e c t . On the usual assumptions about Y x, i t is positive <53>. According to the results summarized in c o r o l l a r y 3.2 above, however, a more appropriate s p e c i f i c a t i o n of the harvesting function i s : Y(E(e,x),x). 312 Adopting t h i s s p e c i f i c a t i o n of the harvesting function in the aggregative f i s h e r i e s model above yiel d s the following p r o f i t maximizing equilibrium solution, assuming i t e x i s t s : (31) G x+C eY x/H e+C eY eE x/H e = r. The t h i r d term of t h i s expression may be ca l l e d the marginal stock-effort effect in equilibrium. Its interpretation i s analogous to that of the marginal stock ef f e c t of Clark and Munro <54>. The sign of the stock-effort e f f e c t , however, i s normally negative, i . e . i f E x(e,x)<0. It thus both complements and modifies the usual stock e f f e c t . A l t e r n a t i v e l y we may think in terms of a decomposition of the grand or t o t a l stock effect into the sum of the usual Clark-Munro stock effect and the stock-effort e f f e c t : (32) C eY x/H e=C e (Ye E^+Y^J/H . Equation (32) highlights two points: F i r s t l y , the usual stock effect i s obtained only in the case that E x(e,x)*0. In other words i t is precisely the effect of the stock l e v e l on the e f f i c i e n c y of economic inputs in producing f i s h i n g e f f o r t or fi s h i n g mortality that creates the stock-effort e f f e c t . Secondly, due to the stock-effort effect the p o s s i b i l i t y of a negative t o t a l stock effect does not seem as farfetched as before. A negative t o t a l stock effect e s s e n t i a l l y means that, in a p r o f i t maximizing equilibrium, or, more generally, for some x, 313 an increase in biomass w i l l actually induce a reduction in the catch l e v e l . In a decentralized fishery this s i t u t i o n c l e a r l y has some interesting s t a b i l i t y implications. As i n t u i t i o n predicts, the existence of a nonzero stock-e f f o r t e f f e c t not only modifies the optimal equilibrium conditions. It also modifies the optimal approach paths. We may conclude that policy recommendations that do not take account of the stock-effort relationship w i l l , in general, be unsound. Most l i k e l y they w i l l understate the optimal use of economic inputs for a l l biomass levels <55>. 314 Footnotes, 1. This assumption i s so common that i t may be regarded as standard in the f i e l d . For basic references on t h i s see Schaefer, 1954 and 1957, Beverton and Holt 1957 and Gulland 1969. For examples from the current economic l i t e r a t u r e see the i n f l u e n t i a l paper by Clark and Munro, 1975, and Clark, 1976. 2. While the fishing mortality production function does not normally appear e x p l i c i t l y in the usual fishery economics models of the Schaefer genus, these models nevertheless i m p l i c i t l y define a f i s h i n g mortality production function as w i l l be shown below. 3. This i s e.g. demonstrated in section 6.4. 4. See e.g. Beverton and Holt, 1957, pp 29-30, 89-91, and Gulland's interpretation, 1969. It should be pointed out that t h i s interpretation of the Beverton-Holt f i s h i n g mortality function does not do f u l l j u s t i c e to their rather extensive discussion of the potential complications of the relationship. This omission, however, i s in accordance with subsequent work and practice by fishery b i o l o g i s t s see e.g. Gulland, 1969, pp. 45-47. 5. See Schaefer, 1954 and 1957. 6. See e.g. Clark, 1976 (p. 14) and Schnute, 1977 (p." 585). 7. See section 4.1.2 above. 8. This is a s l i g h t l y generalized version of the harvesting function in the basic Clark-Munro model. See Clark and Munro, 1975. 9. For examples see Schaefer, 1957, Beverton and Holt, 1957, Hannesson, 1974, Anderson, 1979, and M i t c h e l l , 1980. 10. See Arnason, 1977, and Dasgupta and Heal, 1979. 11. Rothschild, 1977, provides a more detailed discussion of t h i s . 12. This function is suggested by Rothschild, 1977. 13. Referring to the cohorts by the numbers 0 ,1 , 2,....etc. according to age, a fixed number of cohorts for each species is c l e a r l y not r e s t r i c t i v e since the number of individuals in the older cohorts may be zero. 14. See e.g. Henderson and Quandt, 1958, pp 67-72. 15. See e.g. Beverton and Holt, 1957, pp. 29-31, Gulland, 1969, pp. 45-47, and ICES: Report of the North Western Working Group, 1976. 315 16. See e.g. Beverton and Holt's discussion of standardized fi s h i n g time in Beverton and Holt, 1957, p. 29. 17. See section 6.1 above, esp e c i a l l y equation (2). 18. This has by no means gone unnoticed by fishery b i o l o g i s t s . See e.g. Beverton and Holt 1957, pp. 29-31 and pp. 94-96. 19. Examples are provided by e.g. ICES, 1976, and Clarke 1976, respectively. An example to the contrary is Hannesson, 1974, pp. 19-30. His awareness of the problem, however, does not carry into his empirical work. 20. This i s further discussed in appendix 6.2-A. 21. This can in fact be inferred from the arguments in appendix 6.2-A. 22. If some search in fact takes place during actual f i s h i n g i t may be regarded as increasing the e f f i c i e n c y of fis h i n g without changing n f ( t ) . 23. The estimated harvesting cost equations in that section are functions of c a p i t a l variables, t o t a l operating time and catch but not fi s h i n g time. 24. See the introduction to t h i s chapter. 25. During the late f i f t i e s and early s i x t i e s for instance Icelandic trawlers engaging in the redfish fishery off Newfoundland spent 10- 12 days out of a t o t a l t r i p of 12-16 days s a i l i n g to the fi s h i n g grounds and back. 26. This i s e.g. discussed in Beverton and Holt, 1957 pp. 94-95. 27. This p a r t i c u l a r decomposition of the fishing a c t i v i t y i s , of course, by no means unique and perhaps not even exhaustive. However, l i t t l e additional insight seems to be gained from further refinement in thi s respect. 28. A good exposition of consumer search theory i s given by Rothschild, 1974. For a survey of job search theory the reader is referred to the a r t i c l e s by Lippman and McCall, 1976a and 1976b. 29. To simplify the notation we have omitted some of the variables included in NF(.,.,.) in section 6.2. 30. Theoretically t h i s corresponds to a certain case in job search theory, in which the worker i s assumed not to be able to r e c a l l previous wage o f f e r s . See e.g. Hay, 1979 pp 108-110. 31. This stochastic process i s a variant of the binomial d i s t r i b u t i o n c a l l e d the geometric d i s t r i b u t i o n the moment generating function of which i s M(k)=p(1 - ( 1 - p ) e x p ( k ) . For 316 further d e t a i l s see e.g. Hogg and Craig, 1970, p 90. 32. See section 6.2.2. 33. For d e t a i l s on the data, see Appendix 1: Data (section 6.3.1). 34. As the trawler f l e e t currently accounts for roughly 2/3 of the t o t a l demersal catch this may not be too damaging. 35. This holds especially for the formulation of fishery policy which often involves cohort s p e c i f i c e f f o r t prescriptions. 36. See Gulland, 1969 pp. 45-55, and Schnute, 1977. 37. See Box and Tidwell, 1962 and, in p a r t i c u l a r , our previous discussion in appendix 5.2-C. 38. Due to the numerical properties of the Box-Tidwell estimation procedure, i t proved troublesome to include s h i f t variables to r e f l e c t environmental conditions. The numerical d i f f i c u l t i e s could, of course, have been overcome but only at the cost of s a c r i f i c i n g some of the s t a t i s t i c a l properties of the estimation process and a considerable amount of e f f o r t . This was not thought worthwhile. 39. A discussion and interpretation of l i k e l i h o o d r a t i o tests in the Box-Cox-Tidwell framework can be found in Judge et a l . , 1980, pp. 308-14. See also appendix 5.2-B above. 40. If the estimation procedure is not maximum li k e l i h o o d , then according to the Neyman-Pearson theory on hypothesis testing (see e.g. Silvey, 1970, pp. 94-108) the l i k e l i h o o d r a t i o tests are not uniformly most powerful. The main point, however, i s that due to the inconsistency of the estimates the meaning of the test s t a t i s t i c s is unclear. 41. This i s a standard procedure on which there are many good references. See e.g. T h e i l , 1971, pp. 451-60. 42. On t h i s see e.g. The i l 1971, pp. 607-15. 43. See e.g. Judge et al.,1980, pp. 531-33. 44. On t h i s see e.g. Schmidt, 1976, pp. 209-16. 45. See Schmidt, 1976, p. 224. 46. On the significance of this see e.g. Schmidt, 1976, p. 211. 47. Standard econometric theory on errors in variables i s exclusively in terms of errors with an expected value of zero. See e.g. T h e i l , 1971 pp. 607-15, and Judge et. a l . , 1980, pp. 521-31 . 317 48. For instance Theil's mixed extimation procedure. See T h e i l , 1971, pp. 346-53 and pp. 670-72. 49. u ~ N ( 0 , a l ) . 50. This also i l l u s t r a t e s that claims of a fixed E(y,x), either 0, see Bjorndal, 1983, or 1, see Clark 1976, while perhaps reasonable for empirically known values of x, can hardly be maintained for a l l values of x. 51. One example is the redfish fishery off Newfoundland in 1957-8 in which the captains frequently sought lesser densities in order to avoid rupturing the trawl and thus not getting any catch. Purse seine f i s h e r i e s hold many similar examples. 52. See Clark and Munro, 1975. 53. See Clark and Munro, 1975, and Arnason, 1977. 54. See Clark and Munro, 1975. 55. An example is provided in figures 8.14 and 8.15 below. 318 PART III MODEL PREDICTIONS 319 7. Simulations. In t h i s chapter the empirical model developed in part II w i l l be employed for simulating the Icelandic demersal f i s h e r i e s during 1960-1980. The main purpose of this exercise is to test in a general, a l b e i t informal, way the a p p l i c a b i l i t y of the model to the empirical situation i t is intended to describe. Clearly, this may, to a certain extent, be accomplished by comparing the endogenous variables generated by the model, i . e . the model "predictions", with actual observations during t h i s period. As the data period constitutes only a part of the simulation period <1>, this comparison should be indicative of the a b i l i t y of the model to discover e f f i c i e n t harvesting programs. In addition to checking the general v a l i d i t y of the empirical model, we w i l l be interested in the effects of some of i t s special features on the accuracy of the predictions. In p a r t i c u l a r , we w i l l consider the effects of the ecological relationships estimated in section 4.2.5 and the process a l l o c a t i o n functions estimated in section 5.4.2 above. As w i l l become clearer below, the simulation process i s somewhat hampered by lack of observations on the relevant variables. This holds both for some exogenous variables required to fuel the model and endogenous variables with which the model predictions are supposed to be compared. Perhaps the most si g n i f i c a n t shortcoming of this nature concerns the f i s h i n g mortality production function. Since neither the t o t a l number of vessels engaged in the demersal f i s h e r i e s nor their individual operating times are accurately known during the simulation 320 period, f i s h i n g mortalities can not be meaningfully simulated. Also, since the economic predictions of the model have no direct correspondences in the recorded economic s t a t i s t i c s , the accuracy of these predictions cannot be precisely assessed. For convenience of exposition the simulation results w i l l be presented in two parts. In section 7.2 we w i l l consider how well the catch predictions, under d i f f e r e n t b i o l o g i c a l s p e c i f i c a t i o n s , f i t the recorded catch l e v e l s . Due to the dynamic structure of the model, th i s focus on the catch predictions should provide a s u f f i c i e n t check on the v a l i d i t y of the b i o l o g i c a l model as a whole. Secondly, in section 7.3, we w i l l compare the economic predictions of the model with the available s t a t i s t i c s . The key variable of interest here is the annual value or rent of the harvesting a c t i v i t y <2>. Before presenting the simulation r e s u l t s , however, we w i l l , in section 7.1, b r i e f l y describe the way in which the simulations are set up. 7.1 The Structure of the Simulations. The empirical model i s e s s e n t i a l l y a system of difference equations composed of a number of endogenous and exogenous variables as well as estimated parameters. The simulations are performed by supplying estimates of i n i t i a l conditions, i . e . at the beginning of 1960, and the values of the exogenous variables to the model. The model subsequently calculates the values of the endogenous variables. 321 According to the general structure of the empirical model <3>, the exogenous variables are the b i o l o g i c a l environmental conditions, economic input and output prices, some of the elements of the processing transformation matrix, the control variables and the i n i t i a l values of the state variables. In the simulations the annual fi s h i n g mortality, f, w i l l also be taken to be exogenous. The reason, as explained above, is that the necessary data for generating f i s h i n g mortality endogenously during the simulation period are simply not available. Also, due to our unab i l i t y to model the determination of natural mortality <4>, this variable w i l l be taken as exogenous in the simulations. The i n i t i a l state of the fis h i n g c a p i t a l , K(0), as well as the control variables, investment in the fi s h i n g f l e e t and the use of economic fishing inputs w i l l be replaced by one exogenously given variable, k ( t ) , which represents the annual number and type of the vessels engaged in the cod, haddock and saithe f i s h e r i e s <5>. F i n a l l y , since, as discussed in section 4.2.5.2, observations on the environmental conditions are largely nonexistent, their role in the simulations w i l l be played by appropriately specified stochastic disturbances. More precisely, the environmental v a r i a b i l i t y w i l l be represented by a stochastic process operating on the recruitment functions of the three species <6>. To summarize the foregoing discussion, the simulated catch and harvesting rents are generated by the reduced form equations: 322 (1 ) y(t)=Y({f},{u},m,X(0)), (2) v(t)=V(y(t),p(t),T,k(t)). ~ ~ Where y(t) and v(t) denote the annual catch and rent of the harvesting a c t i v i t y respectively. y ( t ) , of course, i s a 3-dimensional vector containing the predicted annual catch of cod, haddock and saithe. In the catch equation, {f} represents the time path of fishing m o r t a l i t i e s from 1960 onwards, {u} i s the time path of the stochastic recruitment disturbances, m denotes the appropriately dimensioned vector of natural mo r t a l i t i e s and X(0) represents the i n i t i a l stock sizes of the three demersal species in 1960. In the harvesting value equation, T stands for the matrix of processing transformation parameters while p(t) and k(t) represent the economic input and output prices and the fish i n g c a p i t a l employed in the f i s h e r i e s at time t. The structure of the empirical model i s , of course, r e f l e c t e d in the functions, Y( ) and V( ). 7.2 B i o l o g i c a l Simulations. Four d i f f e r e n t sets of simulations w i l l be considered. In the f i r s t , recruitment and individual weights as well as natural and f i s h i n g m o r t a l i t i e s w i l l be taken as exogenously given and stochastic effects ignored <7>. In the second set of simulations, the ecological nature of individual weights as estimated in section 4.2.5.1 w i l l be recognized. Thus, in thi s set of simulations, the individual weights w i l l become 323 endogenous. In the t h i r d set of simulations, recruitment w i l l also be made endogenous, according to the estimation results in sections 4.2.2.2-4.2.4.2 above. In p a r t i c u l a r , the recruitment of cod and haddock w i l l be described by the Beverton-Holt recruitment functions and the recruitment of saithe by the Ricker recruitment function estimated in these sections <8>. F i n a l l y , in the fourth set of simulations, the stochastic nature of the recruitment process w i l l be e x p l i c i t l y recognized. The stochastic s p e c i f i c a t i o n employed w i l l be according to the estimation results expressed by equations (53)-(55) in section The key distinguishing features of the four simulations are l i s t e d in table 7.1: Table 7.1 The 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 . Stochast ic Simulations Exogenous variables* effects SIM1 f , m , r , w No SIM2 f , m , r No SIM3 f , m No SIM4 f , m Yes * f=fishing mortality, m=natural mortality, r=recruitment, w=individual weights. The results of the simulations for each of the three species are depicted in figures 7.1-7.3 below. Also drawn in these figures, for comparative purposes, is the actual h i s t o r i c a l catch volume for each species. 324 Figure 7.1 Annual Catch of Cod 1960-1980: Simulation Results. CATCH (1000 TONS) 600-400 H 200 H - ACTUAL -• SIM1 •- SIM2 • SIM3 - SIM4 i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — r 60 65 70 75 . _ YEARS " 80 Figure 7.2 Annual Catch of Haddock 1960-1980: Simulation Results. YEARS 325 Figure 7.3 Annual Catch of Saithe 1960-1980: Simulation Results. The f i t of the various simulation s p e c i f i c a t i o n s with the actual recorded catch levels is further summarized in table 7.2: 326 Table 7.2 Catch Simulations 1960-81: F i t with Observed Catch Levels. Percentage-Root-Mean-Square-Errors* Simulations Cod Haddock Saithe Total** SIM1 7.1% 29.3% 24.0% 12.5% SIM2 2.6% 25.3% 8.4% 6.5% SIM3 9.2% 43.0% 26.6% 16.2% SIM4 14.4% 38.3% 26.7% 21.1% * Defined as 100{Z(y(t,pred)-y(t,act)) 2/n)**0.5/y(mean,act)} t where y(t,pred) denotes predicted and y(t,act) actual catch, y(mean,act) denotes the actual mean catch over the observat-ions and n i s the number of observations. ** I.e. the catch of a l l three species. Among the results of the simulations, the following should be especially noted: F i r s t l y , the catch paths predicted by the model are generally f a i r l y similar to those actually observed. Secondly, the accuracy of the catch predictions i s considerably improved by allowing individual weights to be ec o l o g i c a l l y determined. This holds even outside the sample period i . e . during 1978-80, thus providing added support for the ecological weight relationships estimated in part I I . Thirdly, making recruitment endogenous generally reduces the accuracy of the predicted catch. Since, according to the properties of the VPA method, the h i s t o r i c a l recruitment measurements should be very accurate while the estimated recruitment functions are poorly determined, th i s result i s as expected. The important point to notice, however, is that, 327 although the prediction errors become generally larger, the predictions do not show any signs of straying away from the true h i s t o r i c a l l e v e l s . Fourthly, incorporating the estimated stochastic process in the recruitment functions does not adversely a f f e c t the o v e r a l l f i t of the predictions except in the case of the cod. The predicted paths, on the other hand, become more r e a l i s t i c , in the sense of exhibiting a similar v a r i a b i l i t y over time as do the actual catch l e v e l s . This attribute of the stochastic simulations i s brought out even more c l e a r l y in figures 7.4 below where, as an example, the actual recruitment l e v e l of cod as well as that predicted with and without stochastic disturbances are depicted. Figure 7.4 Recruitment of Cod 1960-81. Actual and Predicted Levels. YEARS 328 7.3 Economic Simulations. The central economic variable of the empirical model i s the annual value or rent of harvesting cod, haddock and saithe. As already mentioned, however, comparison of simulated harvesting rents with the actual ones i s somewhat problematic, as s t a t i s t i c s on the economic rents in these f i s h e r i e s separately are simply not a v a i l a b l e . As discussed above, notably in sections 5.1 and 5.3.1, the Icelandic f i s h i n g f l e e t engages in a number of f i s h e r i e s other than those for cod, haddock and saithe. Whithin each year the fish i n g vessels t y p i c a l l y switch repeatedly from one fishery to another. In certain periods the representative vessel may be engaged in fis h i n g for cod, haddock or saithe, while at others i t may be engaged in e.g. the redfish fishery or, in the case of the multipurpose f l e e t , even pelagic or crustacean f i s h e r i e s . S i m i l a r l y , a s i g n i f i c a n t share of the output of most processing plants consists of species other than cod, haddock and saithe. By contrast, the recorded s t a t i s t i c s are limited to the operating results of the various classes of f i s h i n g vessels and processing plants for the year as a whole. These s t a t i s t i c s , moreover, are only available from 1971 onwards. Hence, there are no o f f i c i a l s t a t i s t i c s on the separate value of the cod, haddock and saithe f i s h e r i e s with which to compare our simulations. However, r e c a l l i n g that cod, haddock and saithe constitute over 90% of the value of the t o t a l demersal catch, a reasonable estimator of the appropriate rents in the harvesting sector is provided by concentrating on the ov e r a l l p r o f i t a b i l i t y of the demersal f i s h i n g f l e e t . This can be approximated by judicious 329 exclusion of certain vessel classes from the available s t a t i s t i c s . As regards the processing industry, however, i t is not possible to r e s t r i c t the available data to the processing of demersal species. From t h i s i t should be clear that comparison of simulated rents with the available s t a t i s t i c s provides, at best, a rough indication as to the accuracy of the model predictions. The economic simulations to be presented below are formulated in terms of the so-called p r o f i t a b i l i t y measure. This concept is defined as follows: prof(i)=2-C(i)/R(i) , where p r o f ( i ) i s the p r o f i t a b i l i t y , C(i) the costs and R(i) the revenue of sector i . The last two concepts are as defined in the respective sections of chapter 5 in part I I . Clearly, when C(i)=R(i), prof(i)=1.0. As well as simulating the t o t a l p r o f i t a b i l i t y of the harvesting a c t i v i t y as a whole, separate simulations are run for the harvesting and processing sectors. Four d i f f e r e n t sets of simulations are performed corresponding to the cases l a b e l l e d SIM1, SIM2, SIM3 and SIM4 in section 7.2. These simulations are a l l run for endogenous process a l l o c a t i o n parameters. The appropriateness of the estimated process a l l o c a t i o n functions w i l l , however, be considered at the end of the section. The simulation results for cases SIM2, SIM3 and SIM4 <9>, as well as the information about the actual path of harvesting values are depicted in figures 7.5-7.7 below. 330 Figure 7.5 P r o f i t a b i l i t y of the Harvesting Sector: Simulation Results, HARVESTING PROFITABILITY 1.50-, 1.25H 1.00 —! 0.75-0.50-ACTUAL SIM2 SIM3 — — — SIM4 ~1 I I 1 | I I I 1 1 1 1 1 1 1 1 1 1 1 1 Go 65 70 75 80 YEARS Figure 7 .6 P r o f i t a b i l i t y of the Processing Sector: Simulation Results PROCESSING PROFITABILITY 1.50-1.25H i. oo-H 0.75H •J . 50-ACTUAL ,'TM2 SIM3 SIM4 ~1 I I I | I I I ' I 1 1 1 1 1 1 1 1 1 1 1 S'J ::,r 70 75 80 YEARJ 331 Figure 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. rc TAL PROFITABILITY 1.50 —I 1.25 — 1.00-0 . 75 — u. 50 • ACTUAL SIM2 o!M3 — SIM4 " i — i — i — i — I — i — i — i — r i—r 30 70 YEARS 75 30 The comparative information contained in figures 7.5-7.7 is further summarized in table 7.3 below: Table 7.3 Simulations of Harvesting P r o f i t a b i l i t y 1971-80: F i t with Observed Values. Percentage-Root-Mean-Square-Errors* Simulations Harvesting Processing Total** SIM1: 6.7% 7.7% 6.6% SIM2: 4.7% 7.9% 6.7% SIM3: 5.1% 7.7% 7.3% SIM4: 8.6% 8.5% 10.4% * For d e f i n i t i o n of th i s concept see table 7.2 * I.e. the harvesting a c t i v i t y as a whole. These results do not require much explanation. Judged by 332 the entries in table 7.3, the prediction errors do not appear to be unduly large, especially considering that the errors are calculated r e l a t i v e to a very imperfect estimate of the actual h i s t o r i c a l values. More importantly perhaps, there are no signs of predictive divergence during the simulation period, although most of i t l i e s outside the estimation period. F i n a l l y , comparing tables 7.3 and 7.2 the economic prediction errors are far less sensitive to the b i o l o g i c a l s p e c i f i c a t i o n s than the catch prediction errors. In this sense the economic predictions are r e l a t i v e l y robust with respect to b i o l o g i c a l s p e c i f i c a t i o n s . F i n a l l y , we b r i e f l y consider the effects of the process a l l o c a t i o n functions estimated in section 5.4.2 above. The natural alternative to the process a l l o c a t i o n functions i s to r e s t r i c t the process a l l o c a t i o n parameters to be constant. These might e.g. be estimated as the h i s t o r i c a l sample means during 1974-82 <10>. On the assumption that the estimated process a l l o c a t i o n functions actually r e f l e c t p r o f i t maximization behaviour, the calculated value of the harvesting a c t i v i t y should generally exceed that in the alternative case. To check t h i s , simulated processing rents for (a) the process a l l o c a t i o n functions and (b) constant mean process a l l o c a t i o n parameters, in both cases under SIM2, i . e . the h i s t o r i c a l recruitment and ecological weight relationships, are graphed in figure 7.8. 333 Figure 7.8 Processing P r o f i t s : Different Process Alloc a t i o n Specifications, PROCESSING PROFITS (M. IKR.) 4000 2000 — 0 — -2000 — -4000 FUNCT CONST YEARS The graphs in figure 7 .8 confirm our presumption that calculated harvesting rents are higher in the case of the estimated process a l l o c a t i o n functions than for constant process a l l o c a t i o n parameters. In a l l but two of the periods, processing rents under the functional s p e c i f i c a t i o n exceed those for constant a l l o c a t i o n s . The predictive performance of the former i s also better. Its percentage root-mean square error i s 6 .7% compared with 6 . 9 % for the constant s p e c i f i c a t i o n . 7.4 Conclusions. The simulation results presented above seem generally sa t i s f a c t o r y . The simulated values show a reasonably good f i t 334 with observed values. This holds especially for the economic predictions. The mean error of the catch predictions is somewhat higher, although the f i t is by no means poor by the usual standards of ocean fishery predictions <11>. More importantly, however, neither the economic rent nor catch predictions show any discernible tendency to wander off course over time. In fact the model exhibits a remarkable robustness in this respect, even when random effects of a s i g n i f i c a n t magnitude are included in the recruitment functions <12>. This property of the model i s important from the point of view of designing e f f i c i e n t harvesting programs which, by their very nature, require long range simulations. 335 Footnotes. 1. The b i o l o g i c a l part of the empirical model i s only based on data up to 1977 and the economic submodel i s mainly based on data from the period 1974-77. 2. For a formal d e f i n i t i o n of t h i s concept see section 3.1. 3. See in p a r t i c u l a r section 3.2. 4. For a discussion of t h i s see section 4.2.5. 5. This variable as well as some of the other exogenous variables used in the simulations are l i s t e d in appendix: data (section 7.1). 6. For detailed s p e c i f i c a t i o n s see section 4.2.5.2 especially equations (53)-(55). 7. The natural mortality and individual weights used in the simulations are the o f f i c i a l b i o l o g i c a l estimates given in tables 4.1, 4.4 and 4.7 in part II of t h i s study. The annual recruitment and fishing m ortalities are l i s t e d in appendix: data at the end of this thesis. 8. These are, i t w i l l be remembered, the recruitment functions estimated for each species separately. 9. The graph corresponding to SIM1 i s l e f t out in the interest of vi s u a l c l a r i t y . 10. The relevant data are l i s t e d in Appendix 1: Data. 11. Even one year predictions t y p i c a l l y show errors of a similar or higher magnitude. See e.g. the various ICES publications in the l i s t of references. 12. This r e s i l i e n c e is largely due to the highly compensatory nature of the recruitment and individual weight processes. 336 8. E f f i c i e n t Harvesting Programs. In t h i s chapter we turn our attention to the s p e c i f i c a t i o n and description of e f f i c i e n t harvesting programs. Although our primary concern i s with the nature and properties of harvesting programs that are e f f i c i e n t in the sense defined in section 3.1, progress in this respect requires the preliminary step of i d e n t i f y i n g such programs. In the formal parlance of chapter 3 t h i s amounts to solving the following class of maximization problems: (1) Max V(A). A Where A represents an harvesting program and V(A) i s the corresponding value function. In general, the permissible A's are constrained to a bounded set. The structure of V(A) is defined by the empirical model developed in part II of the study. Now, as should be clear from part II, V(A) i s a very complex function including a multitude of possible control variables. To solve (1), therefore, we have to resort to numerical search. Moreover, i f only to keep computational costs within manageable bounds, the number of control variables over which maximization takes place has to be r e s t r i c t e d . It follows that only approximate solutions to (1) r e l a t i v e to a limited class of possible A's can r e a l i s t i c a l l y be hoped for. How close to the mark the solutions w i l l be depends in a fundamental way on the solution strategy adopted. By t h i s term, "solution 337 strategy", we refer primarily to (a) s p e c i f i c a t i o n of the permissible A's and (b) s p e c i f i c a t i o n of the numerical search techniques. A discussion of the solution strategy adopted in this study is provided in section 8 . 1 . The fundamental objectives of this study suggest a twofold approach to the investigation of e f f i c i e n t harvesting programs. Our primary aim i s to discover and describe e f f i c i e n t harvesting programs for the p a r t i c u l a r empirical case of the Icelandic demersal f i s h e r i e s . In addition to t h i s , however, th i s work i s also concerned with e f f i c i e n t harvesting of f i s h stocks in general. Consequently we would also l i k e to consider, however b r i e f l y , e f f i c i e n t harvesting programs for other, less s p e c i f i c , fishery situations that have' attracted attention in the l i t e r a t u r e . Section 8 .2 w i l l be devoted to the actual case of the Icelandic demersal f i s h e r i e s . Our aim in t h i s section i s to discover harvesting programs that, i f adopted, are l i k e l y to approximate the maximal attainable s o c i a l value of the demersal f i s h e r i e s to the Icelanders. This section i s thus concerned with p r a c t i c a l fishery p o l i c i e s for Iceland. It constitutes, in a sense, the completion of the cost-benefit aspect of t h i s thesis. In section 8 .3 we w i l l consider less s p e c i f i c situations. Without i t s p a r t i c u l a r empirical content, the model developed in part II provides a structural framework for the analysis of any multispecies, multifleet fishery <1>. It follows that, by the appropriate choice of data, t h i s structure can portray various interesting fishery situations. Among other things, the model can be used to suggest e f f i c i e n t harvesting programs and check 338 theore t i c a l propositions without the disadvantage of having to abstract away from most of the complications of actual f i s h e r i e s . 8.1 Solution Strategy. The central components of the solution strategy are (a) the sp e c i f i c a t i o n of the control variables to be included in the maximization exercise and (b) the selection of numerical techniques to find the corresponding maximum. A discussion of the control variables employed as well as the constraints to which they are subject i s provided in section 8.1.1 below. Having spe c i f i e d the control variables we w i l l , in section 8.1.2, consider the techniques employed to locate the maximum of V(A) over the permissible A's. 8.1.1 Control Variables. As discussed in chapter 3, the main categories of control variables defined by the empirical model are: (i) The use of economic fishery inputs, ( i i ) Investment in fi s h i n g c a p i t a l , ( i i i ) A l l o c a t i o n of catch to production processes. These categories comprise hundreds of possible control variables <2>. For computational reasons, therefore, we w i l l have to 339 concentrate on but a few of these to which we w i l l refer as the operative controls. In part II we concluded that the f i s h i n g f l e e t could be adequately described in terms of two vessel classes, namely trawlers and multipurpose vessels. Within each cl a s s , however, we found that vessels d i f f e r e d w.r.t. tonnage, engine size, age, location, gear etc. each of which constitutes a p o t e n t i a l l y controllable variable. By r e s t r i c t i n g our attention to standardized vessels in each c l a s s , however, we may reduce t h i s dimensionality d r a s t i c a l l y . In p a r t i c u l a r , we w i l l assume that each f l e e t consists only of vessels with the c h a r a c t e r i s t i c s of the actual sample mean <3>. The standardized c h a r a c t e r i s t i c s are defined in table 8.1: Table 8.1 Standardized Vessels. Standardized c h a r a c t e r i s t i c s Trawlers Multipurpose vessels Vessel size; tonnage (GRT) Engine size; (HP) No of e l e c t r . f i s h , instrm, Type of f i s h i n g gear Trips to Europe Locat ion 572.2 2105.1 * Trawl 0.8 * * * 87.0 421 . 1 4.4 ** * *** * Not relevant. ** Weighted average. (Weights being days using each gear) *** Weighted average. (Weights being number of vessels in each location) Having standardized the vessel c h a r a c t e r i s t i c s of the two f l e e t s in t h i s way, i t should be clear, i . a . from the discussion in section 5.3.1, that the remaining fishery inputs to be controlled are largely limited to (a) the t o t a l operating days 340 for each vessel and (b) the choice of fishery <4>. Operating days w i l l be limited to two values only. Either the vessel has no operating time, i . e . i s e n t i r e l y i d l e during the year, or i t operates at f u l l capacity. Thus, in e f f e c t , the operating time decisions are transformed into decisions on the number of i d l e vessels each period <5>. The choice of fishery w i l l s i m i l a r l y be expressed in terms of the number of vessels to participate in each fishery. Each vessel can be applied in four d i f f e r e n t ways during the year. It can pursue one of the three demersal f i s h e r i e s or i t can be i d l e . In what follows, we w i l l find i t notationally convenient to interpret idleness as the fourth fishery. The fact that a p a r t i c u l a r vessel i s thus not allowed to engage in more than one fishery during the year i s not p a r t i c u l a r l y r e s t r i c t i v e . After a l l we have assumed that the vessels in each fl e e t are i d e n t i c a l . Hence that part of the f l e e t that pursues a p a r t i c u l a r fishery can also be regarded as the f r a c t i o n of the t o t a l operating time each vessel spends on that fishery. Given these s p e c i f i c a t i o n s , the fishery input controls have been r e s t r i c t e d to the application of two types of vessels, namely standardized trawlers and multipurpose vessels, to one of possible four a c t i v i t i e s ; the cod, haddock or saithe f i s h e r i e s or idleness. In addition to the vessel application controls, investment and, for that matter disinvestment, in standardized f i s h i n g vessels w i l l be an operative control variable for both types of vessels, subject, of course, to the investment r e s t r i c t i o n s discussed in section 5.5.4. 341 Given the i n i t i a l number of vessels in the two f l e e t s , investment (positive or negative) and deterioration determine the time path of f i s h i n g c a p i t a l . This, in turn, imposes an upper bound on the possible application of vessels to d i f f e r e n t f i s h e r i e s . The boundary of the set, from which the fishery selection variables must be chosen, depends, in other words, upon the state variables of the model and thus, i n d i r e c t l y , on the investment controls. In order to avoid the consequent numerical problems <6>, we have elected to represent vessel applications to p a r t i c u l a r f i s h e r i e s , as fractions of the t o t a l available f l e e t s . The main advantage of t h i s i s that these redefined control variables are r e s t r i c t e d to the closed i n t e r v a l [0,1] which i s , of course, independent of the state of the system. The t h i r d class of potential control variables, process a l l o c a t i o n s , i s excluded from the set of operative controls to be considered below. This s i m p l i f i c a t i o n i s not f e l t to be unduly r e s t r i c t i v e . F i r s t , contrary to harvesting decisions, there does not appear to be any theoretical reason to expect individual decisions on process all o c a t i o n s to be suboptimal. Moreover, as explained in section 5.4.2 and i l l u s t r a t e d in section 7.3, these individual decisions have been related, in a seemingly sat i s f a c t o r y manner, to the other variables of the model and are thus already contained therein. We are now in a position to formalize the exposition s l i g h t l y . F i r s t , refer to investment in f l e e t i at time t by 342 I ( i , t) , i = 1,2; t=1,2,...,T, where T denotes the time horizon, not necessarily f i n i t e . Moreover, represent the t o t a l number of vessels in f l e e t i by K ( i , t ) , i=1,2; t=1,2,...,T. F i n a l l y l e t the fractio n of the f l e e t i allocated to fishery j at time t be a ( i , j , t ) , i=1,2; j = 1,2,3,4; t=1,2,...,T, where j=4 stands for the application of the vessel to idleness. With the help of this notation, we can now restate our basic maximization problem in terms of the operative controls and their constraints as follows: 343 T (2) Maximize V(A)= L V ( l ( i , t ) , a ( i , j , t ) d ( t ) , t-O I ( i , t ) , a ( i , j , t ) Subject to <7>: (i) I ( i , t ) > - K ( i , t ) , a l l i and t, ( i i ) 1>a(i,j,t)>0, a l l i , j and t, ( i i i ) £ a(i,j,t)=1, a l l i and t. > This problem includes 10 control variables each period; 2 investment variables (1 for each f l e e t ) and 8 vessel a l l o c a t i o n variables. Each of these may be adjusted in every one of the T periods. However, since 2 of the a l l o c a t i o n variables are redundant due to the equality constraint, i . e . ( i i i ) , (2) constitutes, in fact, an 8T-dimensional control problem. To be able to solve (2) by numerical methods, i t i s c l e a r l y necessary to r e s t r i c t the control period to a f i n i t e number. The p a r t i c u l a r method chosen for that purpose i s to divide the time horizon into two periods; (a) the control period, [0,T1], during which the control variables are r e s t r i c t e d only by the constraints in (2), and (b) the subsequent period, [T1 + 1,°°], during which the c o n t r o l l e r s remain at their T1 values. In spite of t h i s s i m p l i f i c a t i o n , (2) s t i l l contains an i n f i n i t e numer of terms. However, given s u f f i c i e n t s t a b i l i t y of V(A) and provided T1 i s in fact f i n i t e , the annual values generally converge s u f f i c i e n t l y close to equilibrium in f i n i t e time. Let t h i s convergence be attained in T2 periods and l e t V(T2) represent the corresponding equilibrium value. Then the 344 present value of a l l subsequent periods is given by the term V ( T 2 ) d ( T 2 ) / r , where r i s the rate of discount and d ( T 2 ) , i t w i l l be remembered, is the discount rate for T2. On the basis of these considerations we may now rewrite (2) as: T l TZ-1 (3) Max Z V ( I ( i , t ) , a ( i , j , t ) ) d ( t ) + Z V ( I ( i , T 1 ) , a ( i , j , T 1 ) ) d ( t ) t-O -t-Ti'-i +V(T2)d(T2)/r. I ( i , t ) , a ( i , j , t ) Subject to: ( i ) , ( i i ) and ( i i i ) . It i s i n t u i t i v e l y clear that as T1 increases, the solution to (3) w i l l converge the solution to ( 2 ) , the speed of convergence depending on the form of v( ) and the magnitude of d ( t ) . Hence the appropriate choice of Tl must take account of these factors. 8.1.2 Numerical Techniques. Expression (3) constitutes a multidimensional nonlinear maximization problem subject to linear inequality constraints. Although there exist a number of algorithms to deal with these kind of problems <8>, e f f i c i e n c y , in terms of computing time, i s 345 usually enhanced by incorporating the constraints in the objective function with the help of the appropriate transformations. Turning to the problem at hand, the following r e d e f i n i t i o n of the control variables, I ( i , t ) and a ( i , j , t ) , of problem (3) in terms of the unconstrained ones, z ( i , j ) and x ( i , j , t ) , w i l l guarantee that the inequality constraints (i) and ( i i ) w i l l not be violated <9>: (4) l ( i , t ) = - K ( i , t ) + z ( i , t ) 2 , z ( i , t ) e R 1 . (5) a ( i , j,t)=ABS(SIN(x(i, j , t) TT/2 ) ) , x ( i , j , t ) e R 1 . This transformation allows us to rewrite the numerical maximization problem in the following unconstrained form: T l oo (6) Max Z V ( z ( i , t ) , x ( i , j , t ) ) d ( t ) + Z V ( z ( i , T 1 ) , x ( i , j , T 1 ) ) d ( t ) . z ( i , t ) , x ( i , j , t ) There are several numerical algorithms that are, in p r i n c i p l e , able to approximately solve t h i s problem <10>. The choice between them is usually based on the c r i t e r i a of (a) accuracy, (b) r e l i a b i l i t y and (c) computational requirements. Which algorithm works best in these respects depends, generally, on the precise nature of the problem especially the form of the objective function. 346 In this p a r t i c u l a r problem, due to i t s i n s c r u t a b i l i t y , there is very l i t t l e a p r i o r i knowledge about the surface of the objective function. By a l i t t l e experimentation, however, we settled on the use of two main algorithms; (1) Powell's conjugate directions direct search method <11> and (2) a modified Newton gradient search method due to G i l l and Murray <12>. The conjugate directions method <13> i s a d i r e c t search method. As most other maximization algorithms, i t i s guaranteed to be quadratically convergent. Normally t h i s process w i l l ultimately discover conjugate search direc t i o n s , at which point the maximum can be located in one i t e r a t i o n . Having found a solution, the algorithm investigates the p o s s i b i l i t y that i t may be a l o c a l extremum by selecting a new i n i t i a l point and star t i n g the search again. The Gill-Murray quasi-Newton algorithm is b a s i c a l l y an improved gradient search method. The improvement primarily consists of using approximations to the Hessian matrix to a s s i s t in the search. In the case of t h i s algorithm also, a solution candidate can be checked by repeating the search from a d i f f e r e n t i n i t i a l point. In the maximizations performed both of these algorithms generally proved to be accurate and r e l i a b l e . The conjugate directions method, however, was generally more e f f i c i e n t in terms of computational costs. Consequently t h i s algorithm was more frequently employed. In some instances, however, especially when the objective function seemed to have rather sharp ridges 347 at which the maximum was located, the Gill-Murray algorithm proved to be superior. 8.2 Icelandic Demersal F i s h e r i e s . The objective of thi s section is to discover an e f f i c i e n t harvesting program for the Icelandic demersal f i s h e r i e s and examine some of i t s properties. This means that we w i l l seek to adjust a l l the operative controls specified in section 8.1.1 so as to maximize the corresponding value function. This part of the work, as discussed at some length in the introduction, is e s s e n t i a l l y normative. Its aim i s to indicate as accurately as possible within the scope of thi s study, the changes in the current Icelandic demersal harvesting pattern that are l i k e l y to generate the maximal economic contribution of these f i s h e r i e s to the Icelandic people. This task, often referred to as the basic case below, involves the most comprehensive maximization presented in thi s study. The results of this exercise are discussed in some d e t a i l s in section 8.2.2 below. In addition to t h i s fundamental maximization, we w i l l consider some variants of the basic case. In p a r t i c u l a r , we w i l l investigate (a) the effects of reduced control periods, (b) the situation in which species selection is impossible, (c) the effects of having only one harvesting f l e e t , (d) the effects of dif f e r e n t degrees of c a p i t a l m a l l e a b i l i t y , (e) the case of no ecological weight relationships, (f) the case of no stock-effort 348 effect and (g) the effects of stochastic recruitment. In order to reduce computational costs, these variants w i l l be examined within a considerably s i m p l i f i e d maximization framework. Before turning to the maximization exercises, we w i l l , in section 8.2.1, specify the i n i t i a l stock conditions and the assumed values of the exogenous variables. 8.2.1 I n i t i a l Conditions and Exogenous Variables. The i n i t i a l conditions consist of the values of the economic and b i o l o g i c a l stock variables at the beginning of the control period. In accordance with our aim to produce results relevant for policy formulation, the maximization period commences in 1984. Hence the i n i t i a l conditions refer to the stocks at the beginning of 1984. Due to the nature of the b i o l o g i c a l assessment techniques, however, information about the magnitude of b i o l o g i c a l stocks at so recent a date i s unreliable. For t h i s reason, measurements of the b i o l o g i c a l stocks in 1980 are taken as our i n i t i a l stocks and the estimated b i o l o g i c a l model u t i l i z e d to generate the stock sizes in 1984. The estimates of the b i o l o g i c a l stocks at the beginning of 1984 are l i s t e d in table 8.2. Also l i s t e d in table 8.2 are the base fis h i n g mortality vectors used in the maximizations. These, i t w i l l be remembered, define the pattern of f i s h i n g m o r t a l i t i e s that can be l i n e a r i l y expanded or contracted according to the relevant f i s h i n g mortality production function <14>. 349 Table 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 , Cohorts Number of Individuals ( M i l l i o n s , beg. of 1980) Cod Haddock Saithe Fishing M o r t a l i t i e s (Base for maximizations) Cod Haddock Saithe 1 335.7 36.7 89.5 0.00 0.00 0.00 2 203.4 34.8 26.5 0.00 0.00 0.00 3 174.5 73.2 25. 1 0.03 0.02 0.01 4 220.8 99.5 38.8 0.19 0.20 0.08 5 115.7 19.6 34. 1 0.32 0.40 0.20 6 51 .2 11.0 11.0 0.44 0.55 0.30 7 90.6 9.9 9.3 0.57 0.60 0.40 8 19.6 1 .4 3.7 1 .02 0.70 0.45 9 5.6 0.2 1 .0 1.13 0.80 0.50 10 1 .7 0.0 0.5 1 .03 1 .00 0.65 1 1 0.7 0.0 0.5 0.81 0.60 1 2 0.2 0.0 0.4 0.72 0.60 1 3 0.2 0.0 0.4 0.78 0.60 1 4 0.0 0.0 0.1 1 .00 0.60 >urce: VP-analysi s based on catch data per 1983 • The i n i t i a l economic condit ions involve the state of shing f l e e t s at the outset of 1984. The relevant data l i s t e d in table 8.3. Table 8.3 Economic stocks at the Beginning of 1984. Fishing Fleet c h a r a c t e r i s t i c s Trawlers Multi-purpose vessels Number of vessels: Mean age of vessels; 1 03 8.9 307 20.0 Source: F i s k i f e l a g Islands, 1984, The main exogenous variables of the model are environmental 350 conditions, natural mortality, prices, technical c o e f f i c i e n t s and the rate of discount. In searching for an e f f i c i e n t harvesting program for the Icelandic demersal f i s h e r i e s below, these variables w i l l be kept fixed at their expected values, i.e their sample means <15>. This means i . a . that environmental v a r i a b i l i t y and predator-prey relationships w i l l not be considered in t h i s section <16>. The rate of discount is set to 0.0363 in accordance with the conclusion in section 5.6. 8.2.2 E f f i c i e n t Harvesting Programs. We now turn to the task of specifying an e f f i c i e n t harvesting program for the three demersal f i s h e r i e s considered in t h i s study. As this program i s intended to be as r e a l i s t i c a representation of the true optimal policy as possible, a l l the control variables specified in section 8.1.2. w i l l be operative. The control period w i l l , however, be r e s t r i c t e d to 7 years. In terms of the value function, the cost of t h i s r e s t r i c t i o n does not seem to be unduely high <17>. The control period accordingly spans the years 1984-1990. Since there are 8 control variables each year, the t o t a l number of control variables i s 56. In t h i s maximization process, the b i o l o g i c a l and economic sp e c i f i c a t i o n s are those suggested by our empirical results in part II and the s p e c i f i c a t i o n s in section 8.2.1 above. In p a r t i c u l a r , the estimated recruitment functions as well as the ecological weight and process a l l o c a t i o n relationships <18> are included in the value function. In this chapter, however, we r e s t r i c t our attention to the deterministic version of the 351 model. This seems, in the present context, the most appropriate procedure. We are here concerned with the formulation of an a p r i o r i or open loop harvesting program. Hence the exogenous variables should naturally assume their expected l e v e l s . The effects of stochastic recruitment are s p e c i f i c a l l y considered in section 8.2.4, below. F i n a l l y , the c a p i t a l m a l l e a b i l i t y parameter, m, i s set to the seemingly reasonable value of 0.75 <19>. The above s p e c i f i c a t i o n constitutes what we w i l l refer to as the basic case below. The search for a global maximum was an arduous one. The function surface proved i n i t i a l l y to be f a i r l y i r r e g u l a r , with quite sharp ridges on which a number of l o c a l maxima appeared to reside. Both the Powell and the Gill-Murray algorithms were employed in the search process. Both, however, tended to converge to l o c a l maxima and saddle points. As knowledge about the shape of the value function acumulated, however, we were able to f a c i l i t a t e the search process by a more appropriate transformation of the constraints and rescaling of the problem. Nevertheless, a l l counted, over 4000 function evaluations were required before the solution, to be presented below, was located and v e r i f i e d to constitute a s u f f i c i e n t l y good approximation to the global maximum. For comparative purposes we have also calculated the outcome of a reference path. This path i s simply defined as the maintenance, in future periods, of the same size of the f i s h i n g f l e e t and i t s a l l o c a t i o n to f i s h e r i e s as was the case in 1983. This, in a sense, might be regarded as the continuation of the 352 current fishery policy <20>. Due to the multidimensional nature of the problem, i t is not easy to present the results in a concise manner. A summary of the key results is l i s t e d in table 8.4. Also contained in that table are the corresponding features of the reference path. A more complete visu a l presentation of the results i s depicted in figures 8.1-8.7 on the following pages. F i n a l l y , tables containing the detailed results are l i s t e d in appendix 8.2-A below. Table 8.4 E f f i c i e n t Harvesting Program: Some Key Results. Fishing Fleet (Number of vessels) Economic Results Annual undisc. rents (M.IKR, 1974 prices) Year Optimal program Current policy Opt imal Program Current Pol icy Trawl. M-boats Trawl. M-boats 1 983 1 03 307 1 03 307 -1888 -1888 1 984* 101 293 1 03 307 -21 36 -1 726 1 985* 99 279 103 307 3260 -1611 1986* 71 263 1 03 307 3340 -1 546 1 987* 69 247 1 03 307 6034 -1 504 1 988* 68 224 1 03 307 3562 -1 549 1 989* 66 181 1 03 307 7540 -1 549 1990* 65 1 35 103 307 4852 -1567 1991 63 75 1 03 307 4801 -1 579 2009 63 75 103 307 4030 -1 667 Present value (M,IKR): 108,824 -45,425 (M USD): 917 -383 * Control period. Trawl.=Trawlers M-Boats=Multi-purpose vessels, According to the results reported in table 8.4, s i g n i f i c a n t economic improvements can be procured by adopting an e f f i c i e n t 353 harvesting p o l i c y . The difference between the present value of the e f f i c i e n t harvesting program and the current policy i s some 154 B.IKR (or 1.3 B.USD) at the 1974 price l e v e l . In the context of the Icelandic economy thi s i s a very s i g n i f i c a n t amount. In 1983 the Icelandic gross national product was, incidently, also about 1.3 B.USD. Hence the economic benefits to be gained from adopting the above described harvesting policy roughly equal the value of one year's national production. In equilibrium, the annual undiscounted economic value of the e f f i c i e n t program is some 34 M.USD or about 2.6% of the 1983 GNP compared with a negative value of about 1.1% of the 1983 GNP under the current harvesting p o l i c y . It is interesting to note that the e f f i c i e n t harvesting program comes close to dominating the current policy in terms of economic r e s u l t s . Only in the f i r s t year, 1984, are the economic results of the e f f i c i e n t program s l i g h t l y i n f e r i o r to those of the current p o l i c y . One implication of this is that extremely high rates of discount are required to make the current policy preferable to the e f f i c i e n t one <21>. Most of the economic benefits of the e f f i c i e n t program are attained via increased e f f i c i e n c y in the f i s h i n g sector. While the rents generated in the f i s h processing sector are only s l i g h t l y increased under the e f f i c i e n t program, the p r o f i t a b i l i t y of the f i s h i n g sector increases by some 38%. This is further described in the following table: 354 Table 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 * * . E f f i c i e n t Program Current Policy Yaer Harv Proc Total Harv Proc Total 1983 0 .955 0 .954 0 .926 0 .955 0. 954 0 .926 1 984* 0 .420 0 .921 0 .679 0 .965 0. 955 0 .933 1985* 1 .253 1 .001 1 .119 0 .971 0. 957 0 .939 1986* 1 .297 0 .989 1 . 1 64 0 .974 0. 958 0 .942 1 987* 1 .324 1 .014 1 .207 0 .976 0. 959 0 .944 1 988* 1 .324 0 .991 1 .172 0 .976 0. 959 0 .943 1 989* 1 .328 1 .025 1 .216 0 .974 0. 958 0 .942 1 990* 1 .381 0 .997 1 .210 0 .973 0. 958 0 .941 1991 1 .364 0 .998 1 .211 0 .973 0. 958 0 . 941 2009 1 .345 0 .992 1 .191 0 .968 0. 957 0 .937 * Control period. ** Defined as 2-(costs/revenue), where the costs and revenue concepts for the respective sectors are defined in part I I . Harv = Harvesting sector Proc = Processing sector. Total = Fishing industry as a whole. The single most important factor in creating these benefits is the very dramatic reduction in the o v e r a l l l e v e l of f i s h i n g c a p i t a l employed in the f i s h e r i e s . This reduction, however, is neither equally d i s t r i b u t e d over the two vessel classes nor i s i t monotonic over time. Comparing the steady state e f f i c i e n t and current f i s h i n g c a p i t a l l e v e l s , however, i t appears that the trawler f l e e t should be reduced by 38.8% and the multi-purpose f l e e t by 75.6%. The e f f i c i e n t time paths of investment, f l e e t size levels and unused capacity are depicted in figures 8.1 and 8.2 <22>. The control period, as mentioned, covers the years 1984-90. The depicted f l e e t size in 1983 thus represents the i n i t i a l f l e e t s i z e . The post-1990 investment i s replacement investment. This 355 varies over time in response to variations in the deterioration rate due to changes in the mean age of the f l e e t s . 356 Figure 8.1 Trawlers: E f f i c i e n t Investment, Fleet and Idle Vessels, A. Optimal Investment. NUMBER OF VESSELS 10-• 1 0 H -20-INVSTM " 3 °—| ' 1 1 1 1 1 1 1 1 1 1 1 r 82 84 86 88 90 92 94 96 YEARS / B. Fleet Size and Number of Idle Vessels. NUMBER OF VESSELS 150 —! 100-50 H V/////////A IDLE7 i 1 1 1 1 • | • i | i | i | r , 82 84 66 88 90 92 94 96 YEARS 357 Figure 8.2 Multipurpose Vessels: E f f i c i e n t Investment, Fleet and Idle Vessels, A. Optimal Investment. NUMBER OF VESSELS 20—i -20-•40--60' INVSTM ' — r ~ 1 — i — 1 — i — 1 — i — • — i — 1 — i — 1 — i 82 84 86 88 90 92 94 96 YEARS B. Fleet Size and Idle Vessels, NUMBER OF VESSELS 400—1 300 200 — 100 — V/////////A IDLF YEARS 358 The e f f i c i e n t investment paths exhibit some interesting features. In the f i r s t place, i t should be noticed that although fis h i n g c a p i t a l i s dramatically reduced in the long run, the disinvestment process i s not instantaneous but a prolonged one drawn out over the whole control period. Consequently, the e f f i c i e n t investment path involves a considerable excess capacity for most of the control period. This holds for both f l e e t s but especially so for the multipurpose vessels. The reason for thi s prescription i s not d i f f i c u l t to see. In order to exploit to the f u l l e s t good fishing opportunities in the not too distant future <23>, there has to be s u f f i c i e n t fishing capacity at that point. This can be secured by maintaining an excess capacity in the form of id l e vessels over the "lean" years to be activated in the "good" years. However, maintaining i d l e vessels involves extra fixed costs. The same effect could also be attained by the appropriate pattern of disinvestment and investment. This alternative, however, i s also costly under the current m a l l e a b i l i t y assumptions since i t involves repeated instances of disinvestment. Comparing the magnitude of these two costs, i t is found, in thi s case, that maintaining i d l e vessels is superior to the disinvestment-investment a l t e r n a t i v e . This argument explains i . a . why, in 1984, i t i s e f f i c i e n t to maintain a large f l e e t of idle trawlers without engaging in any disinvestment only to disinvest d r a s t i c a l l y one year l a t e r . The reason i s that a large trawler f l e e t i s useful in 1985 i . a . to exploit the haddock fishery. The argument also explains the 359 even more peculiar path of the multipurpose f l e e t controls where the number of idle vessels i s nonmonotonic over time. Another interesting aspect of the optimal investment path is the disproportionate reduction in the two f l e e t s . The multi-purpose f i s h i n g f l e e t i s reduced much more than the trawler f l e e t . The reason for th i s must be attributed to higher e f f i c i e n c y of the trawler f l e e t , either in terms of operating costs or the a b i l i t y to produce f i s h i n g m o r t a l i t i e s or both. This conjecture i s , in fact, supported by the observed investment pattern by individual f i s h i n g firms in the past 10-15 years. During this period, the trawler f l e e t has grown from 20 to over 100 vessels while the multipurpose f l e e t has declined s l i g h t l y . These observed facts are evidence of the comparative e f f i c i e n c y of the trawler f l e e t . The e f f i c i e n t application of the vessels to the three f i s h e r i e s is depicted in figures 8.3 and 8.4 below. The corresponding vessel applications according to the current fishery policy can be inferred from the vessel applications in 1983, i . e . before the start of the control period. 360 Figure 8.3 Trawlers: E f f i c i e n t Allocation to Fisheries, A. The Cod Fishery. k w , w . W A l CURRENT mSSSSi OPTIMAL YEARS B. The Haddock Fishery. VESSELS 80-, 60-40-20 — 82 84 i — • — r 86 88 90 92 94 96 YEARS C. The Saithe Fishery. VESSELS 8O—1 60-40-20-, i CURRENT bSSSSK^ OPTIMAL 1 I | I | I | I | I | 82 84 86 88 90 92 94 96 YEARS 361 Figure 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, A. The Cod Fishery. VESSELS 300-1 200-100-CURRENT OPTIMAL ° 1 1 i 1 T 1 — i — 1 — i — 1 — i — r ~ ~ i — 1 — i 82 84 86 B8 90 92 94 96 YEARS B. The Haddock Fishery, CURRENT SttSttttd OPTIMAL 94 96 YEARS C. The Saithe Fishery. VESSELS 300—| 200-100 — • , CURRENT OPTIMAL 362 As figures 8.3 and 8.4 c l e a r l y demonstrate, a very d e f i n i t e d i v i s i o n of labour between the two f l e e t s seems to be e f f i c i e n t , especially in the long run. According to the e f f i c i e n t harvesting program, trawlers should sp e c i a l i z e in the cod fishery and the multi-purpose f l e e t in the haddock and saithe f i s h e r i e s . This suggests a certain pattern of r e l a t i v e e f f i c i e n c y of the two fl e e t s with respect to f i s h e r i e s . Again this conclusion conforms well with the q u a l i t a t i v e information. The saithe fishery and, in pa r t i c u l a r , the haddock fishery are r e l a t i v e l y shallow water f i s h e r i e s which the multi-purpose vessels can, and do, e f f e c t i v e l y pursue. During the control period, however, there are interesting jumps in the application of vessels to f i s h e r i e s . Thus the trawler f l e e e t , although concentrating on the cod fishery for most of the period, i s p a r t i a l l y shifted to the haddock fishery in 1985 and, to a greater extent, in 1989. S i m i l a r i l y , the multipurpose vessels which concentrate on the saithe fishery for most of the control period, are almost completely shifted to the haddock fishery in 1987 and 1989. In fact, as already mentioned, the greater part of the multipurpose f l e e t seems to have been kept in reserve to be able to make use of the good haddock fishery conditions in precisely these years. The catch and biomass paths corresponding to the above described control are described in figures 8.5-8.6 <24>. 363 Figure 8 .5 E f f i c i e n t Harvesting Program: Catch Levels, A. Cod CATCH (1000 TONS) 600-1 400 — 200-OPTI MAI-CURRENT 0 1 i i i | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I960 1985 1990 1995 2000 2005 2010 2015 2020 YEARS B. Haddock CATCH (1000 TONS) 600—1 400-200-0PTIMAL CURRENT I I I 1 | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I960 1985 1990 1995 2000 2005 2010 2015 2020 YEARS C. Saithe CATCH (1000 TONS) - OPTIMAL •• CURRENT 1980 1985 1990 1995 2000 2005 2010 2015 2020 YEARS 364 Figure 8 .6 E f f i c i e n t Harvesting Program: Biomass Levels, A. Cod BIOMASS (1000 TONS) 3000-1 2000-1000 — OPTIMAL CURRENT I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | 1980 1985 1990 1995 2000 2005 2010 2015 YEARS B. Haddock BIOMASS (1000 TONS) 1000-750-500-250-OPTIMAL CURRENT 0 | i > >> | i i I I | I I I I | I I I I | I I I I | I I I I j I I I I .1980 1985 1990 1995 2000 2005 2010 2015 YEARS C. Saithe BIOMASS (1000 TONS) 600 400-200-OPTIMAL • CURRENT I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | 1980 1985 1990 1995 2000 2005 2010 2015 , YEARS 365 The most s t r i k i n g b i o l o g i c a l result during the control period are the dramatic fluctuations in the catch l e v e l s , especially that of haddock. This, of course, i s the result of corresponding s h i f t s in the application of vessels to the f i s h e r i e s , expecially the haddock fishery, as i l l u s t r a t e d in figures 8.3 and 8.4 above. The reason why thi s harvesting pattern i s e f f i c i e n t i s probably to be found in the interplay between the optimal harvesting age and the actual cohort exploitation pattern. As shown in figures 8.3 and 8.4, the pivo t a l species in thi s harvesting s h i f t process is haddock. As explained in section 4.2.3.3, the haddock's cohorts attain their maximum biomass at an age of about 7. Hence with costless harvesting and zero discount rate, this would be the optimal age of exploitation <25>. The cohort pattern of exploitation as reflected in the base vector of fishing m o r t a l i t i e s <26>, does not, however, f i t this objective very . well. Thus, within the framework of the control problem considered here, the only way to at t a i n a higher average exploitation age is to harvest intermittently. The attractiveness of this policy may be further enhanced by the ecological weight relationships according to which a certain pattern of fluctuations in the haddock stock sizes may increase the average weight by age <27>. The most noteworthy long run b i o l o g i c a l results are some increase in equilibrium biomass levels and a considerable decrease in equilibrium catch for a l l three species. Both of these results are, of course, as predicted by theory <28>. The increase in biomass levels i s perhaps somewhat less than would have been expected. In t h i s , however, we observe the effects of 366 the ecological weight relationships <29>. Due to i t s multidimensional nature, the description of the e f f i c i e n t harvesting program i s , as we have seen, rather awkward. Hence, in an attempt to provide a convenient summary of the results we present below an aggregated state-space diagram <30> for the e f f i c i e n t harvesting program as well as the current p o l i c y . Figure 8.7 E f f i c i e n t Harvesting Program: State Space Diagram. A G G R E G A T E D C A P I T A L 3 0 0 — 1 2 0 0 -100-OPTIMAL CURRENT T — I — i — I — | — I — I — I — l — | — I — I — I — I — | I l l I | 1000 2 0 0 0 3000 4000 AGGREGATED BIOMASS Not much needs to be said about figure 8.7. It mostly summarizes information already discussed above. The current harvesting policy i s close to i t s equilibrium state, while a considerable reduction in fis h i n g c a p i t a l and some increase in biomass i s prescribed by the e f f i c i e n t policy. At thi s stage, however, i t i s illuminating to compare t h i s , rather jagged, e f f i c i e n t approach path with the corresponding very smooth ones 367 i n e.g. the t h e o r e t i c a l a n a l y s i s by C l a r k et a l . and the work by C h a r l e s <31>. A l t h o u g h , the d i f f e r e n c e can, t o some e x t e n t a t l e a s t , be e x p l a i n e d i n terms of d i f f e r e n t t e c h n i c a l f e a t u r e s and assumptions of the r e s p e c t i v e models, i t n e v e r t h e l e s s s u g g e s t s the p r o p o s i t i o n t h a t r e a l w o r l d e f f i c i e n t h a r v e s t i n g p a t h s a r e g e n e r a l l y much more i r r e g u l a r than those d e r i v e d i n s i m p l i f i e d models. 368 Appendix 8.2-A E f f i c i e n t Harvesting Program: A Survey of Results. This appendix contains a detailed l i s t of the properties of the e f f i c i e n t harvesting program for the Icelandic demersal f i s h e r i e s described above. For a discussion of these res u l t s , the reader i s referred to the main text. Table A.1 E f f i c i e n t Harvesting Program: Fishing Fleet S t a t i s t i c s . Note 1: The Investment numbers prior to 1984 are meaningless. Note 2: Stocks evaluated at beginning of each year. Notation: M-Boats = multipurpose vessels. Invm. = investment. Age = mean age of f l e e t . TRAWLERS M-BOATS YEAR INVM. TOTAL ACTIV IDLE AGE INVM. TOTAL ACTIV IDLE AGE 1980 0.0 86.0 86.0 0.0 6.5 0.0 280 .0 280.0 0.0 17.6 1981 0.0 91.0 91.0 0.0 7.2 0.0 295 .0 295 .0 0.0 18.4 1982 0.0 101.0 101.0 0.0 8.0 0.0 307 .0 307.0 0.0 19.2 1983 0.0 103 .0 103.0 0.0 8.9 0.0 307.0 307.0 0.0 20.0 1984 0.0 101.3 18.4 82.8 9.9 -3.3 292.9 30.1 262.8 21.0 1985 -25.8 99.3 99.3 0.0 10.9 0.0 274 .8 27.3 247.5 22.0 1986 -0.1 71.4 71.4 0.0 11.9 0.0 259 .7 35.7 224 .0 23.0 1987 0.6 69.7 69.7 0.0 12.9 -6.9 244 .0 133.3 110.7 24.0 1988 0.1 68.6 68.6 0.0 .13.8 -26 .5 220 .8 32.9 188 .0 25.0 1989 0.3 66 .8 66 .8 0.0 14.8 -3 0.5 177.9 177.9 0.0 26 .0 1990 0.0 65.1 65.1 0.0 15.7 -47.6 132.7 73.1 59.7 27 .0 1991 2.2 63.0 63.0 0.0 16 .7 7.7 72.8 72.8 0.0 28.0 1992 2.3 63.0 63.0 0.0 17.1 6.0 72.8 72.8 0.0 26 .0 1993 2.3 63.0 63.0 0.0 17.5 5.3 72.8 72.8 0.0 24.9 1994 2.4 63.0 63.0 0.0 17.9 4.9 72.8 72.8 0.0 24.0 1995 2.5 63.0 63.0 0.0 18.2 4.6 72.8 72.8 0.0 23.4 1996 2.5 63.0 63 .0 0.0 18.5 4.4 72.8 72.8 0.0 23 .0 1997 2.6 63.0 63.0 0.0 18.7 4.2 72.8 72.8 0.0 22.6 1998 2.6 63.0 63.0 0.0 18.9 4.1 72.8 72.8 0.0 22.3 1999 2.7 63.0 63.0 0.0 19.1 4.0 72.8 72.8 0.0 22.0 2000 2.7 63.0 63.0 0.0 19.3 3.9 72.8 72.8 0.0 21.8 2001 2.8 63.0 63.0 0.0 19.5 3.9 72.8 72.8 0.0 21.6 2002 2.8 63.0 63.0 0.0 19.6 3.8 72.8 72.8 0.0 21.5 2003 2.8 63.0 63.0 0.0 19.8 3.8 72.8 72.8 0.0 21.3 2004 2.9 63.0 63.0 0.0 19.9 3.7 72.8 72.8 0.0 21.2 2005 2.9 63 .0 63.0 0.0 20.0 3.7 72.8 72.8 0.0 21.1 2006 2.9 63.0 63.0 0.0 20.0 3.7 72.8 72.8 0.0 21.1 2007 2.9 63.0 63.0 0.0 20.1 3.7 72.8 72.8 0.0 21.0 2008 2.9 63.0 . 63.0 0.0 20.2 3.6 72.8 72.8 0.0 20.9 2009 3.0 63.0 63.0 0.0 20.2 3.6 72.8 72.8 0.0 20.9 369 Table A.2 E f f i c i e n t Harvesting Policy: Allocation of Vessels to Fisheries and Economic Results. Note 1: Rate of discount 0.0363. Note 2: Pre-2984 discounted values are meaningless. . Notation: M-Boats = multipurpose vessels. Undisc. = undiscounted. Disc. = discounted. Had = haddock. Sai = saithe. NUMBER OF V E S S E L S NET V A L U E A C T I V E TRAVELERS A C T I V E M-BOATS I D L E U N D I S C . D I S C . YEAR COD HAD SA I COD HAD SA I TRAWL MULTI ( M . I K R ) 1980 68.7 8 .0 9.3 223 .8 26 .2 30 .0 0. 0. 438.8 0.0 1901 72.0 10 .0 9.0 23 0.1 33 .6 31 .3 0. 0. 1258.5 0.0 1982 79.0 12 .0 10.0 235.5 38.7 32 .8 0. 0. -7 58.5 0.0 19S3 80.0 12 .5 10.5 23 4.5 39.7 32 .8 0. 0. -1887.5 0.0 1984 15 .9 0.0 2.5 0 .5 0 .0 29 .5 83.263. -2136.3 -2062 .1 1985 66 .6 32.7 0.0 0 .0 1 .0 26.3 0.248. 3260.4 3037 .7 1986 71 .4 0.0 0.0 7 .3 0 .0 28 .3 0.224. 3340.8 3004 .5 1987 69 .6 0.1 0.0 0 .0131 .0 2.2 0.111. 6034.1 5238 .1 1988 68 .5 0.0 0.0 0 .0 0 .0 32.8 0.188 . 3561.7 2984 .4 1989 33 .3 33.5 0.0 0 .0177 .6 0.3 0. 0. 7539.5 6098 .0 1990 65 .1 0.0 0.0 0 .0 40 .2 32.8 0. 60. 4851.9 3787 .9 1991 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4801.2 3618 .0 1992 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3600.8 2619 .1 1993 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3858.1 2708 .8 1994 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3700.4 2507 .8 1995 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3714 .1 2429 .6 1996 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3767.1 2378 .6 1997 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3820.1 2328 .3 1998 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3875.6 2280 .1 1999 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3941.8 2238 .4 2000 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 3974.8 2178 .7 2001 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4015.8 2124 .7 2002 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4041.5 2064 .0 2003 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4060.2 2001 .5 2004 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4068.4 1935 .8 2005 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4072.4 1870 .4 2006 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4067.8 1803 .4 2007 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4056.2 1735 .7 2008 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4043.4 1670 .1 2009 63 .0 0.0 0.0 0 .0 40 .1 32.7 0. 0. 4030.3 1606 .9 MET DISCOUNTED V A L U E OF PROGRAM: S IMULATED PER IOD THEREAFTER TOTAL 64188.4 446 3 5.8 108824 .2 370 Table A.3 E f f i c i e n t Harvesting Program: Vessel Catch S t a t i s t i c s . Note 1: Catch per unit e f f o r t s denotes the following: For trawlers: Catch in tons per hour trawled. For multi-purpose vessels: Catch in tons per operating day. Notation: M-Boats = multipurpose vessels. Had = haddock. Sai = saithe. CATCH CATCH PEP. UNIT, EFFORT TRAWLERS M-BOATS TRAWLERS M-BOATS YEAR COD HAD SAI COD HAD SAI COD HAD SAI COD HAD SAI 1980 243 .2 29 .2 36 .4 162 .1 19 .6 24 .2 1 .032 1 .062 1 .140 2 .693 2 .780 3 .001 1981 298 .9 41 .9 38 .8 198 .3 29 .3 28 .1 1 .210 1 .221 1 .255 3 .204 3 .237 3 .335 1982 266 .9 45 .0 44 .9 16 2 .1 29 .8 30 .8 0 .985 1 .093 1 .309 2 .558 2 .864 3 .490 1983 238 .3 41 .8 39 .7 141 .0 27 .0 25 .5 0 .868 0 .974 1 .101 2 .235 2 .530 2 .887 1984 60 .2 0 .1 10 .2 0 .4 0 .0 25 .6 1 .100 1 .080 1 .217 2 .895 2 .833 3 .226 1985 326 .2180 .8 0 .2 0 .0 1 .2 30 .7 1 .428 1 .610 1 .606 3 .832 4 .362 4 .349 1986 356 .4 0 .1 0 .2 7 .7 0 .0 36 .3 1 .456 0 .967 1 .745 3 .916 2 .497 4 .765 1987 338 .6 0 .4 0 .3 0 .0223 .4 3 .3 1 .417 2 .263 1 .981 3 .801 6 .340 5.443 1988 360 .7 0 .2 0 .3 0 .0 0 .0 50 .0 1 .535 1 .185 2 .066 4 .151 3 .096 5 .662 1989 183 .5227 .3 0 .2 0 .0257 .9 0 .6 1 .607 1 .977 2 .329 4 .353 5 .399 6 .339 1990 3 91 .0 0 .1 0 .0 0 .0 17 .0 60 .6 1 .750 0 .626 2 .533 4.779 1 .569 6 .867 1991 333 .7 0 .3 0 .0 0 .0 59 .7 60 .8 1 .544 1 .975 2 .568 4 .161 5 .541 6 .924 1992 295 .3 0 .2 0 .0 0 .0 43 .1 61 .1 1 .366 1 .493 2 .581 3 .648 3 .998 6 .954 1993 303 .7 0 .2 0 .0 0 .0 46 .5 61 .0 1 .406 1 .591 2 .577 3 .770 4 .314 6 .942 1994 297 .5 0 .2 0 .0 0 .0 46 .2 59 .9 1 .377 1 .584 2 .527 3 .684 4 .287 6 .820 1995 295 .5 0 . 2 0 .0 0 .0 50 .0 59 .1 1 .367 1 .702 2 .489 3 .657 4 .640 6 .730 1996 296 .5 0 '.2 0 .0 0 .0 52 .0 58 .2 1 .372 1 .767 2 .443 3 .671 4 .822 6 .619 1997 298 .3 0 .2 0 .0 0 .0 52 .8 57 .2 1 .380 1 .793 2 .398 3 .696 4 .895 6 .510 1998 301 .0 0 .2 0 .0 0 .0 52 .6 56 .5 1 .393 1 .789 2 .364 3 .732 4 .880 6 .426 1999 303 .6 0 .2 0 .0 0 .0 53 .0 56 .0 1 .405 1 .800 2 .341 3 .767 4 .913 6 .371 2000 305 .3 0 .2 0 .0 0 .0 52 .8 55 .5 1 .413 1 .794 2 .318 3 .789 4 .895 6 .315 2001 306 .6 0 .2 0 .0 0 .0 53 .4 55 .2 1 .419 1 .813 2 .306 3 .807 4 .951 6 .286 2002 307 .8 0 .2 0 .0 0 .0 53 .3 55 .2 1 .424 1 .810 2 .303 3 .823 4 .941 6 .277 2003 308 .5 0 .2 0 .0 0 .0 53 .3 55 .2 1 .428 1 .813 2 .306 3 .833 4 .948 6 .286 2004 308 .8 0 .2 0 .0 0 .0 53 .4 55 .4 1 .429 1 .814 2 .314 3 .837 4 .952 6 .306 2005 308 .8 0 .2 0 .0 0 .0 53 .5 55 .7 1 .429 1 .816 2 .326 3 .837 4.958 6 .337 2006 308 .6 0 .2 0 .0 0 .0 53 .5 56 .0 1 .428 1 .817 2 .338 3 .833 4 .961 6 .368 2007 308 .0 0 .2 0 .0 0 .0 53 .5 56 .2 1 .425 1 .818 2 .349 3 .826 4 .962 6 .396 2008 307 .4 0 . 2 0 .0 0 .0 53 .5 56 .4 1 .423 1 .818 2 .358 3 .818 4 .963 6 .419 2009 306 .8 0 '.2 0 .0 0 .0 53 .5 56 .5 1 .420 1 .818 2 .365 3 .810 4.964 6 .435 371 Table A.4 E f f i c i e n t Harvesting Program: B i o l o g i c a l S t a t i s t i c s . Note 1: A l l quantities in 1000 metric tons. Note 2: Stocks evaluated at the beginning of each year. Notation: Biom. = biomass. Spawn. = spawning stock. SPECIES=COD SPECIES=HADDOCK SPECIES=SAITHE YEAR CATCH BIOM. SPAWN. CATCH BIOM. SPAWN CATCH BIOM. SPAWN 1980 413 .6 1770 .4 611 .7 49 .8 324 .1 237 .1 61 .8 346 .0 147 .6 1981 507 .4 1687 .1 509 .2 72 .6 334 .3 290 .3 68 .2 349 .2 200 .7 1982 437 .7 1538 .1 370 .6 76 .3 334 .0 252 .8 77 .3 367 .7 229 .3 1983 387 .1 1510 .1 381 .6 70 .2 308 .8 187 .9 66 .5 338 .5 175 .8 1984 61 .8 1586 .1 345 .1 0 .2 316 .8 194 .1 36 .6 371 .0 146 .4 1985 332 .9 2078 .3 527 .3 185 .7 591 .8 418 .8 31 .6 465 .5 226 .6 1986 371 .6 2080 .9 615 .3 0 .2 257 .4 174 .4 37 .3 472 .3 258 .1 1987 3 45 .5 1930 .6 635 .4 228 .4 809 .7 595 .7 3 .6 483 .1 293 .6 1988 368 .1 2052 .2 701 .9 0 .2 293 .5 206 .9 51 .3 498 .6 335 .3 1989 187 .3 1966 .2 685 .0 495 .1 92 0 .4 703 .4 0 .8 507 .3 347 .3 1990 399 .0 2218 .0 850 .6 17 .4 218 .6 129 .8 61 .9 578 .2 40 5 .6 1991 340 .5 1914 .1 738 .5 61 .2 681 .3 460 .4 62 .1 578 .4 407 .8 1992 301 .3 1673 .7 648 .2 44 .2 40 9 .9 298 .0 62 .4 577 .5 409.0 1993 310 .0 1735 .5 665 .2 47 .7 405 .8 309 .6 62 .3 57 3 .4 40 9 .2 1994 303 .6 1704 .0 646 .0 47 .4 379 .6 289 .9 61 .2 561 .3 401 .1 1995 301 .5 1699 .4 638 .8 51 .3 396 .3 308 .3 60 .4 550 .6 394 .9 1996 302 .5 1710 .7 639.8 53 .3 404 .8 316 .9 59 .4 539 .8 384 .4 1997 304 .4 1724 .3 6 43 .5 54 .1 407 .9 320 .1 58 .4 530 .8 375 .2 1998 307 .1 1742 .1 649 .3 53 .9 408 .2 319 .9 57 .6 524 .8 368 .5 1999 309 .8 1759 .5 6 55 .3 54 .3 410 .0 321 .5 57 .1 521 .8 364 .0 2000 311 .5 1770 .2 659 .2 54 .1 409 .9 321 .4 56 .6 519 .8 360 .4 2001 312 .9 1778 .5 662 .5 54 .7 412 .3 323 .8 56 .4 519 .5 358 .6 2002 314 .1 17 85 .8 665 .5 54 .6 412 .1 323 .5 56 .3 520 .6 358 .4 2003 314 .8 1790 .0 667 .3 54 .7 412 .5 323 .9 56 .4 522 .6 359 .6 2004 315 .1 1791 .5 668 .1 54 .7 412 .8 324 .2 56 .6 525 .0 361 .4 2005 315 .2 1791 .7 668 .3 54 .8 413 .1 324 .5 56 .8 527 .8 363 .8 2006 314 .9 1789 .8 667 .6 54 .8 413 .3 324 .7 57 .1 530 .2 366 .2 2007 314 .3 1786 T_ 666 .3 54 .8 413 .4 324 .8 57 .4 532 .2 368 .3 2008 313 .7 1782 .3 664 .9 54 .9 413 .5 324 .9 57 .6 533 .6 369 m 0 2009 313 .1 1778 .5 663 .4 54 .9 413 .5 324 .9 57 .7 534 .4 371 !1 372 Appendix 8.2-B The Aggregative State-Space Diagram. The aggregative state-space diagram relates the aggregated le v e l of biomass to the aggregated l e v e l of fi s h i n g c a p i t a l according to any harvesting p o l i c y . Let X(t) denote the aggregated biomass at time t. This variable i s constructed from the individual species biomass levels with the help of the respective f i s h prices as follows: (B.1) X(t)=x(1,t)+ I x ( i , t ) p ( i , t ) / p ( 1 , t ) , I- z where x ( i , t ) and p ( i , t ) represent the biomass and landings price respectively of species i at time t. The aggregated biomass i s thus calculated in units of species 1, i . e . cod by the convention of this study. Aggregated fishing c a p i t a l is s i m i l a r i l y aggregated on the basis of investment prices as follows: (B.2) K(t)=k(1,t)+ I k ( i , t ) p ( i , t ) / p ( 1 , t ) , where K(t) represents the aggregated fishing c a p i t a l at time t and k ( i , t ) and p ( i , t ) are the f i s h i n g c a p i t a l and investment prices for f l e e t i at time t, respectively. In th i s study f l e e t 1 corresponds to trawlers. Hence, aggregative f i s h i n g c a p i t a l i s represented in trawler units. 373 8.2.3 Variants of the Basic Case. In t h i s section we w i l l b r i e f l y consider a few variants of the basic maximization case considered above. This exercise serves a twofold purpose. F i r s t l y , i t provides some indication as to the s e n s i t i v i t y of the above results to d i f f e r e n t model s p e c i f i c a t i o n s . Secondly, one of the variants to be defined provides a convenient s i m p l i f i e d version of the basic case. This s i m p l i f i e d version w i l l be employed in further analysis of e f f i c i e n t programs in later sections. Hence, i t i s useful, at this stage, to compare i t s properties with those of the basic control problem. 8.2.3.1 Shorter Control Periods. We w i l l f i r s t consider the effects of reduced control periods on the attained level of economic rents. Apart from reduced control periods, we w i l l solve the same maximization problem as in the previous section. Control periods ranging from 1 to 5 periods were considered. The solution strategy and convergence c r i t e r i a were the same as before. The search for maxima was easier, however, because of fewer controls. The attained maximum values for di f f e r e n t control periods are l i s t e d in table 8.6 and depicted in figure 8.8. 374 Table 8.6 E f f i c i e n t Harvesting Programs: Different Control Periods, Length of Attained Maximum Control period Value Function Percentage of (Years) (B.IKR) (7-years max) 1 101.1 92.9% 2 105.1 96.6% 3 106.2 97.6% 4 107.5 98.8% 5 108.1 99.4% • • • 7 108.8 100.0% The corresponding graph is drawn in figure 8.8: Figure 8.8 E f f i c i e n t Harvesting Paths: Different Control Periods. ECONOMIC RENTS 110 — 108 — 106 — 104 — 102 — 100 1 l l I | I 1 I I | I I I I | I I I I | I I I I 1 2 3 4 CONTROL PERIOD A key feature of these results i s the r e l a t i v e l y l i t t l e gain, in terms of the value function, of increasing the control period. Equivalently we may say that the shadow cost of r e s t r i c t i n g the control period to less than 7 years seems to be small. This holds in spite of the e f f i c i e n t 7-years harvesting program involving, as we saw in section 8.2.2, rather dramatic 375 s h i f t s in the control variables during the control period. The main reason for this p a r t i c u l a r i n s e n s i t i v i t y seems to be the r e l a t i v e l y low rate of discount which emphasizes the importance of the long run or equilibrium values of the controls rather than their adjustment period values. In addition to having an obvious implication for policy formulation (assuming that the low discount rate is actually appropriate) t h i s result provides a j u s t i f i c a t i o n for analysing e f f i c i e n t harvesing paths in terms of reduced control periods. 8.2.3.2 Nonselective F i s h e r i e s . One of the assumptions in section 8.2.2 above was that each fishery could be pursued separately. In section 5.3.1, th i s was, in fact, asserted to be in accordance with the available q u a l i t a t i v e evidence on the Icelandic demersal f i s h e r i e s . In t h i s section we w i l l consider the opposite extreme, i . e . the one where fishery s e l e c t i v i t y i s impossible. Within the framework of the model of t h i s study, we w i l l represent t h i s case by s t i p u l a t i n g that the same fraction of both f l e e t s must be applied to each of the three f i s h e r i e s as was the case in 1983. This variant can be interpreted in at least two ways. In the f i r s t place, i t may be regarded as an attempt to represent the actual empirical situation to the extent that fishery s e l e c t i v i t y is in fact not f e a s i b l e . The results to be presented w i l l accordingly provide a rough test of the s e n s i t i v i t y of e f f i c i e n t programs to a p a r t i a l or complete nonselectivity. 376 A l t e r n a t i v e l y , t h i s s p e c i f i c a t i o n can be regarded as the representation of a p a r t i c u l a r harvesting policy according to which only the o v e r a l l c a p i t a l l e v e l of the two f l e e t s are controlled but individual fishing firms allowed to choose the fishery they prefer. Of course, there i s no presumption that individual firms would in fact maintain unchanged proportionate vessel a l l o c a t i o n in the face of reduced overall c a p i t a l l e v e l s . On the other hand, there i s no reason to expect them to choose the e f f i c i e n t a l l o c a t i o n <32>. They might, in fact, choose an even more i n f e r i o r one. In any case, the corresponding results provide an example of the possible outcome of a harvesting program under which only the o v e r a l l l e v e l of active f i s h i n g c a p i t a l is controlled. A summary of the results of 5-year rent maximization paths under (i) selective f i s h e r i e s and ( i i ) nonselective f i s h e r i e s i s provided in table 8.7. Table 8.7 E f f i c i e n t Harvesting Programs: Selective and Nonselective Fi s h e r i e s . Non-Selective selective Fisheries Fisheries Present value of rents (B.IKR) : 108.078 56.507 Equilibrium Quantities: Trawler Fleet (vessel number): 61.9 59.4 Multi-purpose f l e e t (vessels): 69.8 0.0 Catch of cod (1000 MT) : 310.9 266.1 Catch of haddock (1000 MT) : 53.1 50.8 Catch of Saithe (1000 MT) : 57.6 52.3 The most s t r i k i n g item in table 8.7 is the very s i g n i f i c a n t reduction in the attainable l e v e l of economic rents when fishery 377 selection i s not allowed. This observation may be explained with reference to arguments used by Gordon in his seminal work on fishery economics in 1954 <33>. The three demersal f i s h e r i e s considered here are not equally productive <34>. In fact, the cod fishery is by far the most productive. The competitive exploitation pattern, however, has a tendency to equalize the average rents of the f i s h e r i e s . Hence, under competitive u t i l i z a t i o n , the degree of economic overexploitation is highest for the most productive species and vica versa <35>. It follows that f u l l harvesting e f f i c i e n c y requires not only a reduction in the over a l l c a p i t a l l e v e l but also a disproportionate reduction with respect to di f f e r e n t f i s h e r i e s <36>. This result should hold q u a l i t a t i v e l y for any multispecies f i s h e r i e s although the relevant magnitudes w i l l vary from one case to another. In the case of the Icelandic f i s h e r i e s the shadow cost of r e s t r i c t i n g a harvesting program to equi-proportionate changes in vessel a l l o c a t i o n to f i s h e r i e s , whether due to a suboptimal policy or biotechnical necessity, is evidently very high. Another interesting feature of the long run e f f i c i e n t harvesting policy under nonselective f i s h e r i e s i s the phasing-out of the multi-purpose vessels. This result can be explained e n t i r e l y within the framework of the model as follows: The trawlers are, as we have seen above, more e f f i c i e n t in harvesting the most productive species, i.e. cod. Due to the vessel application proportionality requirement, however, a suitable application of trawlers to the cod fishery implies that 378 the trawlers are at the same time f u l l y exploiting the haddock and saithe fishery. Consequently, there i s no room for the multi-purpose f l e e t although i t seems to be more e f f i c i e n t in the haddock and, especially, the saithe fishery. The properties of the e f f i c i e n t adjustment path in t h i s case can, to some extent, be inferred from the corresponding aggregative state-space diagram presented in figure 8.9 below. Also included in figure 8.9, for comparative purposes, i s the corresponding 5-year e f f i c i e n t harvesting program for the basic case. Figure 8.9 E f f i c i e n t Harvesting Programs: Selective and Nonselective Fi s h e r i e s . 2 0 0 — 1 SELECT NONSEL A G G R E G A T E D 150 — 100 — C A P I T A L 50 0 — | — i — i — i — i — | — i — i — i i | r 1000 2 0 0 0 3 0 0 0 0 AGGREGATED BIOMASS 379 8.2.3.3 A Single Fishing Fleet. Above we have investigated the effects of reduced control periods and impossibility of fishery s e l e c t i v i t y . In order to f a c i l i t a t e the investigation of other features of e f f i c i e n t harvesting paths, i t i s useful to simplify the model s t i l l further. In p a r t i c u l a r , we w i l l , in this section, consider the effe c t s of r e s t r i c t i n g the control set to one f l e e t only. Our approach i s to l i m i t our attention to trawlers only. For t h i s purpose we transform the t o t a l demersal f i s h i n g f l e e t in 1983 into a uniform trawler f l e e t . The method for doing this is to calculate how many additional trawlers with precisely the same c h a r a c t e r i s t i c s as the actual ones would have been needed in 1983 to generate the t o t a l actual cod, haddock and saithe catch in that year. This number, i t turned out, was 55 trawlers. Hence 158 trawlers would have been able to generate the t o t a l demersal catch in 1983. This then constitutes our i n i t i a l single f l e e t . We w i l l now b r i e f l y consider e f f i c i e n t harvesting paths for this single f l e e t s p e c i f i c a t i o n and compare them with the correponding ones for the two fl e e t s p e c i f i c a t i o n . Two sets of e f f i c i e n t harvesting paths are considered: In the f i r s t set we compare the results of f u l l e f f i c i e n t 3-year control paths <37> and the single f l e e t s p e c i f i c a t i o n defined above. In the second set, we impose the nonselective f i s h e r i e s r e s t r i c t i o n of the previous section and consider the effects of the single f l e e t r e s t r i c t i o n under these circumstances. The main results are l i s t e d in table 8.8. 380 Table 8.8 E f f i c i e n t Harvesting Programs: Single Fleet Investigation. Selective Non-selective f i s h e r i e s f i s h e r i e s (1) (2) (3) (4) Present Value (B.IKR): 1 06. 2 88.7 56. 5 54.9 Equilibrium levels Aggregate f l e e t * : 83. 1 66.6 59. 4 60.3 Catch of cod : 311. 9 305.8 266. 1 265. 1 Catch of haddock : 49. 7 52.0 50. 8 51 .5 Catch of saithe : 56. 6 0.0 52. 3 53. 1 (1) 3 period f u l l e f f i c i e n t program. (See section 8.2.3.1). (2) 3 period single f l e e t program. (3) 5 period f i s h e r i e s program. (See section 8.2.3.2). (4) 5 period single f l e e t f i s h e r i e s program. * Trawlers in cases (2) and (4). Aggregated trawler units acording to the method in appendix 8.2-B in cases (1) and (3). Consider f i r s t the selective f i s h e r i e s case in table 8.8. There, the r e s t r i c t i o n to a single f l e e t implies a reduction in attainable economic rents of about 16.5%. Hence, t h i s i s the cost of r e s t r i c t i n g the control set to the trawler f l e e t . Notice that t h i s i s a much lesser cost than when fishery non-s e l e c t i v i t y was imposed in the previous section. Secondly, i t i s interesting to notice that, in th i s case, the saithe fishery is not pursued under the single f l e e t r e s t r i c t i o n . The implication, of course, i s that the saithe fishery is simply not productive enough for the trawler f l e e t <38>. Thirdly, when the control set is r e s t r i c t e d to a single f l e e t , the t o t a l f i s h i n g c a p i t a l , as measured, i s considerably smaller than in the unrestricted basic case. Since the economic f l e x i b i l i t y inherent in two complementary f l e e t s i s now l o s t , t h i s result i s i n t u i t i v e . 381 Turning to the nonselective f i s h e r i e s case, the central result i s that i t i s now f a i r l y immaterial whether the multipurpose f l e e t i s available or not. This i s , of course, no surprise. After a l l i t was one of the results of the previous section that under f i s h e r i e s nonselectivity the multi-purpose f l e e t was not employed in the long run at a l l . Hence imposing a r e s t r i c t i o n to the same effect w i l l not be co s t l y . What difference there exists between columns (3) and (4) derives e n t i r e l y from the e f f e c t s of the adjustment period, during which multi-purpose vessels are employed in case (3) <39>. 8.2.4 The Simplified Case. In the interest of computational and expositional s i m p l i c i t y we w i l l continue the analysis of e f f i c i e n t harvesting paths with the help of the single f l e e t nonselective f i s h e r i e s model discussed in the previous section. This we w i l l refer to as the si m p l i f i e d case. In the s i m p l i f i e d case there are only 3 control variables in each period, namely investment, number of active vessels and number of i d l e vessels. Moreover, since a l l available vessels must be either active or i d l e , there are only two independent control variables each period. Hence, the search for a global maximum i s much more manageable in the simple case than the basic case. This, of course, i s i t s main advantage. The disadvantage, on the other hand, i s that the si m p l i f i e d case ignores many of the d e t a i l s of the basic case. This by i t s e l f i s unimportant. The c r i t i c a l question i s whether th i s 382 s i m p l i f i c a t i o n w i l l result in q u a l i t a t i v e l y d i f f e r e n t e f f i c i e n t aggregative paths. Although a d e f i n i t e answer to thi s question cannot be supplied, the results in the previous section do not suggest so. 8.2.4.1 Different Degrees of M a l l e a b i l i t y . In our discussion of fi s h i n g c a p i t a l in section 5.4, we spec i f i e d a certain kind of c a p i t a l m a l l e a b i l i t y in terms of disinvestment costs. In section 8.2.2, on e f f i c i e n t harvesting path for the Icelandic demersal f i s h e r i e s , the corresponding m a l l e a b i l i t y parameter, m, say, was set to the "seemingly reasonable value" of m=0.75 <40>. In t h i s subsection we w i l l , in order to investigate the effe c t s of di f f e r e n t degrees of ma l l e a b i l i t y on the properties of e f f i c i e n t harvesting programs, consider a few other m a l l e a b i l i t y parameters. This investigation w i l l be performed on the basis of the si m p l i f i e d case specified above. The following m a l l e a b i l i t y s p e c i f i c a t i o n s are considered: I. m=0.0. In t h i s case the disinvestment costs equal the depreciated value of the vessel. Hence to get r i d of unwanted vessels, and the accompanying fixed costs, their undepreciated value must be written off e n t i r e l y . In technical terms, this can be referred to as the case of no resale markets but free disposal. 383 II . m=0.75. This i s the basic case considered in section 8.2.2 included here for the sake of completeness. This case corresponds to disinvestment costs of 25% of the depreciated value of the vessels. II I . m=1.00. For th i s m a l l e a b i l i t y parameter there are no disinvestment costs. Depreciated vessel values can be realized at any time <41>. Moreover, since investment i s also costless, t h i s case might be regarded as that of perfect m a l l e a b i l i t y <42>. Within the framework of the si m p l i f i e d case, 5-year e f f i c i e n t harvesting paths for each of these m a l l e a b i l i t y assumptions were discovered. A summary of the results in the form of the aggregated state-space diagrams i s presented in figures 8.10 and 8.11. The diagram in figure 8.10 relates aggregate biomass and t o t a l f i s h i n g c a p i t a l . The diagram in figure 8.11 relates aggregate biomass to active f i s h i n g c a p i t a l . 384 Figure 8.10 Different Degrees of M a l l e a b i l i t y : Total C a p i t a l . A G G R E G A T E 0 C A P I T A L I. m=0. II. m=0.75. I l l . m=1.0. 200—, 150-100-50 — I I I I I I i I I I | I I I I | I I I i | I I I I | I I I I | 1000 1500 2000 2500 3000 3500 AGGREGATED BIOMASS Figure 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 . A G G R E G A T E D C A P I T A L 200-150 — 100 — 50-I I I I I I I l l l | I l l l | l l I I | I i i i | i i i i | , 1000 1500 2000 2500 3000 3500 AGGREGATED BIOMASS I. m=0. II. m=0.75 III . m=1.0 385 Figures 8.10 and 8.11 demonstrate that the state of the fis h i n g vessel resale market can have a s i g n i f i c a n t effect on the properties of e f f i c i e n t harvesting programs. In th i s p a r t i c u l a r case the difference is greatest during the adjustment period. There, as a general rule, the higher the degree of m a l l e a b i l i t y the lower the c a p i t a l l e v e l . The difference in equilibrium magnitudes is considerably smaller. As mentioned previously, however, th i s long run result i s probably highly dependent on the rate of discount. 8.2.4.2 Ecological Weight Relationships. We w i l l now investigate the effects of the estimated ecological weight relationships on the properies of e f f i c i e n t harvesting programs. For th i s purpose, consider f i r s t e f f i c i e n t harvesting programs for (I) the si m p l i f i e d case including ecological weight relationships and (II) the same case without ecological weight relationships. As before, the control period w i l l be r e s t r i c t e d to 5 years, i . e . from 1984-88. The corresponding aggregate state space diagrams are presented in figures 8.12 and 8.13. The f i r s t of these diagrams relates the e f f i c i e n t path of biomass and t o t a l c a p i t a l , the second biomass and active c a p i t a l . 386 Figure 8.12 E f f i c i e n t Harvesting Programs: Total C a p i t a l . 2 0 0 - 1 A G 6 R E G A T E D C A P I T A L 150 — 100 -5 0 -: V -• i -- I I i i i i i i i 2000 2500 l | l l l l | I 3000 3500 l l | I l l l | 4000 4500 AGGREGATED BIOMASS I. Ecological weights included. II . Ecological weights excluded. Figure 8.13 E f f i c i e n t Harvesting Programs: Active Capital A G G R E G A T E 0 C A P I T A L 200—1 1 5 0 -1 0 0 -50-7 I I I i i i i [ i r P T I i i i r | i i i i | i i i i | 2000 2500 3000 3500 4000 4500 AGGREGATED BIOMASS I . II . Ecological weights included. Ecological weights excluded. 387 As figures 8.12 and 8.13 show, there i s a considerable difference between the two paths during the control period. In the long run, however, the difference in fishing c a p i t a l levels becomes r e l a t i v e l y small. Biomass differences, on the other hand, grow even bigger <43>. From the point of view of policy formulation, the c r u c i a l issue i s the costs, in terms of the value function, of erroneously ignoring ecological weight relationships in specifying a supposedly e f f i c i e n t harvesting program. We w i l l now b r i e f l y explore t h i s . Let the label "program (1)" refer to the ex ante e f f i c i e n t harvesting program on the assumption that there are no ecological weight relationships. S i m i l a r i l y l e t "program (2)" denote the ex ante e f f i c i e n t harvesting program assuming ecological weight relationships indeed e x i s t . The le v e l of attainable economic rents under programs (1) and (2) as well as some pertinent equilibrium magnitudes are l i s t e d in table 8.9. Also contained in table 8.9 are the results of applying program (1) erroneously to the case where ecological weight relationships actually e x i s t . The difference between the two programs in that case provide an indication as to the s e n s i t i v i t y of harvesting results to the ecological weight s p e c i f i c a t i o n . 388 Table 8.9 Ecological Weight Relationships and E f f i c i e n t Harvesting Programs. Ecological weight relationships nonexistent Program (1 ) Ecological weight r e l a t ionships existent Programs (1) (2) D i f f e r -ence (1)-(2) Present value (B.IKR) 1 29.2 50.4 55. 1 -4.7 Equilibrium: Fishing c a p i t a l Catch (1000 MT): Cod Haddock Saithe 55.3 353.6 81.1 72.7 55.3 249.4 49.3 50.4 59, 262 51 51 -4, -13, -1 -1 Program (1): E f f i c i e n t harvesting path assuming no ecological 'weight relationships. Program (2): E f f i c i e n t harvesting path assuming ecological weight relationships. According to the results in table 8.9 and, in fact, figures 8.12 and 8.13 too, there i s not a great difference between the conrol paths in the long run. Policy makers that erroneously apply program (1) to a situation of ecological weight relationships are nevertheless in for a nasty surprise. The attainable economic rents in t h i s case are only some 42% of those attainable i f the reverse case were true. The wrong policy, moreover, reduces the attained economic rents by an additional 4.7 B.IKR or 40 -M.USD. 389 8.2.4.3 The Stock-effort E f f e c t . In t h i s section we w i l l investigate the influence our estimated stock-effort effect <44> has on the nature of e f f i c i e n t harvesting paths. For th i s purpose we w i l l compare such paths with and without the stock-effort e f f e c t . The stock-e f f o r t e f f e c t w i l l be excluded by r e s t r i c t i n g the vessel s p e c i f i c stock c o e f f i c i e n t s in the estimated f i s h i n g mortality production function to be zero. Consequently, there w i l l be a linear relationship between the number of active vessels and the fishing m o r t a l i t i e s produced, irrespective of the volume of the f i s h stocks. Our approach is similar to that in previous subsections. We w i l l consider 5 period e f f i c i e n t harvesting programs for the simp l i f i e d case, on the one hand excluding the estimated stock-e f f o r t e f f e c t , on the other hand including i t . We refer to control programs based on the f i r s t assumption as program (1) and the second as program (2). Aggregated state-space diagrams for these two situations are presented in figures 8.14 and 8.15. 390 Figure 8.14 Different Stock-effort Assumptions: Total C a p i t a l . A 6 6 R E G A T E D C A P I T A L 200—1 1 5 0 -100-5 0 --• I •- I I T 2000 2200 I ' I 2400 2600 2800 3000 3200/ AGGREGATED BIOMASS I. Stock-effort effect included. I I . Stock-effort effect excluded. Figure 8.15 Different Stock-effort Assumptions: Active C a p i t a l . A G G R E G A T E D C A P I T A L 200—1 150 — 100 — 50 — -- I I I 2000 i — 1 — i — 1 — r 2200 2400 2600 T 2B00 AGGREGATED BIOMASS 3000 3200 I. Stock-effort effect included. I I . Stock e f f o r t effect excluded. 391 According to figures 8.14 and 8.15, e f f i c i e n t harvesting programs are not much dependent on which stock-effort s p e c i f i c a t i o n i s adopted. This vis u a l impression i s v e r i f i e d in table 8.10 below. Table 8.10 Stock-effort Effect and E f f i c i e n t Harvesting Programs. Stock-effort re l a t ionship nonexistent Program (1 ) Stock-effort re l a t ionship existent Programs ^ (1)1 ,' (2) , D i f f e r -ence (1)-(2) Present value (B.IKR): 64.2 Equilibrium: Fishing c a p i t a l : 56.3 Catch (1000 MT): Cod: 265.4 Haddock: 51.9 Saithe: 55.0 54.7 56.3 253.0 49.7 50.6 55. 1 59.5 262.9 51.1 52.7 -0.4 -3, -9, -1 . -2, Program (1): E f f i c i e n t harvesting program assuming no stock-effort e f f e c t . Program (2): E f f i c i e n t harvesting program assuming stock e f f o r t e f f e c t . According to table 8.10 there i s l i t t l e difference between the long run e f f i c i e n t l e v e l s of fishing c a p i t a l under the two stock-effort s p e c i f i c a t i o n s . Consequently, i t i s of no great significance, in this case <45>, which of these two spe c i f i c a t i o n s are adopted. The loss in attainable economic rent when program (1) is erroneously applied to the situation of stock-effort relationship i s correspondingly small or 3-4 M.USD. 392 8,2.5 Stochastic Recruitment, In discussing environmental influences on the b i o l o g i c a l subsystem in section 4.2.5.2. in part II, we found that observed deviations of actual recruitment from that predicted by the estimated recruitment functions could, to some extent, be represented by a mixed autoregressive moving average process <46>. U n t i l now, however, we have only worked with the deterministic case, i . e . the one where the stochastic recruitment process has be r e s t r i c t e d to i t s expected value. In this section we w i l l relax t h i s r e s t r i c t i o n by considering some examples of the effects stochastic recruitment may have upon the properties of e f f i c i e n t harvesting programs <47>. Our plan, in this section, i s to investigate the answers to two related questions. The f i r s t question concerns the costs, in terms of loss of value, of ignoring the stochastic nature of recruitment in formulating supposedly e f f i c i e n t harvesting programs. To answer th i s question we w i l l f i r s t specify an e f f i c i e n t harvesting program for the deterministic case. This program we w i l l refer to as program I. With program I imposed, we w i l l run the model several times including the estimated stochastic recruitment process. The arithmetic mean and variance of the outcomes from these runs provide estimates of the corresponding moments of the stochastic d i s t r i b u t i o n . The second question concerns the extent to which e f f i c i e n t harvesting programs can take advantage of stochastic variations in recruitment. To investigate t h i s we w i l l include the stochastic recruitment process in the model and, for given values of the relevant random variates, find the corresponding 393 e f f i c i e n t harvesting program. We w i l l refer to the program defined in t h i s manner as program I I . Since the random shocks are known and invariant during the search process, program II i s r e a l l y equivalent to a f u l l information or perfect foresight program. As such i t defines an upper bound on the attainable l e v e l of economic rents under stochastic recruitment. The detailed procedure w i l l be as follows: We w i l l work within the framework of the s i m p l i f i e d model. The control period w i l l be r e s t r i c t e d to 10 periods. Hence the number of control variables i s 20; 10 investment controls and 10 vessel application controls. The stochastic recruitment process w i l l be operative during the same 10 periods. After that, in order to reach equilibrium within a reasonable time, the stochastic process is r e s t r i c t e d to i t s expected l e v e l . To test the result of the deterministic e f f i c i e n t control, i . e . program I, under conditions of stochastic recruitment, the model w i l l be run 100 times. For the perfect foresight program, i . e . program I I , under which a new maximal path must be found for each set of stochastic recruitment, the number of r e p l i c a t i o n s i s 10. A summary of the most pertinent results of t h i s exercise i s provided in table 8.11: 394 Table 8.11 Effects of Stochastic Recruitment: A Summary. Deterministic Rec rui tment Stochastic Recruitment Program I Program I Program II (100 repl.) (10 repl.) Present Value (B.IKR): Standard Deviation : 54.846 0. 54.793 0.811 55.622 0.298 Control period Investment (RMSD*) : Idle Vessels (RMSD*): 0. 0. 0. 0. 5.1 0.3 Program I: Efficient, deterministic harvesting program. Program I I : E f f i c i e n t perfect foresight stochastic program. * RMSD stands for root-mean-square-deviation defined as follows: where y(t) i s the lev e l of the respective variable according to program I in year t and x(t) the lev e l of the same variable according to program I I . T, of course, equals 10. RMSD, in other words, i s the average deviation, over the 10 re p l i c a t i o n s , of the respective variables according to program II from those of program I. For further i l l u s t r a t i o n we present, in figure 8.16, graphs of the annual investment during the control period according to program I and a ty p i c a l path of investment according to program II. In figures 8.17 and 8.18 we, moreover, i l l u s t r a t e the corresponding time paths of the t o t a l f l e e t and the number of idle vessels for both programs. RMSD=( Z (x ( t ) - y ( t ) ) 2 ) / T ) * * 0 . 5 , t-i 395 Figure 8.16 E f f i c i e n t Investment. Deterministic and Stochastic Recruitment NUMBER OF VESSELS 5 0 — 1 -50-•100-82 I / 84 86 88 YEARS 90 -• DETERM -- STOCH ~r~~'—I 92 94 Figure 8.17 E f f i c i e n t Total Fleet: Deterministic and Stochastic Recruitment NUMBER OF VESSELS 200-150 H i o o H 50 * \ DETERM STOCH ~ r — T — — 1 — T ™ " 1 T ' 1 ' i 1 I 82 84 86 88 90 92 94 96 YEARS 396 Figure 8.18 E f f i c i e n t Idle Vessels: Deterministic and Stochastic Recruitment. NUMBER OF VESSELS 82 84 86 88 90 92 94 96 ^ YEARS The main conclusion to be drawn from the above results seems to be the following: There i s not much difference between the e f f i c i e n t deterministic and e f f i c i e n t stochastic harvesting paths, neither in terms of the attainable l e v e l of economic rents nor in terms of the application of the control variables. Most s t r i k i n g l y , perhaps, with perfect foresight (program II) the expected present value of the f i s h e r i e s i s increased by less than 2% compared to the deterministic harvesting program. These small magnitudes, however, obscure an important q u a l i t a t i v e difference between the two programs. In the expectation of occasionally favourable recruitment, the stochastic program maintains considerably more vessels during the adjustment period, even at the cost of some of them being kept i d l e , than does the deterministic program. The degree to 397 which th i s i s e f f i c i e n t obviously depends on the empirical s i t u a t i o n , i . a . the size of the random shocks. Hence, under other empirical circumstances, the quantitative difference between the two situations might be much greater. Perfect foresight does not, of course, exist in the real world <48>. A r e a l i s t i c e f f i c i e n t harvesting program would therefore consist of a closed loop policy according to which the control variables for the remainder of the control period would be annually revised according to the most recent recruitment information. Thus, at the beginning of the f i r s t period, a deterministic maximization problem with a l l future random terms at their expected values would be solved. At the beginning of the next period, i t would turn out that, due to the stochastic nature of the situ a t i o n , the state of the system would be somewhat d i f f e r e n t from what was expected. Hence a new deterministic e f f i c i e n t program for the remainder of the control period would be worked out etc. Although this closed-loop procedure i s conceptually simple, i t i s obviously very burdensome computationally. Hence, i t is omitted here. It may, however, be pointed out that the economic results of t h i s procedure w i l l be bounded from above by those of the perfect foresight program and the deterministic program from below. Since, as we have seen, there is l i t t l e difference between the value of these two programs, the value of the closed-loop policy is already well determined. 398 8.2.6 Icelandic Demersal Fi s h e r i e s : A Summary. In t h i s section we f i n a l l y attempt to summarize the central results concerning e f f i c i e n t harvesting of the Icelandic demersal f i s h e r i e s . (1) The current demersal harvesting policy i s very i n e f f i c i e n t involving excessive use of fi s h i n g c a p i t a l . (2) In terms of the Icelandic economy there are very s i g n i f i c a n t economic gains to be had from adopting the e f f i c i e n t harvesting program specified in section 8.2.2. Compared with the current harvesting policy the long run annual gains constitute a s i g n i f i c a n t f r a c t i o n of the current Icelandic GNP. (3) The optimal adjustment of fis h i n g c a p i t a l to the long run e f f i c i e n t l e v e l i s rather irregular. In order to take advantage of good fis h i n g opportunities during the control period, disinvestment is slow and, consequently, a great deal of i d l e vessels are kept for a number of years. (4) In equilibrium both f l e e t s , i . e . the trawler f l e e t and the multipurpose f l e e t , are operative. The multipurpose f l e e t , however, i s reduced proportionately more than the trawler f l e e t implying that the l a t t e r i s more e f f i c i e n t . (5) In equilibrium a l l three species, cod, haddock and saithe, are exploited. There i s , however, a very clear s p e c i a l i z a t i o n of the fi s h i n g f l e e t s with respect to the fi s h e r i e s with the trawler f l e e t s p e c i a l i z i n g in cod and the multipurpose f l e e t s p e c i a l i z i n g in haddock and saithe. (6) The e f f i c i e n t harvesting program implies considerably 399 lower catch levels and somewhat higher biomass levels in the long run than would be the case under the current harvesting p o l i c y . (7) Tests of the s e n s i t i v i t y of the above results to various model spe c i f i c a t i o n s revealed the following: (i) Neither the attainable l e v e l of economic rents nor the long run l e v e l of f i s h i n g c a p i t a l i s very sensitive to the length of the control period. ( i i ) The attainable l e v e l of economic rents i s highly dependent upon fishery s e l e c t i v i t y , i . e . the a b i l i t y to allocate vessels to p a r t i c u l a r f i s h e r i e s . ( i i i ) The attainable l e v e l of economic rents i s also very sensitive to the a b i l i t y to manipulate the two f l e e t s separately. This, however, is c l o s e l y related to fishery s e l e c t i v i t y , since i f the l a t t e r i s impossible i t is optimal to phase the multipurpose f l e e t out. (iv) The properties of e f f i c i e n t harvesting paths are quite sensitive to the l e v e l of c a p i t a l m a l l e a b i l i t y , as measured by the state of the f i s h i n g vessel resale market. (vi) The existence of ecological weight relationships has a dramatic effect on the l e v e l of attainable economic rents as well as the catch and biomass l e v e l s . Moreover, a harvesting program formulated on the erroneous assumption of no ecological weight relationships generates s i g n i f i c a n t l y lower economic rents than the appropriate e f f i c i e n t program. 400 ( v i i ) The stock-effort e f f e c t , as a part of the estimated fishing mortality production function, does not have a very s i g n i f i c a n t effect on the attainable l e v e l of economic rents or the path of optimal controls. The stock-effort effect can consequently be ignored in policy formulation without s i g n i f i c a n t costs. ( v i i i ) The estimated stochastic recruitment effect also has l i t t l e effect on the optimal control path as well as the l e v e l of attainable economic rents. Generally speaking, the above results do not hold many surprises. As predicted by theory, the current competitive u t i l i z a t i o n has been found to be extremely i n e f f i c i e n t , employing- excessive c a p i t a l and maintaining too low l e v e l s of the f i s h stocks. In s t r i c t l y quantitative terms, the loss in economic rents stems primarily from excessive c a p i t a l since i t turns out that, due to the highly compensatory nature of the ecological weight relationships, biomass levels are r e l a t i v e l y l i t t l e increased in e f f i c i e n t equilibrium. This r e s u l t , of course, r e f l e c t s the r e s i l i e n c e of the Icelandic demersal species to predation and may not generalize to other f i s h e r i e s . 401 8.3 More General Cases. We w i l l now depart from the s p e c i f i c empirical context of the Icelandic demersal f i s h e r i e s to examine more general harvesting cases. Two topics w i l l be considered. In section 8.3.1 we w i l l perform a somewhat more detailed analysis of e f f i c i e n t adjustment paths than hitherto. We w i l l be p a r t i c u l a r l y interested in the combinations of controls employed along e f f i c i e n t paths. In section 8.3.2 we w i l l , on the other hand, specify simple predator-prey relationships and consider some e f f i c i e n t harvesting paths under those circumstances. The analysis in t h i s section w i l l be mostly carried out within the framework of the s i m p l i f i e d case sp e c i f i e d in section 8.2.4 above. However, in order to concentrate on the special c h a r a c t e r i s t i c s of the predator-prey case we w i l l allow fishery s e l e c t i v i t y in section 8.3.2. On the other hand, the analysis in that section w i l l be r e s t r i c t e d to two species only. 8.3.1 E f f i c i e n t Adjustment Paths. In this section we propose to examine in some d e t a i l s the application of controls along adjustment paths towards equilibrium. For t h i s purpose, we w i l l consider 4 e f f i c i e n t paths which d i f f e r only with respect to the i n i t i a l values of the state variables. More precisely, the paths w i l l be characterized in terms of low and high levels of aggregate biomass and f i s h i n g c a p i t a l , where the adjectives low and high are r e l a t i v e to the respective e f f i c i e n t equilibrium l e v e l s . 402 These paths are further sp e c i f i e d in table 8.12 below: Table 8.12 E f f i c i e n t Harvesting Paths: I n i t i a l Conditions. Paths I n i t i a l Conditions Fishing Capital Aggregate Biomass Path 1: Path 2: Path 3: Path 4: No c a p i t a l High c a p i t a l High c a p i t a l No c a p i t a l High biomass* High biomass* Low biomass Low biomass * In fact, v i r g i n stock equilibrium. As already mentioned, the harvesting paths considered in th i s section are e f f i c i e n t r e l a t i v e to the s i m p l i f i e d model. Hence, there are two independent control variables each period; investment in the fis h i n g f l e e t and vessel application to fi s h e r i e s ( f l e e t u t i l i z a t i o n rate). Therefore we may, rather a r b i t r a r i l y , p a r t i t i o n the control space into the following nine regions <49>: Table 8.13 Control Regions, Investment Negative (-): None (0): Positive ( + ) : Fleet U t i l i z a t i o n Rate' None (0) P a r t i a l (P) F u l l (F) I IV VII II V VIII III VI IX According to table 8.13, there are nine conceivable control regions, l a b e l l e d I to IX. Each control path must pass through one or more of these regions. It is of some interest to find out whether any of these regions do not contain e f f i c i e n t control 403 paths. Some of them, for instance VII and VIII according to which investment should take place under limited u t i l i z a t i o n of current capacity, seem rather unlikely candidates for e f f i c i e n c y <50>. However, since the analysis i s r e s t r i c t e d to examples of e f f i c i e n t control paths, we w i l l not be in a position to derive propositions to thi s e f f e c t . We w i l l , however, with the help of these examples, be able to assert the reverse i . e . that, under the specified conditions at least, there are e f f i c i e n t control paths that pass though the respective regions. The relevant results of 5-period e f f i c i e n t harvesting paths for the 4 d i f f e r e n t i n i t i a l conditions are l i s t e d in table 8.14 below: Table 8.14 E f f i c i e n t Paths: Different I n i t i a l Conditions. Path •j ** Path 2 * * Path 3 * * Path 4 * * Period I nv App Reg Inv App Reg I nv App Reg I nv App Reg 1 + F* IX P II 0 I + F* IX 2 - F III P II - 0 I + F IX 3 - P II 0 I - P II + F IX 4 - F III F III 0 F VI + F IX 5 — F III F III + F IX + F IX 6 (equil. ) 0 F VI 0 F VI 0 F VI 0 F VI * Arbitrary, since no f l e e t a v a i l a b l e . ** See table 8.13. Code: Inv = Investment; + = po s i t i v e investment, - = negative investment, 0 = no investment. App = U t i l i z a t i o n of f l e e t ; F = f u l l u t i l i z a t i o n , P = p a r t i a l u t i l i z a t i o n , 0 = no u t i l i z a t i o n . Reg = Control region; see table 8.13. Path 1 corresponds to the i n i t i a l conditions of no c a p i t a l 404 and v i r g i n state f i s h stocks. This case then represents e f f i c i e n t harvesting of previously unexploited species. The e f f i c i e n t path involves very high i n i t i a l investment and thereafter annual disinvestment u n t i l c a p i t a l equilibrium i s reached in period 6. There i s f u l l u t i l i z a t i o n of the available f l e e t in every period except period 3. There, in the f i r s t period after the i n i t i a l investment becomes operative, only a fraction of the available f l e e t , in fact some 20%, are employed. The reason may be the following: Foreseeing a blocked in t e r v a l in the form of a s i g n i f i c a n t loss in the biomass of the oldest cohorts, i t is e f f i c i e n t to overinvest in the f i r s t period and overharvest in the second (both r e l a t i v e to the e f f i c i e n t long run l e v e l s ) . Consequently, in period 3, f u l l u t i l i z a t i o n of the available f l e e t would excessively exploit the biomass, now very s i g n i f i c a n t l y reduced. A higher rate of disinvestment in period 2 i s , on the other hand, i n e f f i c i e n t because a larger f l e e t w i l l be needed in period 4, when the f i s h stocks have recovered somewhat. Path 2 corresponds to a high i n i t i a l l e v e l of both f i s h i n g c a p i t a l and f i s h stocks. This case thus may be taken to represent a fishery in which the biomass has been protected by controls but the acccumulation of fis h i n g c a p i t a l has gone r e l a t i v e l y unchecked <51>. In this case, e f f i c i e n t harvesting implies c a p i t a l disinvestment in every period u n t i l an equilibrium i s reached. The u t i l i z a t i o n of the f l e e t i s p a r t i a l in the f i r s t two periods and drops to zero in the t h i r d . Thereafter, u t i l i z a t i o n remains f u l l . The explanation for the sin g u l a r i t y in period 3 i s related to that for path 1. To be 405 able to exploit the large cohorts passing through the fishery in the f i r s t two periods, a high rate of exploitation is optimal. In fact, much higher than in the f i r s t case since a very large fl e e t i s already available. Hence the f i s h stocks, in period 3, have been driven to such a low l e v e l that i t is now optimal to delay further harvesting. Path 3 corresponds to a high i n i t i a l l e v e l of fishing c a p i t a l and low f i s h stocks. It may therefore be taken to represent the t y p i c a l state of competitive f i s h e r i e s . The e f f i c i e n t harvesting path from t h i s i n i t i a l state consists of disinvestment for the f i r s t 3 periods, no investment in the 4th period and some reinvestment in the f i f t h to bring c a p i t a l to the long run e f f i c i e n t l e v e l . The f l e e t u t i l i z a t i o n rates are monotonically increasing during the adjustment period. In the f i r s t two periods the u t i l i z a t i o n rate i s zero, becoming positive in the t h i r d and f u l l thereafter. The reason why i t i s optimal to disinvest excessively ( r e l a t i v e to equilibrium) f i r s t and then reinvest at the end is seeminly straight forward. The cost of the "excessive" disinvestment i s simply exceeded by the cost of maintaining the corresponding i d l e vessels for two more periods. Path 4 corresponds to i n i t i a l conditions of low biomass and no fi s h i n g f l e e t . The empirical si t u a t i o n to which th i s i n i t i a l state might correspond i s a previously unexploited species but in a poor b i o l o g i c a l shape due e.g. to ecological reasons. The e f f i c i e n t harvesting paths in t h i s case c a l l s for continual investment, hand in hand with the growth of the biomass u n t i l equilibrium i s reached. Capital u t i l i z a t i o n , correspondingly, i s 406 f u l l throughout the control period. A state space diagram further i l l u s t r a t i n g these four paths is provided in figure 8.19. Figure 8.19 E f f i c i e n t harvesting Paths: Aggregate State-Space Diagram, A 6 G R E G A T E D C A P I T A L 300 2 0 0 -100 — I I I I | I 1*1 T j I 0 1 0 0 0 2 0 0 0 I I | I I I I' | I I I 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 AGGREGATED B I O M A S S According to the results l i s t e d in table 8.14, seven of the nine control regions defined actually contain e f f i c i e n t paths. Only the regions that combine underutilization of f i s h i n g c a p i t a l with investment are not featured in the results. As mentioned at the outset, however, a categorical exclusion of these regions as e f f i c i e n t would be premature as they might be e f f i c i e n t under di f f e r e n t circumstances <52>. 407 8.3.2 Predator-prey Relationships. In t h i s section we w i l l b r i e f l y explore a simple predator-prey relationship. For thi s purpose we w i l l impose upon our basic model the following predation s i t u a t i o n : Consider two species. Refer to them as species 1 and species 2 respectively <53>. Let the species with lower equilibrium biomass, i.e . haddock, prey on the one with higher equilibrium biomass, i . e . cod <54>. Let the predation r e l a t i o n be defined by the following l o g i s t i c function <55>: (7) m(1)=an(2)/(n(2)+bexp(-n(2))), n ( l ) e R i , where m(l) is the natural mortality of species 1, n(2) i s the to t a l number of mature individuals of species 2 and a and b are nonegative parameters. Clearly, the s p e c i f i c a t i o n of a and b determine the quantitative nature of the predation s i t u a t i o n . This equation describes how the natural mortality of the prey, species 1, depends on the stock size of the predator, species 2. Under the assumptions made, m(1) is monotonically increasing in n(l) with m(l) r e s t r i c t e d to the int e r v a l [0,a) <56>. Equation (7) i s b a s i c a l l y an extension or generalization of the basic empirical model employed in the previous sections <57>. However, in the interest of si m p l i c i t y we w i l l , in what follows, r e s t r i c t the analysis to the above-mentioned two species and a single f l e e t . This s p e c i f i c a t i o n we w i l l refer to as the predation case. The predation case w i l l be used to examine two related issues. F i r s t , we w i l l demonstrate that under predation 408 e f f i c i e n t harvesting programs may be r a d i c a l l y d i f f e r e n t from those appropriate to the case of no predation. For this purpose we w i l l specify 3 di f f e r e n t sets of predation parameters and compare the resulting e f f i c i e n t paths. Secondly, we w i l l use the predation case to i l l u s t r a t e the potential peculiar relationship between the output price ratios of the two species and the corresponding e f f i c i e n t harvesting pattern <58>. 8.3.2.1 E f f i c i e n t Harvesting Paths. To represent the d i f f e r e n t predation cases we specify the following 3 sets of predation parameters: Case I. a=0.2, b=0. Here, i t i s easy to v e r i f y , m(1) i s constant at 0.2. Hence this i s a case of no predation. Therefore, with these parameter sp e c i f i c a t i o n s the predation case reduces to the basic case with only two species and a single f i s h i n g f l e e t . Case I I : a=0.4, b=0.1279. This is a case of predation. m(l) goes from zero to 0.4 as n(2) goes from zero to i n f i n i t y . The value of b i s chosen so that m(1)=0.2 when n(2) i s at i t s h i s t o r i c a l average l e v e l . Case I I I : a= 0 . 6 , b=0.2558. This i s a case of more intensive predation than I I . m(l) goes from zero to 0.6 as n(2) goes from zero to i n f i n i t y . Again, b i s chosen so that m(l)=0.2 when n(2) i s at i t s h i s t o r i c a l 4 0 9 a v e r a g e l e v e l . We now c o n s i d e r e f f i c i e n t h a r v e s t i n g p r o g r a m s c o r r e s p o n d i n g t o t h e s e t h r e e p r e d a t i o n c a s e s . T h e l e n g t h o f t h e c o n t r o l p e r i o d i s 5 y e a r s . T h e r e a r e 3 i n d e p e n d e n t c o n t r o l v a r i a b l e s e a c h p e r i o d ; i n v e s t m e n t ( p o s i t i v e o r n e g a t i v e ) a n d a p p l i c a t i o n o f v e s s e l s t o f i s h e r i e s 1 a n d 2 . T h e i n i t i a l c o n d i t i o n i s o n e o f v i r g i n s t o c k e q u i l i b r i u m a n d n o f i s h i n g c a p i t a l . T h e c e n t r a l r e s u l t s o f t h i s e x e r c i s e a r e i l l u s t r a t e d i n t h e f o l l o w i n g a g g r e g a t e s t a t e - s p a c e d i a g r a m : F i g u r e 8 . 2 0 D i f f e r e n t P r e d a t i o n R e l a t i o n s . A G G R E G A T E 0 C A P I T A L 100-75-5 0 -2 5 -1000 J I I I I I I I I I I I 2000 3000 AGGREGATED BIOMASS I 1 1 1 ' I 4000 5000 A s f i g u r e 8 . 2 0 s h o w s , e f f i c i e n t h a r v e s t i n g p a t h s a r e h i g h l y d e p e n d e n t u p o n t h e p r e d a t i o n r e l a t i o n b e t w e e n t h e s p e c i e s i n q u e s t i o n . T h e m o r e i n t e n s i v e t h e p r e d a t i o n t h e l o w e r t h e o p t i m a l a g g r e g a t e c a p i t a l a n d b i o m a s s l e v e l s . S i n c e p r e d a t i o n , a s 410 specified here, b a s i c a l l y implies less marginal productivity of the aggregate biomass <59>, this result i s not surprising. More detailed information i s provided by the following disaggregated state-space diagrams: Figure 8.21 E f f i c i e n t Fishing Capital Levels. -20 0 20 40 60 80 100 S P E C I E S 1 VESSELS 41 1 Figure 8.22 E f f i c i e n t Biomass Levels. 1000—i 0 1 I I I I I I I I I I I I I l I I I I j I I I I I I I I I i 1000 1500 2000 2500 3000 3500 4000 SPECIES 1 BIOMASS Figures 8.21-22 show that as predation becomes more intense, the optimal a l l o c a t i o n of vessels to the predator fishery, i . e . 2, increases and i t s biomass decreases. The reason is not d i f f i c u l t to see. In addition to i t s market value, each unit of predator caught reduces the natural mortality rate of the prey. Hence, to each such unit there corresponds a positive shadow value r e f l e c t i n g this predation e f f e c t . In terms of e f f i c i e n t harvesting, therefore, the existence of predation enhances the unit harvesting value of the predator <60> just as an increase in i t s market price would. 412 8.3.2.2 Relative Prices. We w i l l now consider how, in the predation case, e f f i c i e n t harvesting may respond to changes in the r e l a t i v e prices of the two species. For t h i s purpose l e t us define the following output price index for the two species: (8) P=w(1)p(1)+w(2)p(2), where P denotes the output price index, p(l) and p(2) are the output prices of the two species and W(1) and w(2) are the respective weights <61>. Also define the r e l a t i v e output price of species 2, the predator, as: p(2)=p(2)/p{ 1 Now, in order to concentrate on r e l a t i v e price changes and not s h i f t s in the output price l e v e l , we f i x P to i t s 1983 l e v e l <62>. Moreover, since we require p(l) and p(2) to s a t i s f y (8), these two absolute prices are c l e a r l y determined once P(2) is chosen. We w i l l now calculate the e f f i c i e n t a l l o c a t i o n of vessels to both f i s h e r i e s for 10 d i f f e r e n t levels of P(2). The predation s p e c i f i c a t i o n i s as in case III in the previous subsection. The resulting e f f i c i e n t equilibrium vessel a l l o c a t i o n levels are i l l u s t r a t e d in figure 8.23. 413 Figure 8.23 E f f i c i e n t Vessel Allocation to Fisheries under Predation. VESSELS APPLIED 0 2 4 6 8 10 RELATIVE PRICE OF PREDATOR. P (2) As figure 8.23 shows there is a rather alien-looking nonmonotonic relationship between the r e l a t i v e prices of the two species and the respective e f f i c i e n t equilibrium harvesting intensity. This, however, i s readily explained in terms of the current b i o l o g i c a l s p e c i f i c a t i o n . F i r s t consider the curve for the predator, i . e . haddock. From figure 8.23 we see that, after a certain point, the harvesting of haddock i n t e n s i f i e s as i t s r e l a t i v e price decreases. In fact, at zero absolute output price of haddock i t s harvesting intensity reaches a maximum. Recalling that lower P(2) corresponds uniformly with higher absolute price of the prey, i . e . cod, this observation becomes understandable. As the absolute price of cod increases, the shadow value of a unit of haddock catch, due to i t s predation on cod, increases. For the 414 right magnitudes of the relevant data this shadow value may dominate the output price with the result that increased harvesting coincides with an absolute, and r e l a t i v e , price f a l l . This e f f e c t may, in fact, persist as long as any harvesting of the prey i s p r o f i t a b l e . This i s precisely the case in figure 8.23. The harvesting curve for cod is somewhat harder to explain. Since the cod does not impose any e x t e r n a l i t i e s on the haddock one would have expected harvesting for cod to decline monotonically with i t s absolute pr i c e . The reason for the i n i t i a l r i s e in exploitation intensity has to do with a combination of the cohort configuration of the cod stock and the predation. As the price of cod decreases and the haddock stock goes up, the natural mortality rate of cod increases. Hence i t s optimal harvesting age goes down <63>. However, the only way to reduce the actual mean harvesting age in this model, is by increased harvesting intensity. Consequently, for the suitable parameter values, the result may be as depicted in figure 8.23. These re s u l t s , as well as those of the previous subsection, c l e a r l y demonstrate that the existence of ecological relationships may invalidate many of the simple harvesting rules derived on the basis of single species models. 415 Footnotes. 1. Or, for that matter, harvesting of other b i o l o g i c a l populations. 2. From the fact that the model involves 3 species of f i s h , each consisting of 9-12 exploited cohorts, 2 f i s h i n g f l e e t s each consisting of multidimensional vessels, 4 production processes for each species of f i s h etc., the v a l i d i t y of this statement should be c l e a r . 3. For the basic data, see chapter 5.3 and especially appendix: data (section 5.3). 4. Other possible control variables that can be regarded as fishery inputs are e.g. cohort s e l e c t i v i t y . Given the current f i s h i n g technology in the Icelandic demersal f i s h e r i e s t h i s , however, does not seem p r a c t i c a l . In other cases, on the other hand, cohort s e l e c t i v i t y might constitute the central control variable. 5. Since the harvesting cost functions are nonlinear in operating time, t h i s may be quite r e s t r i c t i v e . 6. This p a r t i c u l a r feature also poses a well known theoretical problem in dynamic programming, see e.g. Leitmann, 1974, ch. 13.10, and Clark et a l . , 1977. 7. It is of course unnecessary as well as numerically impractical to r e s t r i c t I ( i , t ) and K ( i , j , t ) to integers. A f r a c t i o n of these variables may conveniently be interpreted as a correspondingly late investment or an application of a vessel for the corresponding fraction of the year. 8. See e.g. Box et a l . , 1969, and G i l l and Murrey, 1976. 9. The t h i r d constraint, La(i,j,t)=1, can be substituted into the value function in an obvious way. 10. Examples are provided by the widely available NAG c o l l e c t i o n of maximization routines. See e.g. Nag, 1980. 11. See Powell, 1964 and 1965. 12. See G i l l and Murray, 1974. 13. A useful description of t h i s method can be found in Box, Davies and Swann,l969. 14. For further information on t h i s see chapter 6 especially section 6.3 and Appendix 1: Data. 15. For data on these variables see the respective chapters in part I I . 416 16. The case of endogenous natural mortality, i . e . predator-prey relationships i s considered in section 8.3, however. 17. Experiments with shorter control periods, further discussed in section 8.2.2 below, showed that the attained maximum of the value function increased by some 1.8% when the control period vas extended from 3 to 5 years and only by 0.6% when i t was further extended to 7 years. 1 8 . For d e t a i l s on these components of the empirical model see sections 4.2.2.3-4.2.4.3, 4.2.5.1 and 5.4.2. 19. This means, as explained in section 5.5.4, that disinvestment costs amount to 25% of the undepreciated value of the vessel. This number can be taken as a measure of c a p i t a l m a l l e a b i l i t y . Other m a l l e a b i l i t y assumptions w i l l be considered in section 8.2.3.4 below. 20. Since the demersal f l e e t has grown at an annual rate of over 5% during the last decade, t h i s i s perhaps an optimistic interpretation of the current p o l i c y . 21. The required rate of discount i s , in fact, over 1100% per annum. 22. More disaggregated results can be found in appendix 8.2-A. 23. In this p a r t i c u l a r case i t i s primarily the haddock fishery that offers these opportunities. See e.g. figures 8.3-8.6 below. 24. The corresponding disaggregated results are given in appendix 8.2-A tables 3 and 4. 25. See e.g. Clark, Edwards and Friedlander, 1973, and Clark, 1976, pp 276-285. 26. See table 8.2. It should be noticed that t h i s p a rticular pattern i s largely determined by the fishing technology. 27. See section 4.2.5.1. Note especially the dynamic weight relationship for haddock in table 4.11. 28. See e.g. Clark, 1976, and Howe, 1979. 29. See section 4.2.5.1 for d e t a i l s . The e ffects of the ecological weight relationships are further explored in section 8.2.3. 30. The construction of this diagram is explained in appendix 8.2-B. 31. See Clark et a l . 1979, and Charles, 1981a. 32. In fact, t h i s is most unl i k e l y , especially i f the f i s h e r i e s are not equally productive, as i s the case here. Moreover, 417 consider the very variable optimal harvesting path for haddock in section 8.2.2 above. 33. See Gordon, 1954. 34. For a precise d e f i n i t i o n of t h i s concept, see Gordon, 1954. For empirical d e t a i l s see the respective sections in part II above. 35. For a more complete development of t h i s argument see e.g. Gordon, 1964, pp 130-31. Actually Gordon is concerned with one fishery and establishing the tendency to excessive f i s h i n g e f f o r t in that context. His arguments, however, f i t the current case perfectly. 36. This i s demonstrated by the results of section 8.2.2 above, where e f f i c i e n t harvesting of Icelandic demersal f i s h stocks implied some 60% long run reduction in the aggregate fl e e t allocated to the cod fishery but only 49% and 50% reduction in the f l e e t a llocations to the haddock and saithe f i s h e r i e s respectively. 37. I.e. as the basic case in every respect except that the control period is r e s t r i c t e d to 3. 38. This i s confirmed by the results in section 8.2.2, where i t was found that the trawler f l e e t was p r a c t i c a l l y never applied to the saithe fishery. 39. The annual employment of multi-purpose vessels during the control period was (0.,36.6,48.0,1.7,50.8,0.) vessels respect i v e l y . 40. This means, i t w i l l be r e c a l l e d , that disinvestment costs amount to 25% of the depreciated investment cost of the vessel in question. 41. But with a time lag of one year. See section 5.5.4. 42. Notice, however, that t h i s perfect m a l l e a b i l i t y case is somewhat d i f f e r e n t from the one s p e c i f i e d in Clark et a l . , 1979. where existent c a p i t a l could be sold at investment prices. 43. This i s not surprising. As we saw in section 4.2.5.1 and chapter 7, the estimated ecological weight relationships can have a rather dramatic effect on individual weights and consequently biomass l e v e l s . 44. This, i t w i l l be remembered, refers to the e f f e c t s of the aggregate biomass on the production of f i s h i n g m o r t a l i t i e s . In part II t h i s effect was found to be mostly negative. For further d e t a i l s see chapter 6, especially sections 6.3 and 6.4. 45. To some extent t h i s i n s e n s i t i v i t y to the stock-effort s p e c i f i c a t i o n can be attributed to the ecological weight 418 re l a t i o n s h i p . If that were nonexistent, the e f f i c i e n t policy would imply a much greater r i s e in the vessel s p e c i f i c catch and, consequently, a higher stock-effort e f f e c t . 46. See, in p a r t i c u l a r , equations (53)-(55) in section 4.2.5.2. 47. It should be emphasized that we are here not engaged in anything l i k e a f u l l y fledged stochastic analysis of e f f i c i e n t harvesting programs. Such an exercise would by far exceed the scope of t h i s work. 48. More to the point perhaps, to the extent i t can be foreseen, recruitment is not stochastic. 49. These should be compared with the regions in Clark et a l . , 1 979. 50. Notice, however, that investment becomes operative with a time lag of one year. 51. Examples are provided by the Icelandic and North-sea herring f i s h e r i e s and the P a c i f i c halibut and salmon f i s h e r i e s . 52. Consider for instance a single cohort fishery, e.g. the Icelandic capelin fishery, in a state of very low cohort to be followed by a very big one. Since investment becomes operative with a lag of one year in t h i s model, th i s s i t u a t i o n would probably imply regions VII and VIII. 53. In terms of our empirical model, the role of these two species w i l l be played by cod and haddock respectively. 54. Thus, the situation resembles several common predation situations, e.g. that of the Icelandic cod and capelin. 55.. Notice that this natural mortality function is somewhat dif f e r e n t from that specified in section 4.2.5.1 in part I I . 56. More precisely, m(l)=0 when n(2)=0, and m(l) converges to a when n(2) approaches i n f i n i t y . Notice that, in spite of equation (7), we retain the assumption that m(1) becomes i n f i n i t e at the maximum age of species 1. 57. Notice, however, the l i m i t a t i o n s of the generalization, es p e c i a l l y that the relationship s p e c i f i e d i s one of pure predation with the predator gaining d i r e c t l y no additional weight from the predation. 58. Hannesson, 1983, has recently explored t h i s issue within the confines of a simple theoretical model. 59. Since, assuming reasonably symmetrical increase in the respective biomasses, m(l) increases with the aggregate biomass. 60. Assuming, of course, that the prey i s valuable. 419 61. In fact we set W(1)=0.8 and w(2)=0.2 to r e f l e c t the diferences in h i s t o r i c a l biomass and catch l e v e l s . 62. See Appendix 1: Data (section 5.4). 63. See section 4.2.3.3. 420 PART IV CONCLUSIONS 421 9. Summary and Conclusions. We have, in th i s essay, constructed a detailed model of the harvesting of interdependent self-renewable natural resources. The major components of the model are 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 defining the link between the b i o l o g i c a l and economic submodels. The model i s designed to describe the population dynamics, including ecological interactions, of any f i n i t e number of s e l f -renewable natural resources within a given environmental framework. The ecological relationships can be either deterministic or stochastic or both. The model can s i m i l a r l y accomodate any f i n i t e number of harvesting technologies. Consequently, in the jargon of fishery economics, the model may be characterized as a multi-species, m u l t i - f l e e t stochastic f i s h e r i e s model. The pa r t i c u l a r empirical case of the Icelandic demersal f i s h e r i e s , consisting of 3 species of f i s h and 2 types of fishing vessels, was s p e c i f i c a l l y examined. The development of the empirical model involves a number of items that may be of general relevance to the economics of s e l f -renewable natural resources. These include the following: (i) A c r i t i c i s m of the use of aggregative models to describe the growth of multi-cohort resources. ( i i ) A formal extention of the Beverton-Holt population growth model to include f a i r l y general ecological relationships. 422 ( i i i ) An empirical estimation of certain ecological relationships for the Icelandic demersal f i s h e r i e s using widely available f i s h e r i e s data. (iv) A detailed examination and c l a r i f i c a t i o n of the rela t i o n s h i p between economic inputs and f i s h i n g m o r t a l i t i e s summarized in the so-called fi s h i n g mortality production function. (v) A d e f i n i t i o n of the stock-effort e f f e c t , an empirical v e r i f i c a t i o n of i t s existence and an examination of i t s theoret i c a l implications. The completed model was employed to simulate the Icelandic demersal f i s h e r i e s during 1960-80 and the simulated values compared with available observations. The outcome of this exercise was generally sa t i s f a c t o r y suggesting that the model is capable of imitating, reasonably accurately, the path of the Icelandic demersal f i s h e r i e s . The objective of the harvesting a c t i v i t y was a r b i t r a r i l y defined to be the maximization of i t s contribution to the economic well-being of the Icelandic people. A f a i r measure of t h i s , i t was argued, is provided by the economic rents generated by the f i s h e r i e s . Harvesting paths having the property of maximizing economic rents were c a l l e d e f f i c i e n t . The model was then used to assess the e f f i c i e n c y of the current harvesting pattern and to discover, with the help of 423 numerical search techniques, e f f i c i e n t harvesting paths. This maximization exercise involved b a s i c a l l y two sets of control variables; investment in the f i s h i n g f l e e t and the a l l o c a t i o n of existing vessels to f i s h e r i e s and/or idleness. The main results of the maximization were as follows: (1) The current competitive demersal fi s h i n g pattern was found to be exceedingly i n e f f i c i e n t , thus confirming t h e o r e t i c a l predictions as well as the conclusions of other empirical studies. (2) The e f f i c i e n t harvesting program included i . a . a dramatic reduction in the o v e r a l l size of the fishing f l e e t , a reallocation of the 2 f i s h i n g f l e e t s to the 3 f i s h e r i e s and a very irregular application of the control variables along the adjustment path towards equilibrium. Having described the e f f i c i e n t harvesting path, an examination of i t s s e n s i t i v i t y to a number of model sp e c i f i c a t i o n s was carried out. It was found that the l e v e l of attainable economic rents was highly sensitive to (i) the p o s s i b i l i t y of fishery selection, ( i i ) the a v a i l a b i l i t y of two f i s h i n g technologies, ( i i i ) the existence of ecological weight relationship between the demersal species and (iv) the degree of fishing c a p i t a l m a l l e a b i l i t y . The e f f i c i e n t path of the control variables was, moreover, found to be very sensitive to the f i r s t two of these factors but less so to the l a s t two. On the other 424 hand, i t was found that the l e v e l of attainable economic rents was r e l a t i v e l y insensitive to (i) the length of the control period, ( i i ) the estimated stock-effort effect and ( i i i ) the estimated stochastic element in the recruitment process. F i n a l l y , leaving the s p e c i f i c context of the Icelandic demersal f i s h e r i e s , the nature of e f f i c i e n t harversting paths given, on the one hand, d i f f e r e n t i n i t i a l conditions and, on the other hand, d i f f e r e n t predator-prey relations were investigated. The main results were: ( 1 ) Given the appropriate i n i t i a l conditions, hardly any combination of controls can be ruled out beforehand as inef f ic ient. (2) The dramat ic existence of predator ef f e c t on the properties -prey relationships can of e f f i c i e n t harvesting have a paths. 425 _LIST OF REFERENCES 426 References. Allen, R.G.D., Macro-Economic Theory, MacMillan, London, 1973. Almon, S., "The Distributed Lag Between Capital Appropriations and Expenditures", Econometrica 33, 178-96, 1965. Anderson, L.G., The Economics of Fisheries Management, Johns Hopkins University Press, Baltimore, 1977. Arnason, R., "Economics of Replenishable Natural Resources: The Case of the Icelandic Cod Fishery", M.Sc. di s s e r t a t i o n , London School of Economics, 1977. Arnason, R., "A Rational Competitive U t i l i z a t i o n of a Replenishable Natural Resources", (unpublished paper), University of B r i t i s h Columbia, 1979. Baerends, G.P., "De rationeele e x p l o i t a t i e van den Zeevischstand, in het bijondervan den Vischstand van de Noordzee". English translation in Special S c i e n t i f i c Report - Fisheries, No 13, U.S. Department of Inte r i o r , Fish and W i l d l i f e Service, Washington D.C., 1950. Beverton, R.J.H. and Holt, S.J., On the Dynamics of Exploited Fish Populations, Fishery Investigations, Series II, London, 1957. Bjorndal, T., "The Optimal Management of North Sea Herring", Programme in Natural Resource Economics, Resources paper no. 87, University of B r i t i s h Columbia, 1983. Box, G.E.P. and Cox, D.R., "An Analysis of Transformations", Journal of the Royal S t a t i s t i c a l Society, Series B, vol 26, 211-43, 1964. Box, G.E.P. and Tidwell, D.W., "Transformation of Independent Variables", Technometrics, vol 4, 531-49, 1962. Box, M.J., Davies, D. and Swann, W.H., Non-linear Optimization Techniques, ICI, Monograph no. 5, Oliver and Boyd, Edinburgh, 1969. Burmeister, E., Capital Theory and Dynamics, Cambridge University Press, U.S.A, 1980. Charles, A., "Optimal Fisheries Investment: Comparative Dynamics for a Deterministic Seasonal Fishery", Programme in Natural Resource Economics, Resources paper no. 85, University of B r i t i s h Columbia, 1981a. Charles, A., "Optimal Fisheries Investment Under Uncertainty", Programme in Natural Resource Economics, Resources paper no. 86, University of B r i t i s h Columbia, 1981b. 427 Chiang, A.C., Fundamental Methods of Mathematical Economics, (2nd ed.), McGraw-Hill, Tokyo, 1974. Clark, C.W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley, U.S., 1976. Clark, C.W., "Towards a Predictive Model for Economic Regulation of Commercial Fisheries", Canadian Journal of Fisheries and Aquatic Sciences 37, 1111-29, 1980. Clark, C.W., Clarke, F.H. and Munro, G.R., "The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment", Econometrica 47, 25-49, 1979. Clark, C.W., Edwards, G. and Friedlander, M., " Beverton-Holt Model of a Commercial Fishery: Optimal Dynamics", Journal of Fisheries Research Board of Canada 30, 1629-40, 1973. Clark, C.W. and Munro, G.R., "The Economics of Fishing and Modern Capital Theory: A Simplified Approach", Journal of Environmental Economics and Management 2, 92-106, 1975. Crutchfield, J.A. and Zellner, A., "Economic Aspects of the P a c i f i c Halibut Fishery", Fishery Industrial Research 1, no. 1, U.S. Departm. of Interior, Washington D.C., 1962. Dasgupta, P.S. and Heal, G.M., Economic Theory and Exhaustible Resources, Cambridge University Press, U.K., 1979. Dasgupta, P.S., Marglin, S., and Sen, A., Guidelines for Project Evaluation, U.N., New York, 1972. Deaton, A. and Muellbauer J., Economics of Consumer Behavior, Cambridge University Press, U.S.A., 1980. Debreu, G., Theory of Value, Cowles Monograph 17, Yale University Press, New Haven, 1959. Diewert, W.E., Duality Approaches to Microeconomic Theory, Discussion paper 78-09, Departm. of Economics,.University of B r i t i s h Columbia, 1978. Emlen, J.M., Ecology: An Evolutionary Approach, Addison Wesley, U.S., 1973. F i s k i f e l a g Islands, Skra y f i r Islensk Skip 1977, F i s k i f e l a g Islands, Reykjavik, 1978a. F i s k i f e l a g Islands, Utvegur 1977, F i s k i f e l a g Islands, Reykjavik, 1978b. F i s k i f e l a g Islands, Skra y f i r Islensk Skip 1984. F i s k i f e l a g Islands, Reykjavik, 1984. Gapinski, J.H., Macroeconomic Theory, McGraw-Hill, Tokyo, 1982. 428 G i l l , P.E. and Murrey, W., "Quasi-Newton Methods for Unconstrained Optimization", Journal of the Institute of Mathematics and i t s Applications 9, 91-108, 1973. G i l l , P.E. and Murray, W., "Minimization Subject to Bounds on the Variables", National Physical Laboratory Report No. NAC 72, 1976. Goldfeld, S.M. and Quandt, R.E., Nonlinear Methods in Econometrics, North-Holland, Amsterdam, 1972. Gordon, H.S., "Economic Theory of a Common Property Resource: The Fishery", Journal of P o l i t i c a l Economy 62, 124-42, 1954. Graham, M., "Modern Theory of Exploiting a Fishery and Application to North-Sea Trawling", Journal du Conceil International pour 1'exploration de l a Mer 10, 264-74, 1935. Graham, J.A. and Edwards, R.L., The World Biomass of Marine Fishes, (mimeograph), FAO. Rome, 1961. Gulland, J.A., Fishing and the Stocks of Fish off Iceland, Fishery Investigations, Series II, London, 1961. Gulland, J.A., "Estimation of Mortality Rates", Annex to the Artie Fisheries Working Group Report, International Council for the Exploration of the Sea, 1965. Gulland, J.A., Manual of Methods for Fish Stock Assessment. Part I: Fish Population Analysis, FAO, Rome, 1969. Hafrannsoknarstofnun, Astand Nytjastofna a Islandsmidum og Aflahorfur 1980, Hafrannsoknarstofnun, Reykjavik, 1980. Hafrannsoknarstofnun, Astand Nytjastofna a Islandsmidum og Aflahorfur 1981, Hafrannsoknarstofnun, Reykjavik, 1981. Hafrannsoknarstofnun, Astand Nytjastofna a Islandsmidum og Aflahorfur 1982, Hafrannsoknarstofnun, Reykjavik, 1982. Hafrannsoknarstofnun, Astand Nytjastofna a Islandsmidum og Aflahorfur 1983, Hafrannsoknarstofnun, Reykjavik, 1983. Hahn, F.H. and Matthews, R.C.O, "The Theory of Economic Growth: A Survey", Economic Journal 74, 779-902, 1964. Hannesson, R., Economics of Fis h e r i e s : Some Problems of Ef f i c i e n c y , Studentlitteratur, Lund, 1974. Hannesson, R., "Optimal Harvesting of Eco l o g i c a l l y Interdependent Fish Species", Journal of Environmental Economics and Management 10, 329-45, 1983. Hay, J.D., Uncertainty in Microeconomics, New York University Press, New York, 1979. 429 Henderson, J.M. and Quandt, R.E., Microeconomic Theory: A Mathematical Approach, (2nd ed.), McGraw-Hill, New York, 1971. Hendry, D.F. and Anderson, G.J., "Testing Dynamic Specifications in Small Simultaneous Systems: An Application to a Model of Building Society Behaviour in the United Kingdom", in M.D. I n t r i l i g a t o r (ed.) Frontiers of Quantitative Economics-IIIA, North-Holland, Amsterdam, 1977. Hogg, R.V. and Craig, A.T., Introduction to Mathematical S t a t i s t i c s , (3rd ed.), MacMillan, New York, 1970. Howe, C.W., Natural Resource Economics: Issues, Analysis and Policy, John Wiley, U.S., 1979. ICES, "Report of the Saithe Working Group", International Council for the Exploration of the Sea, 1974/F:2, Charlottenlund, 1974. ICES, "Report of the North Western Working Group", International Council for the Exploration of the Sea, 1976/F:6, Charlottenlund, 1976. ICES, "Report of the Saithe Working Group", International Council for the Exploration of the Sea, 1977/F:3, Charlottenlund, 1977., ICES, "Report of the Saithe Working Group", International Council for the Exploration of the Sea, 1978/G:3, Charlottenlund, 1978. ICES, "Report of the Saithe Working Group", International Council for the Exploration of the Sea, 1980/G:11, Charlottenlund, 1980a. ICES, "Report of the Ad Hoc Working Group on Multispecies Assessment Model Testing", International Council for the Exploration of the Sea, 1980/G:2, Charlottenlund, 1980b. I n t r i l i g a t o r , M., Econometric Methods, Techniques and Applications, Prentice-Hall, New Jersey, 1978. Jakobsson, J., "The North Icelandic Herring Fishery and Environmental Conditions 1960-68", Symbosium on the B i o l o g i c a l Basis of Pelagic Fish Stock Management, no. 30, 1978. Jensson, P, Stokastisk Programmering. Del II Methodologiske Overvejelser og Anvendelser, Lingby, 1975. Jonsson, J. and Schopka, S., " Vertidarrannsoknir 1973", Aegir 66, 441-4, 1973. Jonsson, G., Islenskir Fiskar, F j o l v i , Reykjavik, 1983. Jonsson, S., "Dreifing Soknar Skuttogara 1975-1976", Aegir 72, 430 311-19, 1979. Judge, G.J., G r i f f i t h s , W.E., H i l l , R.C. and Lee, T.C, The Theory and Practice of Econometrics, John Wiley, New York, 1980. Kerry Smith, V. and K r u t i l l a , J., "The Economics of Natural Resource Scarcity: An Interpretive Introduction", in V. Kerry Smith (ed.) Scarcity and Growth Reconsidered, Resources for the Future, Washington D.C., 1979. Lackey, R.T., "Fisheries and Ecological Models In Copenahagen's Fisheries Resource Management", in C S . Russel (ed.) Ecological Modelling In a Resource Management Framework, Resources for the Future, Washington D.C, 1975. Leitmann, G., The Calculus of Variations and Optimal Control, Plenum Press, New York, 1981. Levin, S.A. (ed.), Ecosystem Analysis and Prediction, Society for Industrial and Applied Mathematics, Philadelphia, 1975. Lippmann, S.A. and McCall, J.J., "Job Search in a Dynamic Economy", Journal of Economic Theory 12, 365-90, 1976a. Lippmann, S.A. and McCall, J.J., " The Economics of Job Search: A Survey. Part I: Optimal Job Search P o l i c i e s " , Economic Inquiry 14, 155-89, 1976b. Ludwig D. and Walters, C.J., "Optimal Harvesting with Imprecise Parameter Estimates, Ecological Modelling 14, 273-92, 1982. Malinvauld, E., Lectures in Microeconomic Theory, North-Holland, Amsterdam, 1972. Marshall, A., Principles of Economics, (8th ed.), MacMillan, London, 1930. May, R.M., S t a b i l i t y and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973. McKelvey, R., "The Fishery in a Fluctuating Environment: Coexistence of S p e c i a l i s t and Generalist Fishing Vessels in a Multipurpose Fleet", Journal of Environmental Economics and Management 10, 287-309, 1983. Mendelssohn, R., "Managing Stochastic Multispecies Models", Mathematical Biosciences 49, 249-61, 1980. M i t c h e l l , C.L., "Stock Adjustment Models, Canada's East Cost Groundfish Fisheries", Ph.D. d i s s e r t a t i o n , University of Ottawa, 1 979. NAG, "Library Manual. Mark 10", Numerical Algorithm Group, Oxford, 1983. 431 NEDI, Fiskiskipaaaetlun I; Haglysing Flota of Veida, Framkvaemdastofnun R i k i s i n s , Reykjavik, 1977. Neher, P.A., "Notes on the Volterra-Quadratic Fishery", Journal of Economic Theory 8, 34-49, 1974. Nerlove, M., "Returns to Scale in E l e c t r i c i t y Supply", in C F . Christ et a l . (ed.), Measurement in Economics: Studies in Mathematical Economics and Econometrics in Memory of Yehuda Grunfeld, Stanford University Press, 1963. Ng, Y., Welfare Economics, John Wiley, New York, 1980. Palsson, O.K., "The Feeding Habits of Demersal Fish Species in Icelandic Waters", Journal of the Marine Research Institute 7, 1-60, 1983. Parsons, L.S. and Parsons, D.G., "An Evaluation of the Status of ICNAF Divisions 3P, 30 and 3LN Redfish", ICNAF Research B u l l e t i n 1 1 , 1975. Patten, B.C., "The Relation between S e n s i t i v i t y and S t a b i l i t y " , in S.A. Levin (ed.) Ecosystem Analysis and Prediction, Society for Industrial and Applied Mathematics, Philadelphia, 1975. Pearse, P., "The Optimal Forest Rotation", Forestry Cronicle 43, 178,95, .1967. Pe l l a , J . J. and Tomlinson, P.K., "A Generalized Stock Production Model", Inter-American Tropical Tuna Commission B u l l e t i n 13, 421-96, 1968. Plourde, C.G., "A Simple Model of Replenishable Resources Exploitation", American Economic Review 60, 518-22, 1970. Pope, J.G., "An Investigation of the Accuracy of Vintage Population Analysis", ICNAF Research Document 71/116, 1-11, 1 971 . Powell, M.J.D., "An E f f i c i e n t Method of Finding the Minimum of a Function of Several Variables without Calculating Derivatives", The Computer Journal 7, 155-62, 1964. Powell, M.J.D., "A Method of Minimizing a Sum of Squares of Nonlinear Functions without Calculating Derivatives", The Computer Journal 8, 303-7, 1965 Quirk, J.P. and Smith, V.L., "Dynamic Economic Models of Fishing", in A.D. Scott (ed.) Economics of F i s h e r i e s : A Symbosium, 3-32, 1970. Reed, W.J., "A Stochastic Model for Economic Management of a Renewable Animal Resource", Mathematical Biosciences 22, 313-37, 1974. 432 Ricker, W.E., "Computation and Interpretation of B i o l o g i c a l S t a t i s t i c s of Fish Populations", B u l l e t i n of the Fishery Research Board of Canada 191, Ottawa, 1975. Ricker, W.E. and Foerster, R.E., "Computation of Fish Production", B u l l e t i n of Bingham Oceanographic College 11, 173-211, 1948. Riffenburgh, R.H., "A Stochastic Model of Interpopulation Dynamics in Marine Ecology", Journal of Fishery Research Board of Canada 26, 2843-80, 1969. Rothschild, B.J., "Fishing E f f o r t " , in J.A. Gulland (ed.) Fish Population Dynamics, FAO, Rome, 1977. Rothschild, M.,"Search for the Lowest Price when the Di s t r i b u t i o n of Price i s unknown", Journal of P o l i t i c a l Economy, 689-711, 1974. Russel, C S . (ed.), Ecological Modelling in a Resource Management Framework, Resources for the Future, Washington D.C, 1975. Saemundsson, B., F i s k a r n i r , Reykjavik, 1926. Schaaf, W.E., "Fish Population Models: Potential and Actual Links to Ecological Models", in C S . Russel (ed.) Ecological Modelling in a Resource Management Framework, Resources for the Future, Washington D.C, 1975. Schaefer, M.B., "Some Aspects of the Dynamics of Populations Important to the Management of Commercial Fish e r i e s " , Inter- American Tropical Tuna Commission B u l l e t i n 1, 1954. Schaefer, M.B., "Some Considerations of Population Dynamics and Economics in Relation to the Management of Marine Fisheries", Journal of the Fisheries Research Board of Canada 14, 669-681, 1 957. Schmidt, P., Econometrics, Marcel Dekker, New York, 1976. Schnute, J., "Improved Estimates from the Schaefer Production Function", Journal of the Fisheries research Board of Canada, 583-603, 1977. Schopka, S., "Urn Thorskinn", Aegir 65, 204-13, 1972. Scott, A.D., "The Fishery: The Objectivs of Sole Ownership", Journal of P o l i t i c a l Economy 63, 116-24, 1955. Shone, R., Microeconomics: A Modern Treatment, MacMillan, London, 1975. Sigurdsson, J., "Stjorn Fiskveida", Aegir 72, 583-90, 1979. 433 S i l v e r t , W. and Smith, W.R., "Optimal Exploitation of a Multispecies Community", Mathematical Biosciences 33, 121-34, 1977. Silvey, S.D., S t a t i s t i c a l Inference, Chapmann and H a l l , London, 1970. Smith, V.L., "Economics of Production from Natural Resources", American Economic Review 58, 409-31, 1968. Solow, R.M., "A Contribution to the Theory of Economic Growth", Quarterly Journal of Economics 70, 65-94, 1956. Southwick, C.H., Ecology and the Quality of the Environment, Van Nostrand, New York, 1976. Swamy, P.A.V.B., "Linear Models with Random C o e f f i c i e n t s " in P. Zarembka (ed.) Frontiers of Econometrics, Academic, 1974. Takayama, A., Mathematical Economics, Dryden Press, I l l i n o i s , 1974. T h e i l , H., Pri n c i p l e s of Econometrics, John Wiley, New York, 1971 . Tierney, S., "Monte Carlo Studies of the Schaefer Model", (unpublished paper), University of B r i t i s h Columbia, 1979. Uhler, R.S., "Least Squares Regression Estimates of the Schaefer Production Model: Some Monte Carlo Simulation Results", Programme in Natural Resource Economics, Resources paper no 23, University of B r i t i s h Columbia, 1978. Ulltang, 0., "Sources of Errors in and Limitations of V i r t u a l Population Analysis", - in International Council for the Exploration of the Sea, CM. 1976/H:40, 1976. Verhulst, P.F., "Notice sur la Loi que la Population Suit dans son Accroissement", Correspondence Mathematique et Physique 10, 113-21 , 1838. Wallace, T.D. and Hussain A., "The Use of Error Components Models in Combining Cross Section with Time Series Data", Econometrica 37, 55-72, 1969. Warming, J., "Om Grundrente af Fiskegrunde", Nationalokonomisk T i d s k r i f t , 499-505, 1911. Warming, J., "Aalegaardsretten", Nationalokonomisk T i d s s k r i f t , 151-62, 1931. White, K.J., "Estimation of the Li q u i d i t y Trap with a Generalized Functional Form", Econometrica 40, 193-9, 1972. Wilen, J.E., "Common Property Resources and the Dynamics of 434 Overexploitation: The Case of the North P a c i f i c Fur Seal", Programme in Natural Resource Economics, Resources paper no. 3, University of B r i t i s h Columbia, 1976. Zellner, A., "An E f f i c i e n t Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias", Journal of American S t a t i s t i c a l Association 57, 348-68, 1962. 435 APPENDICES 436 Appendix 1. Data. The following constitutes the bulk of the unpublished data used in the thesis. The data are arranged according to the sections in which they are primarily employed. 437 Section 4.2.5: Data. 1. Recruitment Data. Cod Haddock Saithe Year (1 ) (2) (3) (4) (1 ) (2) (3) (4) (1 ) (2) (3) (4) 1955 258 1039 701 161 1 956 305 1039 623 167 1 957 1 52 1236 505 214 1958 189 1 1 40 549 189 1 959 1 30 883 649 143 1 960 1 62 825 542 1 38 1 1 4 240 339 1 55 84 223 70 18 1 961 289 641 531 1 04 91 447 1 06 251 55 244 76 19 1 962 253 597 461 98 80 331 143 172 94 253 70 23 1 963 271 748 390 1 32 65 226 181 95 70 306 1 22 23 1 964 325 589 522 98 77 181 151 93 68 352 1 36 22 1965 171 461 578 80 42 1 50 1 26 79 60 441 178 24 1966 252 315 653 52 64 1 05 128 55 88 553 176 42 1 967 185 257 766 44 36 94 1 04 51 66 624 185 49 1 968 1 77 508 642 97 40 98 95 54 50 676 1 58 75 1 969 1 35 572 634 1 07 31 84 86 44 26 741 1 78 79 1 970 301 667 555 124 64 95 67 55 22 731 1 73 75 1 971 1 68 606 490 1 1 0 47 84 63 44 23 695 1 57 67 1 972 262 446 410 77 59 72 88 40 27 587 107 67 1 973 426 327 527 57 108 65 97 34 35 500 73 65 1 974 141 228 484 41 64 77 62 47 25 407 57 52 1 975 217 1 95 534 36 84 80 1 55 47 57 355 56 43 1 976 278 151 745 28 423 92 1 52 54 48 319 68 33 1 977 1 75 187 628 36 297 1 49 1 36 92 25 280 68 24 (1) Recruitment size of the respective cohort in mi l l i o n s of individuals. (2) Spawning stock in thousand metric tons. (3) M i l l i o n s of immature individuals during pre-recruitment period. (4) M i l l i o n s of mature individuals during pre-recruitment period. Sources: VPA analysis on catch data up to 1983. 438 11• Weight Data• Cod Haddock Saithe Year (1) (2) (1) (2) (1 ) (2) 1 955 -.007 2767 1 956 -.042 2568 1957 -.007 2325 1958 .042 2218 1 959 .058 2131 1 960 .037 1945 -.538 521 .202 223 1961 .031 1826 -.516 467 .058 244 1 962 -.020 1 689 -.249 427 -.069 253 1 963 .013 1 555 -.319 362 -.147 306 1 964 .057 1 545 -.089 294 -.010 352 1965 .059 1 503 -.058 245 -.027 441 1 966 .110 1599 -.018 198 -.017 553 1967 .108 1833 .015 175 -. 1 73 624 1 968 -.036 1914 .027 166 -. 1 78 676 1969 -.070 1 965 .094 151 -.232 741 1970 -.090 1848 -.011 1 45 -.317 731 1 971 -. 1 45 1636 -.033 131 -.388 695 1 972 -. 1 60 1 333 .117 1 33 -.302 587 1973 -.052 1 284 .092 1 38 -.203 500 1 974 -.004 1 165 .094 145 .068 407 1975 .014 1 208 . 1 25 188 .058 355 1976 .029 1 454 .045 210 .012 318 1 977 .044 1 469 .028 245 .018 280 (1) Relative weight divergence, i . e . R(j)-1, (see 4.2.5.1). (2) Standardized biomass. (See 4.2.5.1). Sources: VPA on catch data up to 1983. Estimateded individual weights, see tables 4.1, 4.4 and 4.7, and o f f i c i a l s t a t i s t i c s on yearly catch volumes. 439 Section 5.2: Data The data used to estimate the processing cost functions in section 5.2.2 are l i s t e d in the following 3 tables. The data are panel data covering 108 plants over the 3 year period 1974-6. The data in the f i r s t table are arranged by types of firms and years. Units numbered 1-41 are multiprocess firms. Units numbered 1-18 are specialized freezing plants, and units numbered 1-26 are specialized s a l t f i s h and stockfish plants. The second table contains the data on the f i s h meal and o i l producing firms. The t h i r d table l i s t s the input and output price indices. A l l the cost data is in m i l l i o n s of old Icelandic crowns. The inputs are in quantity terms measured in tons. The sources of the data are: (1) F i s k i f e l a g Islands (The Icelandic Fisheries Association) that provided the data on the input quantities. (2) Thjodhagsstofnun (The National Economic Institute) from whose records the data on the costs of individual firms and the input and output cost indices were extracted. 440 Table 1 . Processing Cost Data: Freezing, S a l t f i s h and Stockfish Processes. Variable Code: ( 1 ) Index for units. (2) Quantity of demersal species, excluding f l a t f i s h , used as inputs into the freezing process. (3) Quantity of f l a t f i s h used as inputs in the freezing process. (4) Quantity of pelagic species used as inputs into the freezing process. (5) Quantity of crustaceans used as inputs into the freezing process. (6) Quantity of demersal species used as inputs into the s a l t -f i s h process. (7) Quantity of demersal species used as inputs into the stock-f i s h process. (8) Quantity of pelagic species used as inputs to the s a l t f i s h process. (9) Total processing costs less taxes. A l l quantities in metric tons of catch. Costs in mil l i o n s of old Icelandic crowns (Ikr.) (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 502a - 33 857 0 1 100 602 0 151 , 0 " 2 ' li 0 0 5 52 3 76"~ _ o - U6 "0 " q ... 68, 2 3 1293 U7 3ia 19 227 69 0 ao, 9 a 163 1 1 132 25 ea3 0 0 2a, 8 5 1890 13 227 0 25 200 0 ao, • o 6 1202 3 536 •0 1 168 0 52, ,2 7 315« 1.8 705 101 731 216 0 90 , 3 8 " 3361 "6 5 " S«5 "" - 9 j 771 90- 0 88 7 • 9 2508 18 505 18 9oa 1 U9 0 85 3 1 0 837 1 a 262 1«1 266 0 0 37 8 1 1 2074 U9 316 57 177 b 0 0 77, r 6 12 220a 95 B26 21 2617 80 0 97 ,3 . 13 703 1 1 2a5 80 628 0 0 33. 6 "1 a 7 76 3 7 '2 8 0 177 l loa 0 " 0 63 ,8 15 369 8 2a2 0 238 0 0 17 ,3 1 6 0 0 a5 0 1202 ao 0 25 0 17 2928 19 226 2 1 6«3 75 0 fit ,5 IS 1734 66 0 0 190 8 90 0 61 , 1 19 a783 339 0 0 2769 0 0 120, 0 20 2333 61 0 0 6 1 a 15a - 0 70 , 0" 21 997 0 0 90 1 1706 76 0 73, 0 22 199a 60 n 0 93 7a 0 a5, b 23 227 1 a23 1 9 0 7ia 75 0 80 , 6 2« 1316 119 1 0 0 0 175 0 53, , 6 25 2607 *>5 0 0 1308 15 0 72 ,8 26 a508 518 72 0 0 13 0 137 2 27 33 oa 32 0 111 a9o 0 0 76 3 28 ?.«21 5? o 0 9 0 0 5« ,7 29 2155 62 0 0 a5 0 0 a3, , a 30 2752 39 n 0 33 17 0 59 , 1 31 1811 30 0 0 7 1 6 125 0 a7 , a 441 32 5228. 157 0 0 773 0 0 125.9 33 1815 2a 0 0 171 0 0 aa,9 •, 3a 882 22 .0 0 1170 0 0 "2.3 35 2311 131 132 0 1152 0 0 71.5 36 3022 69 1027 537 2e>29 138 0 16a,0 37 1271 39 1 06 6a 595 0 0 _.39.5 38 1519 __ "321 . . . . . 0 T o 5 7 " o" "0 60 . 7 39 7675 285 1101 5a 1055 5 0 225.3 ao 13302 308 1 357 21 3102 0 0 357_t 3 1 ai 713" 250 702 60 1202 0 0 188.a 1 9 0 55 58 561 0 0 0 0 135." 2 7466 "3 735 0 0 0 0 153.0 3 19 1 7 a 30 7 56 0 " o 0 "0,9 a 86" 39 "61 2a 0 0 0 32. 0 5 1678 a 281 0 0 0 0 ( I f l , 0 6 3796 33 676 0 0 0 0 90.7 7 1 188 1 a 276 0 0 0 0 32.0 8 6318 a I 3 1?0 a73 0 0 0 1U7.2 q "38 0 1 6 2 2 0 0 0 0 75. 1 10 7326 1 79 0 0 0 0 0 1 aa,8 1 1 6863 92 58 0 0 0 0 12a.5 12 1 1370 290 0 0 0 0 0 182.9 13 3935 1 36 0 0 0 0 0 103.9 1 a 2303 30 0 0 0 0 0 55.3; 15 22"2 1 aa 0 "0 " 0 o 0 35.5 16 5"a5 36" 533 0 0 0 0 113.7 17 3815 I 0 7 0 0 0 0 0 o/S.a 16 4650 29 539 0 0 0 0 99,6 r _ .... 0 0 0 2a 76 1230 0 fib", 2 2 0 0 0 0 801 338 0 18,0 3 0 0 0 0 12"C 0 . 0 1 1.3 a 0 0 0 0 5"5 0 0 7.3 5 0 0 0 0 898 82 0 3 . a 6 0 0 0 0 2316 0 0 25,8 • 0 0 o .. 0 783 " ' o ~ 0 1 5,6" 8 0 0 0 0 1 a«i 0 0 22.5 9 0 0 0 0 132" 0 0 18.0 1 0 0 0 0 0 565 0 la 11.2 1 1 0 0 0 0 2577 0 0 27,3 12 0 0 0 0 156 0 0 a .7 1 3 0 0 0 0 1398 122 0 ""28,5 1« 0 0 0 0 1 06 a6 0 a, 8 15 0 0 0 0 109-5 0 0 13,9 1 6 0 0 0 0 2636 0 0 29.5 17 0 0 0 0 1657 0 0 21.3 1 R 0 0 0 0 3176 0 0 39,5 19 "0 0 0 1 157 2a6 0 23,0 20 0 0 0 0 2131 1 « 7 0 0 0 2 52 209 101 508 137 0 203 11 8 2363 137 128 122 678 336 395 160 r2 9 2010 06 119 53 737 227 0 ioe ,9 10 160 21 59 130 639 9 191 55 • 6 11 2685 72 180- 78 1002 367 v 0 102 i 1 12 26 0 0" 165 "050 """OH 1769 350 "285" "235, ,1 .13 2085 36 0 1 31 0 0 67 7 10 662 20 130 196 700 28 0 92 3 15 319 12 57 52 103 06 0 25, ,0 16 0 0 13 0 683 153 272 01 , 6 17 2321 05 16 20 1120 620 0 103, 5 18" 2552 i a 0 9 0 925 115 """" 0 106, 6 19 3215 266 22 0 20oo 07 0 232, 1 20 2755 1 09 0 0 722 76 0 166, 3 21 828 0 51 3390 823 266 0 165, 5 22 1875 52 0 2 9 030 0 90, 5 23 2908 171 •o 0 90 710 0 170, 8 2" "259 2 98 0 2 0 ' 6 3 1 "' 0 139, 1 25 3396 115 21 0 589 71 0 153, ,6 26. 5813 209 105 0 0 180 0 299 ,7 27 3597 60 0 192 927 25 0 175, 1 28 2012 5 3 0 0 066 606 0 137 0 29 2302 80 0 0 79 1 1 0 83 18 30 2556 187 0 0 2? 117 0 109, 6 31 3013 1"2 0 0 3 5 0 3 6 0 0 133 3 32 33"5 170 0 0 1337 268 0 202 ,3 444 33 2704 47 0 0 735 173 0 163,8 3a 1302 9 0 2 643 140 0 96,8 35 2103 24 0 0 1582 27 0 137,9 36 " 3527" 220 T o 50 "503 4673 426 0 "461 ,8 37 1219 33 40 63 82 46 0 74,3 38 2123 75 7 38 1 003 0 327 147,2 39 9307 335 58 89 0 0 0 358,6 ao 8442 271 83 68 2162 0 722 414,2 a i 9786 345 129 103 .56 0 0 1100 470, 1 i o T l 9 6 0' 151 69 0 n 0 0 322,4 20 7096 149 187 0 0 0 0 282.4 30 1643 16 163 107 0 0 0 70.1 ao 1457 45 152 51 0 0 0 71,0 50 1770 25 8 49 0 0 0 77 ,4 60 4765 93 29 0 _ 0 0 0 185,8 70 1037 25 1 17 0 " " 0 0 51,9 80 1 0049 90 122 055 0 0 0 378.9 90 4 4 4 3 69 0 0 0 0 0 156,2 10 9340 135 0 0 0 0 0 342,6 1 1 9145 184 74 0 0 0 0 333,3 12 14934 407 0 0 0 0 0 52b, 2 13 ""• a 104 "285 139 4 26 0 0 0 "226,3 14 2068 83 0 0 0 0 0 8-8,7 15 1467 254 38 0 0 0 0 74,5 16 5423 195 173 0 0 0 0 239,4 17 4 4 4 0 335 0 74 0 0 0 202,8 1 8 5881 48 66 0 0 0 0 2bl , 0 1 0 0 0 b 321 1 288b 72 0 152,0 2 0 0 0 0 1 0U9 7"2 0 49,3 3 0 0 0 . 0 903 285 0 34,0 • 4 0 0 0 0 1181 154 0 24,8 : 5 0 0 ~~0 " •.- 0- 4 99 ^ . . . 229" "15". 9 6 0 0 0 0 2265 38 566 a5.« 7 0 0 0 0 1 182 170 0 28. 1 8 0 0 0 0 1725 0 266 45.6 9 0 0 0 0 1039 16 0 26.4 10 0 0 0 0 1031 0 0 24,0 1 1 0 0 0 0 2989 1 0 4""' 30 8" '58.4 12 0 0 0 0 269 328 0 17,9 13 0 0 0 0 1462 1512 661 8 3,7 I" 0 0 0 0 3« 1 4<4 0 11.2 15 0 0 • 0 0 1 135 0 173 32.a 16 0 0 0 0 1964 6 0 43.3 17 0 "o o b 1 19 0 o ' 0 37.8 16 0 0 0 0 3232 0 0 64,3 19 0 0 0 0 1380 864 0 52.3 20 0 0 0 0 2326 959 0 72.5 21 0 0 0 0 820 332 0 36.2 22 0 0 0 0 72 30 0 23 ~ 0 o o ' 0 1572 398 '"y 37,6 24 0 -0 0 0 1944 22 0 53.« 25 0 0 0 0 1743 531 0 62.0 26 0 0 0 0 1285 1 U4 121 56.7 445 Table 2 Processing Cost data: Fish Meal and O i l firms. Variable Code: (1) Index for units. (2) Quantity of primary demersal inputs excluding redfish (3) Quantity of redfish inputs. (4) Quantity of other demersal inputs (5) Quantity of pelagic inputs. (6) Total processing costs less taxes. A l l quantities measured in metric tons. Costs measured in millions of old Icelandic crowns (Ikr.) (1) (2) (3) (4) (5) (6) 1 2 0 3 7 4 1 6 0 0 5 2 6 2 3 1 8 5 . 2 n 1 1 6 3 0 0 0 1 6 0 7 5 7 3 . 0 ' 3 6 4 2 1 0 0 1 2 7 4 5 4 2 . 9 4 2 2 9 8 3 0 0 2 7 8 7 9 1 1 3 . 3 c «J 9 6 2 2 1 0 7 3 8 0 4 9 7 5 2 1 6 1 . 6 6 4 2 6 2 3 9 4 2 0 2 0 5 7 7 5 4 . 7 7 5 3 2 0 1 4 9 0 0 2 7 7 3 5 6 1 . 2 3 6 4 9 4 0 0 0 2 3 . 6 9 1 5 5 5 0 0 2 5 2 4 1 0 . 3 1 0 2 9 3 2 0 0 S3 4 1 3 . 1 11 2 2 4 4 1 2 7 0 0 7 3 3 2 2 3 . 7 1 2 6 0 4 3 . 0 0 0 1 0 . 1 1 3 1 0 S 1 0 0 0 1 0 . 1 1 4 3 4 5 5 6 4 0 0 0 1 1 , 6 1 5 4 8 0 9 7 5 0 1 2 9 3 6 6 3 . 0 1 6 3 6 2 6 0 0 ' 0 1 4 . 7 1 7 3 6 4 5 0 0 0 1 0 . 6 I S 2 1 5 0 0 0 7 2 2 5 2 5 . 1 1 ? 0 0 0 2 3 0 3 3 7 2 . 3 2 0 O T 0 0 1 2 2 0 5 4 6 . 0 2 1 2 5 9 6 0 0 1 3 9 6 3 3 6 . 6 n n 1 7 9 9 0 0 9 7 4 1 2 4 . 3 *-t —r 2 4 6 4 r\ A 1 5 5 7 3 C" *? •*? »_ O U V 1 1 3 0 1 3 0 6 3 5 2 1 0 6 " 2 7 9 " . 3 1 1 3 7 3 0 •-> tr .J 1 6 0 6 0 1 0 1 . 1 3 7 9 5 3 0 1 7 1 1 0 9 9 7 2 . 0 4 2 4 5 7 4 0 3 3 1 S 2 2 2 1 6 2 . 3 c- S 7 5 0 6 6 2 0 4 0 2 6 S 9 4 1 9 6 . 7 6 5 3 5 3 4 1 4 1 2 ^ 1 3 1 3 7 9 4 . 1 - 5 3 6 0 1 0 3 2 3 1 1 6 3 0 4 8 1 . 4 O 7 4 3 9 0 6 0 3 1 . 3 9 3 2 3 1 0 3 0 1 2 . 3 1 0 3 7 9 3 0 3 0 7 1 4 . 1 1 1 4 8 2 1 3 0 2 1 0 5 9 S 3 4 5 . 5 1 2 1 1 0 2 3 0 9 0 3 5 . 3 1 3 1 6 1 1 0 3 0 9 . 3 1 4 3 7 3 1 3 3 3 0 0 1 6 . 3 1 5 5 1 4 1 0 2 1 1 7 7 6 1 1 0 9 . 8 1 6 2 3 3 9 0 n 0 1 4 . 9 1 7 3 2 0 1 0 3 0 1 8 . 0 446 I S 1 9 9 4 • 0 1 6 1 4 6 5 7 6 4 . 7 1 ? 0 0 3 5 3 5 1 6 7 1 4 8 . 8 2 0 1 1 4 4 0 . 2 6 2 5 0 1 5 8 1 . 5 2 1 3 4 6 6 0 3 3 2 7 2 7 5 9 2 . 0 2 2 2 3 6 0 0 1 2 1 0 3 6 4 4 2 . 6 2 3 3 7 3 8 0 1 8 1 4 8 3 1 8 0 . 4 1 2 5 7 1 4 0 4 7 2 6 4 5 9 " 2 2 7 . 1 •-> 2 0 6 0 0 0 2 5 8 8 0 2 1 3 4 . 7 3 1 1 5 4 9 0 2 0 1 0 4 8 4 8 4 . 8 A 2 3 7 8 0 0 4 1 2 1 5 2 8 2 0 8 . 8 c-u 8 6 1 8 7 3 8 1 5 2 3 7 1 8 S 3 0 2 . 9 6 4 5 0 3 4 3 9 5 2 6 1 7 9 3 6 1 3 5 . 7 7- 6 1 7 0 1 0 4 6 2 6 1 9 8 3 0 1 1 4 . 7 8 4 9 0 3 0 4 0 3 6 . 2 9 3 1 1 0 0 3 0 1 6 . 0 1 0 3 9 7 0 0 3 0 1 9 . 4 11 5 6 1 4 4 7 0 2 6 2 1 1 5 3 1 1 8 . 3 1 2 1 2 2 0 1 0 1 0 0 4 4 . 7 1 3 1 7 4 4 0 1 0 1 3 . 4 1 4 3 7 4 1 3 S 6 3 0 2 6 . 2 1 5 4 5 3 5 7 0 6 6 6 2 0 3 2 3 4 2 . 3 1 6 1 5 7 5 0 1 0 1 7 , 6 1 7 3 4 2 1 0 3 0 1 6 . 5 1 3 2 0 4 3 0 1 9 1 6 9 4 8 1 0 1 . 3 1 9 0 0 1 9 1 9 4 4 6 1 3 1 . 7 2 0 9 4 5 0 9 8 7 7 5 7 5 . 5 2 1 3 7 2 5 0 1 5 1 1 7 7 1 7 1 . 1 -I n 2 5 6 3 0 9 7 0 5 9 4 2 . 9 2 3 5 S 1 S 0 1 3 S 6 5 1 8 0 . 3 447 Table 3 Input and Output Price Indices. Input Price Indices Output Price Indices Fish meal Other Free- Salt- Stock- Meal, Year and o i l processes zing f i s h f i s h o i l 1974 1.00 1.00 1.00 1.00 1.00 1.00 75 1.48 1.38 1.40 1.27 0.96 0.88 76 2.04 1.99 1.96 1.40 2.22 1.62 448 Section 5.3: Data. The available data relevant to the estimation of the harvesting cost functions cover a number of cross section observations on the Icelandic demersal fi s h i n g f l e e t yearly during .1974-77. These data were made available by the Icelandic Fisheries Association which is the Government agency responsible for c o l l e c t i n g t h i s kind of information. The data include observations on the physical a t t r i b u t e s of each vessel, the operating time i t allocates to each of five possible types of fishing gear, i t s catch of each demersal species and t o t a l costs during the year. Observations on the time spent on each demersal fishery was not available. The data on the physical attributes of the vessels include their type, size, engine power, age and number of electronic instruments. There are two main types of f i s h i n g vessels; (i) multi-purpose vessels, usually under 400 tons, and ( i i ) deep sea trawlers most of whom are over 400 tons. Operating time i s measured in days at sea. The fishing gear are trawl, g i l l n e t s , longline, handline and other gear (mostly purse seine). The deep-sea trawlers use trawl exclusively. The cost estimate i s simply t o t a l bookkeeping costs less taxes. The cost concept thus involves the cost of c a p i t a l in the form of interest payments and depreciation. The l a t t e r , however, were adjusted (by the Icelandic Fisheries Association) to r e f l e c t more accurately economic r e a l i t i e s . The data on input and output prices consist of one Laspeyre's input price index and 5 Laspeyre's indices for the output (or landings) price for each type of demersal catch. The data only cover vessels that were exclusively engaged in demersal f i s h e r i e s during each year. Many of the multi-purpose vessels participate in pelagic and crustacean f i s h e r i e s during certain seasons of the year. These vessels were l e f t out of the sample. The number and identity of vessels in the sample varies considerably from year to year. Thus, only 25 multi-purpose vessels are included in the data in 1974 but 75 in 1976. There are no observations on the deep-sea trawlers in 1975. 449 Variables: Code. Code Variables 1 Reference number. 2 Tonnage of vessel. 3 Engine power (HA). 4 Age of vessel. 5 Number of electronic instruments. 6 Type of vessel (l=Deep-sea trawler, 2=steel multi-purpose vessel, 3=wooden multi-purpose vessel). 7 Location of vessel (1=South-West, 2=West, 3=North-West, 4=North-East, 5=East). 8 Days fi s h i n g with trawl. 9 " " " g i l l n e t s . 10 " " longline. 11 " " handline. 12 " " " other gear (especially purse seine). 13 Catch of cod in metric tons. 14 Catch of haddock in metric tons. 15 Catch of saithe in metric tons. 16 Catch of redfish in metric tons. 17 Other demersal catch in metric tons. 18 Total costs in m i l l i o n s of old Icelandic crowns (Ikr. ) . 19 Labour costs ' 20 Number of landings t r i p s to Europe. 450 T a b l e 1 Deep-sea T r a w l e r s : D a t a . 3 1 0 1 1 1 2 1 4 1! 1 6 1 7 1 8 1 9 3 7 1 2 7 1 2 7 1 3 7 1 1 355 1" 34 9 1 263 J 2b3 D" D 0 0 BW 6 7"2 3000 1 6 1 1 2 0 5 0 0 0 0 78b 7 7"2 3000 l_7 1 1_134 0 0 __0 0 211 "" 8 "726 2169 3 7 1 1 *B '" 0""" 0" 6 0 78 9 7U2 3000 \ 4, t 1 i T 7 0 0 0 0 622 10 «62 1800 6 * 1 « " 5 * T 0 0 0 0 2135 201 105 62 76 1618 955 530 056 1727 20 o2 2283 1827 177 171 iep 170 135 120 103 119 57 13 8 72 100 958 773 307 55 076 333 1500 1003 736 135 733 893 161 77 75 '""30 90 283 TV? 78 52 43 61 88 11 781 1 "62 296 06 2 062 "62 2200 2000 1500 20 00 2000 2000 7 6 3 5 3 S ? * 3 5 4 3 0 6 1 3 3 9 1 3 3 9 2 3 5 2 3 3 3 4 3 3«0 0 0 0 0 0 0 0 0 0 "0 0 0 0 1""6 0 652 0 1336 b 23 30 0 1902 0 1823 ao 002 157 274 228 131 0 5 4 1276 823 0 05 376 256 1180 828 646 154' 230 632 216 277 1 49 210 20U 257 67 95 76 96 93 91 6 2°9 7 "07 6_062 9 "61 1 969 2 969 1500 1750 2000_ 2000 2820 ?«20 3 1 3 7 5 o 5 1 5 1 6 1 <~1 7 1 7 1 3 33U 4 344 5 310 "5 321 1 360 1 371 0 1329 0 282" 0 179b "02073 0 2072 0 1389 T I T 127 "70 26 6 135 169 "T5T 287 1170 312 293 036 289 121 68 20 0 132 205 68 1650 2505 198 221 236 ~77 92 99 91 13" 137 3 969 0 702 5 90? V~7 0 2 7 702 8 726 2820 3000 2800_ 300 0 3000 2169 0 7 1 3 * 1 0_7 1 3 6 1 3 7 1 5 7 1 355 1 355 t 344 1 335 1 304 1 1 355 0 0 0 "0 0 0 9 723 I 0 701 II 781 12'781 13 901 1 0 901 15 "62 I "62 _ 2 "51 3 "51 0 297 5 "51 2169 3000 2200 2200 2800 2800 1800 2000 1700 17 0 0 1500 1700 5 7 1 3 7 1 8 h 1 9 h 1 3 6 1 3 6 1 1 0 1 369 1 75 a 368 4 372 0 308 0 362 6 5 6 "6 •? 6 a 213 1 291 1 365 1 32 7 348 1 1 1 1 332 0 0 0 "0" 0 0 0 0 0 o 0 0 T T 0 0 0 0 0 0 0 0 0 0 0 0 1291 0 1965 0 1807 0 1675 0 1566 0 2171 208 236 230 180 90 86 5 33 623 ouo 596 934 513 2006 889 1585 225 209 170 1082 395 1901 15" 101 170 131 126 100 1 1 8 112 13" 0 0 0 "0" 0 0-0 0 0 0 0 0 0 30 0" 0 "56 0 2802 0 28 3 2 0 3221 0 2928 0 1"39 0 059 0 996 0 1208 0 1102 0 988 173 33 172 236 229 179 13" 316 111 "163 2"5 269 "55T" 3 007 '395 208 500 1 125 29 100" 337 33 290 137 21 118 1060 007 925 231 160 2"0 120 338 171 266 190 310 20fe 1 175 1916 1 182 1173 1606 107 282 161 106 123 123 60 70 86 6 296 7 299 8 ""2 9 ao7 10 "07 1 1 "07 1 5 0 0 1500 1800 1 750 1750 1 7 5 0 5 2 5 5 o 0 1 31S 1 335 1 362 2 35" 2 358 2 336 0 0 0 0 0 0 12 "36 13 256 10 062 15 ""62 16 299 .17. "07 1780 1060 2000 2 0 on 1500 1750 3 6 1 119 1 5 5 1 " 6 1 5 5 1 3 5 1 308 3"5 356 30 0 357 358 0 IbOci 0 2191 0 1295 0 3602 0 "080 0 2"33 "2~b"T 209 156 250 22" 177 •3W 326 051 587 538 0 1 0 858 8 0 8 83 152 17^ 16" 130 169 85 71 85 199 103 202 1 57 139 197 -9~6 89 82 1 Oo 110 93 0 "229 0 1706 0 2193 0 27 08 0 2002 0 3560 "226" 79 99 To 5" 100 213 ~"5"8~~ 139 1"6 "258 169 238 299 57 502 152 180 96 136 55 16* To 2 107 76 TOT 59 60 —79' 86 92 18 500 19 300 20 328 21 328 22 "62 23 "61 2" "51 1800 1500 1200 120 0' ,2000 '2000 170 0 3 6 8 5 10 5 i n S 5 6 1 5 6 1 0 363 0 339 332 316 363 330 M 1 5 3 5 2 0 3807 0 2615 0 2106 01620 0 2"52 0 1736 T65~ 68 307 282 541 332 371 32b 340 263 319 343 133 84 36 147' 61 51 80 91 18 112 55 185 103 90 71 71 85 75 0 2129 278 035 104 339 100,7 451 m T : 1 2 3 9 6 9 9 6 9 7 " 2 2 8 2 0 2 8 2 0 3 0 0 0 6 7 5 7 a 6 9 0 2 7 4 2 7 0 2 "7 26 7 2 3 7 0 1 2 8 0 0 3000 3 0 0 0 2 1 6 9 2 1 6 9 3 0 0 0 5 7 4 6 " 7 6 7 6 7 0 7 1 0 1 1 '2 t i 1" 1 781 781 90 1 90 1 0 6 2 " 6 2 2 2 0 0 2 2 0 0 2 8 4 0 " 2 8 0 0 iaoo 2 0 0 0 9 10 o ' u" 11 h 1 3 4 7 1 3 3 0 1 3 5 2 0 2 1 0 8 0 1 6 9 9 0 2 6 0 0 208 310 346 4 8 5 6 9 7 8 5 0 1646 2049 97T) 3 5 8 2 5 5 2 2 2 1 4 5 , 2 1 3 5 , 5 1 3 0 , 3 1 3 0 3 1 3 1 5 1 3 6 6 1 3 5 4 1 3 5 0 1 3 1 8 0 0 0 "0 0 0 2 0 3 9 2 5 3 9 2 5 6 3 3 5 5 4 1 9 2 1 3 5 4 3 3 1 3 6 " 3 6 8 3 6 2 1 3 2 3 3 7 9 3 3 4 0 1 4 2 6 9 "45 18 2 8 7 3 1 0 1 7 " 2T5 " 1 3 3 1 3 0 "150 2 4 9 2 3 4 3 0 9 3 2 6 2 9 1 2 8 8 2 4 2 3 7 0 7 0 9 8 8 4 1 4 9 5 " 9 3 0 1 0 8 6 7 3 1 ~9TF 6 8 6 6 3 8 " 1 9 5 5 " 1 1 7 6 1 1 2 7 2 1 3 3 2 5 1 12 0 19 2 3 9 2 1 9 1 2 3 , 3 1 1 1 . 4 1 4 3 . 3 1 3 4 , 0 1 3 6 , 9 1 2 6 , 0 5 " 7 2 2 2 191 0 8 1 1 6 2 5 7 1 2 7 7 3 8 8 4 8 " 4 3 8 3 3 5 9 9 2 5 3 9 4 3 0 3 7 6 5 3 8 4 2 9 2 " 2 1 2 4 . 9 1 2 2 . 6 1 3 1 . 5 1 3 5 . 3 8 8 , 3 9 3 . 8 3 5 , 3 5 , 37 T 6 ; 30 3 3 , 3 9 , 41 3 4 , 01 3 9 " 1 0 4 31 30 1 1 5 1 6 2 T 7 9 0 2 1 0 1 2 1 " 1 , 0 , a ,9' .6 ,7 " 5 1 0 8 8 " 5 1 2 9 6 2 9 9 0 0 2 1 7 0 0 2 2 0 0 1 7 0 0 1 5 0 0 1 5 0 0 1 8 0 0 " 6 1 6 " 6 6 5 3 6 6 6 1 3 5 0 1 2 3 2 1 3 3 4 1 3 6 0 1 3 5 3 1 3 0 6 0 0 0 0 0 0 0 "n 8 9 1 o T F 12 13 14 15 16 I T 18 19 " 3 6 " 0 7 " 6 2 " 6 2 0 6 2 2 9 9 1 7 8 0 1 7 5 0 2 0 0 0 "2 0 0 0 2 0 0 0 1 5 0 0 " 0 7 5 0 0 4 2 4 "265 3 2 8 " 6 2 1 7 5 0 1 8 0 0 _2100_ 9 9 0 1 2 0 0 2 0 0 0 6 6 5 T" 6 < 6 h 3 0 6 3 2 2 3 3 3 3 3 2 2 9 7 2 9 9 3 " 0 3 3 5 2 3 0 0 0 0 0" 0 0 2 6 11 «5 6 f> " 4 " 3 2 9 5 3 0 5 5 3 2 0 0 D 0 "CT o 0 0 0 0 "0 0 0 0 2 0 7 2 0 1 0 9 7 0 2 " 3 1 " n 2 0 4 7 0 2 6 9 8 0 1 5 8 1 170 81 2 4 8 2 0 2 2 5 5 1 7 6 2 2 7 3 7 2 1 7 6 0 7 0 0 0 9 2 1 2 7 2 2 3 2 5 5 0 3 6 9 9 7 2 9 5 8 2 3 6 3 1 3 1 3 5 1 2 6 8 2 8 7 " 3 1 8 5 , 1 7 5 , 8 9 1 . 1 1 0 4 , 1 9 2 . 7 8 4 , 8 0 0 0 0 0 0 ~h~ o o 0 5 0 3 9 0 3 3 7 3 0 3 7 5 2 0 3 5 8 3 0 3 8 1 6 0 2 7 8 1 2 5 0 9 0 3 9 9 " 7 8 9 0 1 5 9 0 3 3 3 5 0 0 3 6 1 0 2 7 7 9 0 2 3 8 7 0 2 6 3 1 0 3 0 4 1 181 1 1 0 _ 5 8 1 3 2 167 3 5 1 " 7 1 301 2 " « 110 1 58 170 2 7 0 9 6 71 2 " 5 2 6 0 110 6 5 7 6 4 3 7 3 8 7 3 5 8 1 2 5 . 6 9 5 . 7 1 0 0 , 3 4 9 15 3 1 2 1 6 1 9 8 1 7 0 6 3 , " 8 6 , 2 7 5 . 7 16 3 5 10 "35 70 2 5 2 1 7 2 6 0 1 4 0 122 118 1 3 0 9 1 , 2 1 0 5 , 0 8 2 , 7 ... ? 2 < 9 7 7 , 3 9 3 , 1 2 0 2 1 2 2 2 3 " 5 1 4 6 1 2 9 7 4 6 2 1 7 0 0 2 0 0 0 1 5 0 0 2 0 0 0 a 5 5 5 3 6 6 6 5 3 2 4 5 3 2 2 5 3 2 5 "5 3 1 0 0 0 0 0 0 o o " o " 0 2 9 0 9 0 3 4 6 9 0 1 9 4 7 0 3 2 5 0 1 58 2 1 2 1 9 0 3 0 1 2 7 0 1 "7 3 2 1 2 5 7 46 3 2 1 3 6 28 3 1 3 1 06 82 1 1 4 9 1 . 1 9 0 , 0 8 2 . 6 8 9 , 5 2 9 18 30 3 2 3 " 2 0 55 3 6 " 1 31 3 6 2 6 31 0 2 , 2 8 , 2 " 2 7 , 3 6 , 3 0 , 3 3 , 2 " , 3 5 , r s i ,5 0 , 9 _ 0 ,6 0 ,8 0 ,8 0 ,2 0 ,5 1 0 0 2 0 9 1 2 0 2 1 3 0 0 0 0 2 6 0 8 0 2 0 3 T 452 Summary S t a t i s t i c s NUMBER "OF O B S E R V A T I O N S : : " 9 6 NUMBER OF INPUT VAHIABLES= 20 USING OBSERVATIONS 1 THRU 96 VARIABLE NAME NO. 1 2 " 3 o M E A N 9.5000 "572.18 2105. 1 0.5625 STANDARD DEVIATION 6.1610 223.29 5"7,72 2.3633 VARIANCE 37.958 09859, 0.30000E+06 5.6805 M I N I M U M 1,0000 256,0 0 990.00 1.0000 b.OOOO MAXIMUM 2 4 , 0 0 0 969., 00 3000,0 1UO00 7.0000 1,0000 5*0000 372^00 0.0 0«g 0,0 0*0 5039,0 ~~5oiyoo 1618.0 25Q5.0 "3197 00" 105,20 55,200 6.0000 5 5.9079 6 1.0000 7 _ 2 ' . « 5 8 3 ' 8 3 2 2 . 5 0 9 0.0 10 n.o 11 0.71627 0,0 1.5622 56.515 0,0 0.0 0.5130b 0,0 2,aooo 3193.9 0,0 0,0 1,0000 1.0000 48.000 0,0 0.0 0,0 0,0 78,000 8,00 00 3,0000 10.OOP 18,000 21,300 0,0 _ 0,0 0 . 0 12 0.0 13 __2260,8 to 195.09 15 8 . 2 5 . 1 ToTX 6 , 2 10,6 -"6,9 7 . 3 1 0 , 9 1 0 b 1 0 2 77 28 11 l o , 3 6 , 0 0 5 2 2 3 9 7 3 7 2 5 , 0 1 1 , 7 5 1 6 31 12 ft 26 3 5 . 2 1 3 , 6 2 3 9 5 7 1 8 6 1 1 " 4 5 2 1 . 8 7 . 9 0 5 5 9 2 0 22 2 0 . 1 9 , 1 3 9 0 19 0 0 0 1 8 , 1 fl.O 454 1" 10" 565 74 6 2 1 234 1 0 3 0 0 0 803 34 8 15 09 37 3 13, ,9 15 9b 400 56 0 2 i b 9 3 0 " o b 376 0 b 0 0 20 9 ........ ^ , 4 lb 103 565 70 6 2 I 124 158 0 0 0 619 59 218 45 2 27, 7 12, i o 17 56 370 56 5 3 I 50 95 0 0 56 376 20 3 6 26 19, A— 4 —7i 8, _ — ,3 18 53 280 43 3 3 I 169 0 0 0 0 26 138 23 0 72 9, , 4 3 , 9 19 103 470 63 S 3 I 7 9 93 41 0 0 521 17 6 0 5 23, ,9 9 , 7 20 101 400 59 5 2 I 161 67 0 0 0 315 138 314 0 26 24 ,8 1 0 • 8 21 " 5 2 "350 46 a 3 I 184 " 8 0 0 0 0 338 25 0 1 6 16 • 6 6 , r 6 22 72 458 60 a 3 l 0 145 3 1 64 0 307 13 192 2 13 20 ,9 8, ,3 23 88 400 59 S 2 I 146 67 0 0 0 231 93 309 22 63 26 • 8 9 , 0 -«3T 25 104 565 68 5 2 i 329 0 0 0 0 387 213 466 17 1 1 34 , 4 15 , 7 2b 103 500 72 6 2 I 9 6 8 1 14S 0 0 556 125 21 71 55 37 ,7 12 , 3 27 66 425 56 4 2 I 8 1 9 7 0 0 58 339 12 39 24 26 22, 0 8 , 0 28 61 335 55 "a" " 3 " I 0 1 7 1 3 9 ' " o " " 0 799 0 0 0 3 28, 7 13 ,5 29 55 425 56 4 3 l hM 0 0 0 0 24 56 63 10 122 13, 0 4 i 6 1 1 49 095 6 3 5 2 I 0 1 9 0 3 7 0 0 827 39 13 9 32 28, 3 14 0 2 143 765 75 6 2 I 1 3 5 9 2 3 1 0 0 804 41 29 37 12 51, 7 16, , 0 3 184 600 63 5 2 I 7 8 7 5 1 2 3 0 0 670 82 32 60 42 44, 8 14, 3 4 168 660 63 5 2 I 0 1 8 2 0 0 27 737 0 74 3 55 36, 3 16, 7 5 199 550 61 5 2 I 0 6 9 1 4 a ~ 7 9 0 505 "78 585 0 36 40, 3 16, 0 b 194 450 63 5 2 I 1 3 6 l c -4 0 0 0 462 49 235 11 4 32, 6 13, , 1 7 125 450 72 5 2 I 0 1 1 5 1 0 9 0 43 717 25 9 0 67 39, 5 15, 0 8 181 495 66 5 2 I 1 5 1 1 1 2 0 0 0 346 26 80 7 9 24, 0 8, s 9 109 600 64 5 2 2 0 0 2*»2 0 0 705 55 0 0 359 39, 5 15, 8 1 0 208 660 65 S 2 1 0 2 5 * 0 0 0 524 0 349 6 0 39, 3 13, 2 11 207 660 66 •5 2 1 142 U 0 0 0 144 92 73 39 7 31. 9 12, 0 7f. 1 39 240 47 2 3 1 2 4 0 0 1 2 0 56 15 20 0 3 7, 9 2, 3 2 49 315 54 a 3 1 0 1 1 2 1 6 « * 0 18 396 106 2 0 29 39, 1 17, u 3 4 5 7 8 45 38 42 45 38 25 350 256 350 335 240 220 56 08 46 54 45 68 3 1 3 1 3 1 3 1 3 I 3 3 0 3 0 0 0 7 7 3 9 JS3 1 1 0 1 2 7 1 2 6 1 3 5 8 0 9 7 1 2 5 0 1 0 0 0 b 0 0 9 10 11 12 13 14 45 29 28 26 45 29 230 300 230 230 305 188 68 75 72 72 72 72 2 4 3 5 3_1 3 a 2 4 3 1 288 316 108 112 73 64 0 0 0 127 201 54 0 0 29 58 0 34 0 0 0 ~""o" 0 61 148 0 0 130 4 9 6a 0 0 0 0 0 114 375 125 198 326 386 290 5 9 30 4 75 0 0 13 0 " 0 1 72 0 0 0 0 6 1 33 11 0 35 43 22 1 1 17 25 32 22 412 358 152 486 193 279 0 0 17 2 0 12 0 0 1 0 0 0 17 11 28 29 32 13 25 20 20 15 16 _17 18 19 20 24 30 39 36 47 35 230 240 300 20 0 280 250 73 3 73 a 5 4 3 45 ? 06 3 57 3 3 4 3 4 3 1 3 1 3 1 3 1 0 0 195 156 257 0 196 260 _ 0 "b o 85 0 0 0 " 0 0 124 0 0 0 0 8 0 289 296 76 53 131 130 21 22 1 „.. 3 4 38 35 89 103 88 103 240 340 300 47 0 400 495 54 3 09 2 06 5 3 I 3 1 3 I 5 6 7 " 6 9 1 0 76 53 101 " 52 72 59 400 330 400 350 425 425 63 60 60 59 54 59 3 1 2 1 3 1 225 188 0 4 6 79 174 0 0 118 124 101 108 0 89 0 6 7 0 0 0 0 82 17 0 0 99 98 302 585 4 4 0 619 0 0 77 45 153 33 128 171 13 33 70 59 a o 5 1 16 13 0 2 0 0 0 0 0 0 22 17 72 43 26 29 11 8 21 15 5 21 o ""0 315 2 0 28 1 124 20 29 7 50 I 1U 24 28 16 23 31 02 37 39 3 I 3 1 2 1 06 a 58 5 57 3 3 1 3 1 3 1 9 103 128 0 30 238 60 97 82 9 6 69 0 1 06 135 0 0 52 0 7 7 4 8 0 15 0 0 o" 0 16 374 163 542 275 508 180 39 4 102 26 94 405 154 9 315 " " 0 388 225 26 20 2 2 0 23 6 4 0 06 3 136 37 16 46 24 49 48 455 11 71 025 59 5 3 1 0 135 98 06 0 012 03 5 0 5 00 ,3 10 ,8 • ~i 2" ~6"5 "on o 60 ........ 3 T 31 82 0 0 "0 7 257 33 28 8 28 26 • 6 10 ,1 13 75 ooo 56 0 2 l 57 87 56 0 70 370 00 6 10 02 32 ,2 12 ,9 i a 6a 280 56 0 2 l 116 0 0 6 0 65 01 80 0 18 10 • 0 5 ,6 ' 15 loa 555 7a 5 2 l 0 267 0 0 0 097 6 752 10 06 69 ,5 25 ,6 16 101 025 70 5 2 1 81 76 120 0 20 592 02 10 7 51 02 16 16 ,8 17 105 565 72 5 2 l 312 0 0 0 0 393 283 360 39 70 52 . 1 22 ,5 ~ 1 8 " 6 5 ""335 72 "0" "2 l 0 238 0 0 0 "056 " 0 19 0 0 29 ,5 T o 21 ,5 19 loa 500 73 5 2 l 168 71 75 0 0 739 07 15 10 59 53 ,5 ,3 20 loa 565 68 5 2 l 282 0 0 0 0 363 201 01 0 10 38 00 ,3 21 ,0 1 135 705 60 5 2 l 60 131 0 0 0 183 120 6 0 1 26 i 0 6 , 9 2 176 aoo 66 5 2 l 0 105 0 0 96 306 10 152 15 09 00 ,2 10 .6 3 107 750 63 6 2 1 323 0 0 0 0 368 270 201 61 1 00 5 3 ,2 20 , 8 . . . . . a 137" 095 63 2 2 0 55 213 0 0 71 a 7 o 0 96 05 , 0 1 9 , 3 5 18a 600 63 5 2 1 56 75 151 0 0 611 122 33 38 59 51 ,3 19 ,7 6 176 600 60 5 2 1 302 0 0 0 0 381 257 38 02 19 58 , 0 17 ,2 7 177 580 59 6 2 1 226 0 0 0 0 136 62 28 20 1 1 27, 3 8 ,2 8 19a 050 63 5 2 1 112 211 0 0 0 728 12 200 19 8 5 7 , ,0 22 ,2 9 lap 600 6a «5 2 2 0 0 281 0 0 672 06 0 0 370 62 a 22 ,6 129 600 "60" _ . ™ "2" "1 136" 162 0 0 0 201 61 200 11 18 " " 4 1 , ,5 " 1 2 ,"0" 1 1 152 600 62 6 2 1 145 12" 0 0 0 181 92 200 27 13 3 9 , 7 12 9 , 12 127 060 60 5 2 1 0 36 105 78 0 226 61 233 0 27 a l , 9 \ u 0 : 13 126 000 60 2 1 0 69 0 9 0 79 263 56 100 19 35 a3, 0 16 5 10 109 095 63 5 2 1 0 102 61 0 0 827 26 03 a 0 0 0 , 8 19 ,7 15 168 660 63 5 2 1 0 220 0 0 0 618 5 159 12 9 5 a , ,8 22 ,6 16 "193" " 095" 6a 5 2 2 0 0 "270 0 0 " 5 37 ""26 0 0 "07 2 "60 , "8" 23 , 1 17 189 600 63 5 2 2 0 0 278 0 0 578 36 0 0 560 66, 9 23, 2 18 170 690 68 5 2 2 0 0 275 0 0 652 31 0 0 575 69, 3 26 ,0 21 12a 000 69 5 2 1 120 10b 0 0 0 69b 37 119 11 15 61 , 7 20, 2 22 118 565 73 6 2 1 101 191 0 0 0 352 56 335 15 20 56, 3 i a 5 23 126 650 73 6 2 1 120 152 0 0 0 539 192 3 6 0 10 70 69, 6 22 ,3 , 2" 125" "565 72 6 2 1 191 125 0 0 0 " 097 29 "126 31 2"o" ' b o , 0 19 ,0 25 142 765 73 6 2 1 0 131 163 0 0 1100 03 03 3 91 76, 0 29 ,2 I 26 138 750 7a 6 2 5 131 161 0 0 11 712 100 206 1 29 58, 0 2 1 , ,6 2 7 1«2 750 7" 6 2 5 112 188 0 0 0 768 70 018 0 0 73, 3 28 0 28 143 765 75 6 2 1 0 86 207 0 0 775 88 3 0 39 60, 3 23 3 29 ia3 765 75 6 2 1 09 118 107 0 0 890 25 22 10 30 55, 1 19, 3 . 30 2a9 800 59 5 2 5 0 210 0 0 0 571 32 002 3 ' " 3 58, 9 20, 6 31 233 800 6a 5 2 1 251 0 0 0 0 292 106 30 29 57 09, 1 13, 7 32 207 660 66 S 2 1 193 89 0 0 0 013 83 26 13 5 39, 8 i a , 9 33 208 660 65 5 2 1 0 233 0 0 0 375 2 96 7 0 0 3 , 2 10, 5 3a 259 800 67 6 2 2 0 0 297 0 0 716 28 0 0 556 67, 8 26, 0 35 308 750 60 5 2 1 328 0 0 0 . . . . . 0 510 673 365 OOO 1 1 0 101 , 3 31 , 5 IG7-V 1 05 220 07 3 3 3 0 239 0 9 257 3 0 0 16 2 2 , 5 1 0 , 02 28 150 55 3 3 1 0 60 0 133 0 133 0 22 0 0 16, 6 7, 0 3 39 305 5a 3 3 1 0 65 0 0 122 221 a 0 0 1" 2 2 , 3 9, 0 u 05 350 56 3 3 1 0 107 56 12 115 565 30 0 0 31 51 . 5 25, 3 1 5 a3 200 06 2 3 1 0 263 0 0 0 619 0 2 0 0 07 , 7 20, 6 6 35 350 56 3 3 1 0 130 80 0 318 37 6 18 33 06, 9 16, 9 7 36 220 50 3 ? 3 1 0 59 118 6' L 5 7 569 76 0 0 55 08, "o 25, 0 8 38 200 06 3 3 0 198 0 0 0 160 0 0 0 0 16, 8 7, 1 9 a5 335 50 3 3 1 0 125 0s? 0 1.29 012 01 0 1 38 05, 3 17, 0 1 0 38 2ao 05 3 3 1 0 83 126 0 0 0 5 3 111 . 0 0 33 07, 0 2 2 , 6 1 1 05 230 68 0 2 0 0 210 0 0 0 008 0 0 0 0 31, 0 13, 0 12 07 230 70 0 2 1 0 73 166 0 0 018 138 0 0 30 09 , 0 2", 6 1 3 08 290 70 3 3 1 0 215 ii 6" 0 035 0 0 0 0 0 2 , 8 27, 6 1 a 29 300 75 a 3 5 0 261 0 0 0 509 0 0 0 0 08 , a 27, 0 15 23 2ao 71 a 3 0 0 223 0 0 3D oxi2 0 0 3 0 0 2 . l 2 0 , 9 456 16 26 230 72 a 3 0 0 80 160 60 0 T 7 " 29 186 72 3 3 1 0 83 01 "0 120 18 29 290 73 0 3 2 0 0 322 0 0 1° 05 3ia 73 9 2 0 0 273 0 0 0 20 21 300 73 0 3 0 0 lob 0 53 65 21 29 300 70 3 3 0 0 0 270 56 0 22 30 200 70 0 3 0 0 206 0 58 0 "28 30 0 75 "3" 3 0 •~"0 235 ~0 " 9" 2« 22 210 75 0 3 3 0 38 169 96 0 25 29 300 75 3 3 0 0 100 165 56 0 26 30 188 75 3 2 0 0 26 20 J 0 50 27 "7 280 08 3 3 I 237 0 0 0 0 28 38 200 50 3 3 1 205 0 0 0 0 r " ' 6 8 " " 000 61 5 2 1 ~ 0 - - 7 9 "26 0 " I f 2 101 00 0 61 3 2 1 0 65 113 0 12 3 90 050 06 5 3 1 280 0 0 0 0 568 "320 308 090 ~2~9"3" 705 090 "062' 160 715 "225" 95 103 "006 007 109 TT3 " 351 558 219 136 217 398 325 321 "339 319 197 2 13 83 0 5 0 0 11 (T 0 0 "0 27 0 1 17 28,8 9.6 "0 20"20.9 "8,3 0 217 30,8 19,2 3 0 35,0 16,3 ~5 20 2T72 T279 -0 03 09,0 26,2 2 0 51,9 25.9 "0 (T~ 5-1,7""29.7" 0 50 23.7 11.5 0 21 "6,5 22.2 r 1 0 - 0 100 1 0 0 6 20 26,6 103 5 0 09 20,0 81 0 0 25 19,6 ' 81 " ' " 5 0 1 3 " "6658,9 87 0 12 127 57,1 231 120 3 32 " " . 2 ~a 55 025 30 a 3 1 207 0 5 58 300 55 0 3 1 0 190 6 _60 058 55 J 2 1 90 73 "7 70 00 0 "58 0 3"1 80 0 8 53 330 50 0 3 1 128 55 9 73 090 56 0 2 1 220 0 IT 0 97 " 0 0 0 0 0 _0 0 0 0 0 7 2 0 0 0 0 "220 TO 00 5 36 13 1 0 3 " 5 3 8 1 220 38 0 17 _0_ "1 1 10 0 0 5 "7 2 50 36 _ 2 0 10 0 10 0 2 31 35 110 27.7 08,6 52.3 38,5 21.9 37.3 00,3 31.9 06,0 "*»2.5 59,3 27.6 10 t 1 12 13 ' 10 15 60 60 80 "52 86 63 370 73 5 000 55 0 39 0 0 6_0 '300"56"" 3 390 51 0 390 59 5 "T 1 1 T 1 3 0 0 100 "65 136 70 115 129 112 29 97 0 57 121 99 70 81 0 0 0 0 0 0 0 "63 0 61 80 0 60 0 8 133 0 100 8 107 "050 129 0 0 0 127 5 155 o T62~ 0 28 168 19 130 21 20 16 101 17 52 18 19 20 21 67 "59 53 61 000 59 5 2 1 113 87 5 0 0 350 06 'J 3 1 0 06 191 65 0 025 00 3 3 1 227 0 0 0 28 "0 2557 3 3 1 2 8 6 " ' " 0 0 0 " 0 330 09 3 3 1 213 0 0 n 0 335 55 0 3 1 0 83 0 71 0 " W 163 158 163 137 278 " 3 T 0 " 000 197 367 350 676 0 8 5 3 . 0 T9„* 0 22 27.0 19 28 33,5 5 6 3 " 5 6 . 0 " 20,^ 0 01 22.8 10.^ > 0 0 33.9 1 30 30,6 i6i*> 9 1 Oft 65,5 2b ^ 0 20 31.0 10.V 17 "35 0 8 ,3 18.9-0 0 31.2 12.* 6 30 57,0 22.-S 22 60 280 59 0 3 2 23 70 000 60 6 3 1 20 60 025 73 0 3 1 2 5 7 5 3 5 0 6 3 5 3 t 26 65 335 72 0 2 1 27 1 03 565 7? t> 2 0 131 7 16 37 0 62 137 0 12 69 108 0 0 67 0 238 0 " 0 8 ? 0 218 0 0 0 1 201 120 0 0 0 0 69 36 7 3 0 59 "37" 0 10 1 6 2 28 100 565 68 29 100 565 70 1 137 095 63 2 180 6006 3 0 193 095 60 5 125 565 72 ^ 2 1 265 0 0 0 0 052 268 1 00 28 8 2 1 205 119 0 0 0 698 82 3 3 S 2 2 0 0 271 0 0 081 6 0 0 5 ;>i "0 59 " 152 0 0 57to 137 " 5 7 15 2 2 0 0 203 0 0 527 05 0 0 6 2 1 170 09 80 0 0 607 66 50 15 12^ 6 102 765 70 6 2 1 toO 66 100 0 0 780 0 7 7 2 7 102 750 7U 7 2 1 138 130 0 0 0 017 82 006 23 P 103 765 75 6 2 1 0 67 227 0 0 671 111 0 0 45 58,2 29,« 08 60,0 25,2 392 73,0 27/» 85 68,2 28,-9 580 72,9 26,« 79,9 28,-6 T2 BTTZ r r ^ r 28 80.9 25.1 «56 88,8 28,« * Building year of vessel ** Not available in 1 9 7 4 . 457 Summary S t a t i s t i c s N U M B E R O F O B S E P V A T I O N S B 2 2 3 N U M B E R O F I N P U T V A R I A B L E S e 2 0 U S I N G O B S E R V A T I O N S 1 THRU V A R I A B L E N A M E M E A N N O . 1 2 3 1 2 . 7 9 4 8 7 . 0 4 5 4 2 1 . 1 4 2 2 3 S T A N D A R D D E V I A T I O N 8 . 3 4 5 2 5 4 , 3 2 9 1 5 4 . 6 5 V A R I A N C E M I N I M U M 6 9 , 6 4 2 2 9 5 1 . 6 2 3 9 1 5 , 1 , 0 0 0 0 2 2 , 0 0 0 1 5 0 . 0 0 3 0 , 0 0 0 2 , 0 0 0 0 2 , 0 0 0 0 1 , 0 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 2 4 , 0 0 0 0 . 0 0 . 0 0 . 0 0 . 0 7 , 7 0 0 0 0 . 0 M A X I M U M ' 3 5 , 0 0 0 3 0 8 , 0 0 8 0 0 , 0 0 7 5 , 0 0 0 7 , 0 0 0 0 3 , 0 0 0 0 5 , 0 0 0 0 3 4 2 , 0 0 3 1 6 , 0 0 3 2 2 , 0 0 1 3 3 , 0 0 1 6 2 , 0 0 1 1 0 0 , 0 6 7 3 , 0 0 7 7 5 . 0 0 4 5 6 7 8 9 6 1 . 9 4 2 4 . 3 8 5 7 _ 2 . 5 1 5 7 1 . 5 0 6 7 8 3 . 0 7 6 9 2 . 9 0 6 9 . 5 7 9 0 1 , 0 8 8 1 0 . 5 0 0 8 8 1 . 1 0 6 4 9 8 , 5 4 8 7 5 , 6 9 4 9 1 , 7 5 8 1 . 1 8 3 9 0 , 2 5 0 8 8 1 , 2 2 4 1 9 7 1 1 , 7 5 7 2 9 . 6 10 1 1 12 1 3 14 15 0 8 , 7 0 0 7 . 1 7 0 4 t 8 . 3 7 7 39 3 . 9 8 6 7 . 9 6 0 7 6 . 3 9 5 7 8 , 9 1 9 2 1 . 2 0 7 3 7 . 8 8 4 2 2 2 , 3 " 8 3 . 8 9 5 1 3 7 . 4 8 6 2 2 8 , 2 4 4 9 , 7 5 1 4 3 5 . 2 4 9 4 5 8 , 7 0 3 8 . 4 1 8 8 9 9 , 16 17 18 "19 1 0 . 3 5 4 0 6 . 5 4 3 3 4 . 8 8 0 1 3 . 2 2 0 3 2 . 2 8 3 8 6 . 7 5 0 1 8 . 3 1 8 8 . 2 3 4 1 1 0 4 2 . 2 7 5 2 5 . 6 3 3 5 . 5 5 6 7 . 8 0 1 4 4 0 , 0 0 5 7 5 , 0 0 1 0 1 , 3 0 3 1 , 5 0 0 458 Table 3 Price Indices. Year Input prices Output Prices Cod Haddock Saithe Redfish Other 1974 1975 1 976 1977 1 .00 1.41 2.01 2.62 1 .00 1 .42 2.19 2.98 1 .00 1.19 1 .74 2.68 1 .00 1 .22 1 .72 2.52 1 .00 1 .25 2.15 2.97 1 .00 1.21 1 .74 2.67 Source: Nat ional Economic Institute (unpublished). 459 Section 5.4: Data. 1. Technical Transformation C o e f f i c i e n t s . 1976 1 978 1979 1 980 c( 1 , 1 ) 0.37 NA 0.38 0.42 C(1 ,2) .435 NA .41 1 0.44 c(1 ,3) 0.17 NA 0.17 0.17 c(1,4) NA • 258 .248 .249 c(2, 1 ) NA • 345 .365 .406 c(2,2) NA NA NA 0.44 c(2,3) NA 0 .17 NA 0.17 c(2,4) NA • 258 .248 .249 c(3, 1 ) NA 418 .438 .485 c(3,2) NA 6 .45 NA 0.45 c(3,3) NA 0 .17 NA 0. 17 c(3,4) NA 258 .248 .249 Source: National Economic Institute 460 2. Quantity of Fi n a l Products, Process Allocation Parameters, Fin a l Product Prices and Prices of Landed Catch. Units: y ( i ) = thousand tonnes p ( i , j ) = Ikr./kg. p(i) = Ikr./kg. Cod. Year y( 1 ) b d ,D b(1,2) b(1,3) b d ,4) P(1 ,1) p(1,2) p( 1,3) P(1,4) 1974 226. 8 0. 508 0.484 0.007 0.001 1 . 589 1 .734 3.715 0.388 1 975 262. 2 0. 522 0.453 0.025 0.000 2. 252 2. 184 5.225 0.343 1 976 279. 1 0. 537 0.386 0.077 0.000 3. 1 75 2.361 8.319 0.630 1 977 323. 6 0. 556 0.371 0.073 0.000 4. 467 2.932 8.820 0.837 1 978 304. 4 0. 639 0.343 0.018 0.000 5. 863 3.958 1 1 .92 1 .024 1979 339. 5 0. 597 0.339 0.064 0.000 8. 641 6.337 1 5.40 1 . 145 1980 405. 0 0. 496 0.335 0. 1 48 0.001 1 2 .69 10.97 25.33 2.038 1981 442. 5 0. 372 0.399 0.228 0.001 20 .66 19.01 39.53 3.327 1 982 367. 3 0. 357 0.490 0.1 53 0.000 31 .39 28.80 64.43 2.910 Haddock. Year y( 1 ) b(2,1) b(2,2) b(2,3) b(2,4) P(1 ,1) p(1,2) p( 1,3) p(1,4) 1 974 29 .5 0. 986 0. 000 0. 014 0 .000 1 .671 0. 000 2.859 0. 388 1 975 31 .5 0. 984 0. 000 0. 016 0 .000 2.247 0. 000 4.021 0. 343 1 976 29 .8 0. 966 0. 000 0. 034 0 .000 3. 1 59 0. 000 6.093 0. 630 1 977 30 .4 0. 954 0. 000 0. 046 0 .000 4.471 0. 000 6.633 0. 837 1 978 33 .2 0. 982 0. 000 0. 018 0 .000 6. 1 47 0. 000 8.513 1 . 024 1 979 41 .5 0. 976 0. 000 0. 024 0 .000 9.516 0. 000 10.74 1 . 255 1 980 31 . 1 0. 919 0. 000 0. 081 0 .000 14.15 0. 000 17.22 2. 038 1 981 50 .7 0. 838 0. 000 0. 162 0 .000 18.91 0. 000 26.88 3. 327 1 982 55 .7 0. 894 0. 000 0. 106 0 .000 32.43 0. 000 41 .43 2. 910 Saithe • • Year y(3) b(3,1 ) b(3,2) b(3,3) b(3,4) p(3, 1 ) p(3,2) p(3,3) p(3,4) 1 974 59 .5 0. 733 0. 264 0. 003 0 .000 0.967 1 . 100 2.916 0. 388 1 975 60 .0 0. 750 0. 238 0. 012 0 .000 1 .278 1 . 492 4.101 0. 343 1 976 52 .8 0. 678 0. 250 0. 072 0 .000 1.711 1 . 859 6.215 0. 630 1977 43 .4 0. 643 0. 260 0. 097 0 .000 2.628 2. 571 6.452 0. 837 1978 39 .0 0. 886 0. 274 0. 015 0 .000 3.901 5. 094 8.578 1 . 024 1979 51 .5 0. 730 0. 1 77 0. 093 0 .000 5.527 7. 534 10.74 1 . 255 1980 44 .8 0. 603 0. 163 0. 234 0 .000 7.207 1 1 .66 17.22 2. 038 1981 49 .5 0. 489 0. 224 0. 287 0 .000 12.95 1 1 .99 26.88 3. 327 1 982 60 .2 0. 515 0. 188 0. 297 0 .000 18.75 20 .07 41 .43 2. 910 Sources: Icelandic Fisheries Association National Economic Institute 461 S e c t i o n 6 . 3 . 1 : D a t a . T h e f o l l o w i n g a r e t h e d a t a u s e d t o e s t i m a t e t h e f i s h i n g m o r t a l i t y a n d f i s h i n g t i m e e q u a t i o n s f o r t h e d e e p - s e a t r a w l e r s a n d t h e m u l t i p u r p o s e f i s h i n g f l e e t . T a b l e 1 . D e e p - s e a T r a w l e r s . D a t a p e r i o d 1 9 7 6 - 8 0 . S o u r c e : I c e l a n d i c F i s h e r i e s A s s o c i a t i o n ( V a r i o u s p u b l i c a t i o n s , e s p e c i a l l y A e g i r , 1 9 7 7 - 1 9 8 1 ) T h e M a r i n e R e s e a r c h I n s t i t u t e . ( U n p u b l i s h e d m a t e r i a l o n f i s h i n g m o r t a l i t i e s d u r i n g 1 9 7 6 - 8 0 ) . V a r i a b l e c o d e : ( 1 ) T o n n a g e o f v e s s e l . ( 2 ) T o t a l n u m b e r o f t r i p s i n t h e y e a r . ( 3 ) T o t a l d e m e r s a l c a t c h d u r i n g y e a r i n m e t r i c t o n s . ( 4 ) t o t a l o p e r a t i n g d a y s d u r i n g y e a r . ( 5 ) T o t a l h o u r s t r a w l i n g d u r i n g y e a r . ( 6 ) E s t i m a t e d v e s s e l s p e c i f i c d e m e r s a l f i s h s t o c k s ( t o n s c a u g h t p e r h o u r t r a w l e d ) . ( 7 ) E s t i m a t e d d e m e r s a l f i s h i n g m o r t a l i t y . (1) (2) (3) ( 4 ) - (5) (6) (7) 4 6 2 . 2 9 . 2 2 8 8 . 3 1 0 . 3 5 1 9 . 0 . 6 5 0 2 0 . 9 9 0 7 0 E - 0 3 4 5 1 . 2 5 . 2 5 6 4 . 3 3 7 , 4 1 0 3 , 0 . 6 2 4 9 0 . 1 1 1 0 2 E - 0 2 2 9 9 . 3 0 . 2 6 8 5 . 3 4 2 . 4 1 0 3 . 0 . 6 5 4 4 0 . 1 1 6 2 6 E - 0 2 4 5 1 . 2 4 . 2 8 2 2 . 2 8 3 . 3 1 7 6 . 0 . 3 8 8 5 0 . 1 2 2 1 9 E - 0 2 2 9 7 . 3 0 . 2 4 5 1 . 3 5 6 . 4 2 1 7 , 0 . 5 8 1 2 0 . 1 0 6 1 3 E - 0 2 4 5 1 . 2 2 . 2 6 1 0 . 2 8 1 . 3 1 9 6 . 0 . 8 1 6 6 0 . 1 1 3 0 1 E - 0 2 2 9 9 . 3 1 . 3 1 0 4 . 3 5 0 . 3 9 3 3 . 0 . 7 8 9 2 0 . 1 3 4 4 0 E - 0 2 4 9 1 . 1 4 . 1 2 7 9 . 1 6 7 . 1 9 4 4 . 0 , 6 5 7 9 0 . 5 5 3 8 1 E - 0 3 2 9 8 . 3 0 , 3 0 9 6 . 3 4 5 , 4 0 5 6 . 0 . 7 6 3 3 0 . 1 3 4 0 6 E - 0 2 4 6 2 . 4 1 . 3 0 4 9 . 3 5 7 . 4 6 4 2 . 0 . 6 5 6 8 0 . 1 3 2 0 2 E - 0 2 4 0 7 . 3 8 , 2 7 1 0 . 3 3 5 , 4 0 8 1 . 0 . 6 6 4 1 0 . 1 1 7 3 4 E - 0 2 3 8 5 . 2 4 . 1 9 9 9 . 2 4 5 . 2 7 7 2 . 0 , 7 2 1 1 0 . 8 6 5 5 7 E - 0 3 5 0 0 . 2 8 . 3 4 3 0 . 3 2 6 . 4 2 0 8 . 0 . 8 1 5 1 0 . 1 4 8 5 2 E - 0 2 4 6 1 . 3 2 . 2 5 6 9 . 3 4 9 . 4 2 4 4 . 0 , 6 0 5 3 0 . 1 1 1 2 4 E - 0 2 3 1 4 . 2 8 . 2 6 0 9 . 3 2 3 . 4 1 7 8 . 0 . 6 2 4 5 0 . 1 1 2 9 7 E - 0 2 4 6 1 . 2 8 . 1 9 8 1 . 3 1 4 , 3 8 4 7 . 0 . 5 1 4 9 0 . 8 5 7 7 7 E - 0 3 3 3 1 . 2 7 . 1 7 9 2 . 2 8 7 , 3 3 7 0 . 0 . 5 3 1 8 0 . 7 7 5 9 4 E - 0 3 4 5 1 . 3 4 . 2 7 0 8 . 3 4 8 . 4 5 4 2 . 0 . 5 9 6 2 0 . 1 1 7 2 6 E - 0 2 3 2 8 , 3 0 . 1 9 7 7 . 3 1 0 . 3 5 6 0 . 0 . 5 5 5 3 0 . 8 5 6 0 4 E - 0 3 4 6 2 . 3 9 . 2 7 2 7 . 3 3 4 , 4 2 6 4 . 0 . 6 3 9 5 0 . 1 1 8 0 8 E - 0 2 3 2 8 . 3 0 . 2 2 1 3 . 3 2 7 , 4 0 0 0 . 0 . 5 5 3 3 0 . 9 5 8 2 3 E - 0 3 4 6 2 . 2 9 . 1 9 1 4 . 3 1 6 , 3 6 4 4 , 0 , 5 2 5 2 0 . 8 2 8 7 6 E - 0 3 2 9 7 . 2 7 . 2 1 3 9 . 3 1 2 , 3 8 6 9 . 0 . 5 5 2 9 0 . 9 2 6 1 9 E - 0 3 7 2 3 . 1 7 . 3 4 4 2 . 2 6 3 , 3 1 1 8 . 1 , 1 0 3 9 0 . 1 4 9 0 4 E - 0 2 7 2 6 . 2 1 . 3 9 9 8 , 3 2 6 , 3 7 1 . 3 . 1 . 0 7 6 8 0 . 1 7 3 1 1 E - 0 2 9 6 9 . 2 1 . 4 0 1 4 . 3 3 1 , 4 2 5 3 . 0 , 9 4 3 8 0 . 1 7 3 8 1 E - 0 2 9 6 9 . 2 2 . 3 9 8 3 , 3 4 1 , 4 1 1 1 . 0 . 9 6 8 9 0 , 1 7 2 4 6 E - 0 2 462 9 6 9 . 2 2 . 7 4 2 , 1 4 . 7 4 2 . 1 8 . 7 4 2 . 1 6 . 7 3 1 . 1 3 . 9 4 2 . 1 8 . 9 4 1 . 2 5 . 9 4 1 . 2 4 . 7 8 1 . 2 5 . 7 8 1 . 2 6 , 4 6 2 . 2 1 . 4 5 1 . 2 5 . 2 9 9 . 2 7 . 4 8 8 . 2 3 . 4 5 1 . 2 4 . 2 9 7 , 2 3 . 4 5 1 . 2 3 . 3 0 2 . 1 5 . 2 9 9 . 3 2 . 3 3 9 . 8 . 3 1 2 . 3 3 . 4 9 1 . 2 0 . 3 7 5 . 4 6 2 . 2 9 . 3 8 5 . 2 1 . 5 0 0 . 2 8 . 4 6 2 . 2 6 . 4 6 1 . 2 4 . 3 0 0 . 2 1 . 4 6 1 . 3 1 . 2 9 9 . 2 6 . 3 3 1 . 3 0 , 4 6 1 . 2 9 . 4 5 3 . 7 . 4 5 1 . 3 3 . 3 2 8 . 3 1 . 4 8 7 . 1 2 . 4 6 2 . 3 0 . 3 2 8 , 3 1 . 7 2 3 . 2 1 . 7 2 6 , 2 2 . 9 6 9 . 2 3 . 9 6 9 , 1 9 . 9 6 9 . 2 0 . 7 4 2 . 2 1 . 7 4 2 . 2 0 . 7 4 1 . 2 1 . 7 4 2 . 2 0 . 7 3 1 . 1 5 . 9 4 2 . 1 4 . 9 4 1 . 2 5 . 9 4 1 . 2 4 . 7 8 1 . 2 4 . 7 8 1 . 2 3 . 4 8 8 . n n 4 5 1 . 2 6 . 2 9 9 , 2 6 . 4 8 8 . -> n 4 5 1 . 2 2 • 2 9 7 . 2 3 . 4 5 1 . 1 3 . 3 9 0 6 , 3 7 2 . 2 1 4 2 . 2 2 5 . 2 7 9 9 . 2 7 5 . 2 6 9 3 . 2 5 3 . 1 6 0 7 , 1 8 9 . 3 2 9 6 . 2 8 5 . 3 9 0 8 . 3 6 2 . 3 3 6 5 . 3 4 8 . 3 9 4 5 . 3 6 8 . 3 9 0 6 . 3 7 2 . 2 2 9 4 . 2 5 7 . 2 4 1 9 . 3 3 1 . 2 5 7 0 . 3 0 0 . 2 4 4 5 . 2 6 2 . 2 8 2 2 . 3 3 4 , 2 0 9 8 . 3 0 0 . 2 6 1 6 . 3 2 3 . 1 6 8 6 . 1 8 5 . 3 3 6 0 . 3 5 3 . 5 4 5 . 1 0 3 . 3 3 3 5 . 3 4 7 . 1 9 0 1 . 2 4 3 . 2 1 8 8 . 2 0 4 . 2 8 3 8 . 2 7 7 . 2 4 9 2 . 2 5 4 . 3 6 7 0 . 3 3 5 . 3 0 3 1 . 3 6 2 . 2 3 5 5 . 2 7 6 . 1 4 5 1 . 2 2 7 . 2 8 6 6 , 3 2 2 . 2 1 9 5 . 2 7 2 . 1 8 0 7 , 2 8 9 . 2 3 5 1 . 3 1 4 . 5 2 8 . 6 8 . 2 7 3 3 . 3 2 4 . 2 3 7 9 . 3 0 5 , 7 4 4 . 1 0 4 . 2 8 4 9 . 3 0 3 . 2 3 5 7 . 3 2 9 . 4 6 2 5 . 3 2 5 . 4 5 9 5 . 3 2 8 . 4 2 8 3 . 3 5 7 . 3 8 7 3 . 3 0 2 . 3 6 4 8 . 3 2 0 . 3 5 7 2 . 3 3 7 . 3 4 8 1 . 2 9 9 . 3 1 6 1 . 2 9 2 . 3 1 4 5 . 3 1 5 . 2 2 7 2 . 2 1 3 . 4 6 8 2 . 3 6 8 . 4 1 6 7 . 3 6 4 . 4 1 3 1 . 3 5 4 , 3 5 6 5 . 3 3 1 . 1 9 5 5 . 2 7 3 . 2 2 9 8 . 3 5 6 . 2 9 5 9 . 2 9 8 . 2 2 7 1 . 2 7 2 . 2 2 3 8 . 2 9 3 . 2 0 9 6 . 2 9 6 , 1 4 5 5 . 1 8 0 . 4 1 0 3 , 0 . 9 5 2 0 2 6 9 0 . 0 . 7 9 6 3 3 6 7 2 . 0 , 7 6 2 3 2 8 2 8 . 0 . 9 5 2 3 2 0 7 0 . 0 , 7 7 6 3 3 4 9 5 . 0 . 9 4 3 1 4 4 1 6 . 0 . 8 8 5 0 4 5 5 2 . 0 . 7 3 9 2 4 2 3 1 . 0 . 9 3 2 4 4 1 0 3 . 0 . 9 5 2 0 3 0 7 3 . 0 , 7 4 6 5 3 9 1 9 . 0 . 6 1 7 2 3 4 9 0 . 0 . 7 3 6 4 2 9 7 6 . 0 . 8 2 1 6 3 7 2 5 . 0 . 7 5 7 6 3 4 4 1 . 0 . 6 0 9 7 3 8 1 6 . 0 . 6 8 5 5 2 0 8 0 . 0 . 8 1 0 6 3 8 8 9 . 0 , 8 6 4 0 1 1 1 5 . 0 , 4 8 8 8 3 7 6 9 . 0 , 8 8 4 9 2 2 8 7 , 0 . 8 3 1 2 2 4 3 9 . 0 . 8 9 7 1 3 1 2 2 . 0 . 9 0 9 0 2 8 1 1 . 0 . 8 8 6 5 4 1 6 6 . 0 . 8 8 0 9 4 3 3 0 . 0 . 7 0 0 0 3 2 0 1 . 0 , 7 3 5 7 2 5 3 2 , 0 . 5 7 3 1 4 1 5 3 . 0 , 6 9 0 1 3 7 3 7 . 0 . 5 8 7 4 3 4 5 9 . 0 . 5 2 2 4 4 1 3 8 . 0 . 5 6 8 1 8 6 7 . 0 . 6 0 9 0 4 3 0 3 . 0 . 6 3 5 1 3 7 0 2 . 0 . 6 4 2 6 1 2 9 3 . 0 . 5 7 5 4 3 9 5 2 . 0 . 7 2 0 9 3 9 7 0 . 0 . 5 9 3 7 4 1 1 0 . 1 . 1 2 5 3 4 1 0 6 . 1 . 1 1 9 1 4 0 6 1 . 1 . 0 5 4 7 4 0 4 0 . 0 . 9 5 8 7 4 1 2 5 . 0 . 8 8 4 4 3 6 4 3 . 0 . 9 8 0 5 4 0 1 6 . 0 . 8 6 6 8 3 5 6 1 . 0 , 8 8 7 7 3 8 3 7 . 0 . 8 1 9 7 2 4 6 9 . 0 . 9 2 0 2 2 6 7 3 . 0 . 9 3 6 0 4 6 6 6 . 1 . 0 0 3 4 4 6 7 3 . 0 , 8 9 1 7 4 4 4 5 . 0 , 9 2 9 4 3 6 6 7 . 0 . 9 7 2 2 3 3 5 7 . 0 . 5 8 2 4 4 3 2 5 . 0 . 5 3 1 3 3 4 9 ' 2 . 0 . 8 4 7 4 3 2 8 1 . 0 . 6 9 2 2 3 2 3 7 . 0 . 6 9 1 4 3 2 3 8 . 0 . 6 4 7 3 2 1 2 3 . 0 . 6 8 5 4 0 . 1 6 9 1 3 E - 0 2 0 . 9 2 7 4 9 E - 0 3 0 . 1 2 1 2 0 E - 0 2 0 . 1 1 6 6 1 E - 0 2 0 . 6 9 5 8 3 E - 0 3 0 , 1 4 2 7 2 E - 0 2 0 . 1 6 9 2 2 E - 0 2 0 . 1 4 5 7 0 E - 0 2 0 . 1 7 0 8 2 E - 0 2 0 . 1 6 9 1 3 E - 0 2 0 . 1 0 0 2 5 E - 0 2 0 . 1 0 5 7 1 E - 0 2 0 . 1 1 2 3 1 E - 0 2 0 . 1 0 6 8 5 E - 0 2 0 . 1 2 3 3 2 E - 0 2 0 . 9 1 6 8 3 E - 0 3 0 . 1 1 4 3 2 E - 0 2 0 . 7 3 6 7 8 E - 0 3 0 . 1 4 6 8 3 E - 0 2 0 . 2 3 8 1 6 E - 0 3 0 . 1 4 5 7 4 E - 0 2 0 . 8 3 0 7 4 E - 0 3 0 . 9 5 6 1 6 E - 0 3 0 . 1 2 4 0 2 E - 0 2 0 . 1 0 8 9 0 E - 0 2 0 . 1 6 0 3 8 E - 0 2 0 . 1 3 2 4 5 E - 0 2 0 , 1 0 2 9 1 E - 0 2 0 . 6 3 4 0 9 E - 0 3 0 . 1 2 5 2 4 E - 0 2 0 . 9 5 9 2 1 E - 0 3 0 . 7 8 9 6 6 E - 0 3 0 . 1 0 2 7 4 E - 0 2 0 . 2 3 0 7 4 E - 0 3 0 . 1 1 9 4 3 E - 0 2 0 . 1 0 3 9 6 E - 0 2 0 . 3 2 5 1 3 E - 0 3 0 . 1 2 4 5 0 E - 0 2 0 . 1 0 3 0 0 E - 0 2 0 . 2 0 2 1 1 E - 0 2 0 . 2 0 0 8 0 E - 0 2 0 . 1 8 7 1 7 E - 0 2 0 . 1 6 9 2 5 E - 0 2 0 . 1 5 9 4 2 E - 0 2 0 . 1 5 6 1 0 E - 0 2 0 . 1 5 2 1 2 E - 0 2 0 . 1 3 8 1 4 E - 0 2 0 . 1 3 7 4 4 E - 0 2 0 . 9 9 2 8 6 E - 0 3 0 . 1 0 9 3 4 E - 0 2 0 . 2 0 4 6 0 E - 0 2 0 . 1 8 2 1 0 E - 0 2 0 . 1 8 0 5 2 E - 0 2 0 . 1 5 5 7 9 E - 0 2 0 . 7 1 5 5 3 E - 0 3 0 . 8 4 1 0 7 E - 0 3 0 . 1 0 8 3 0 E - 0 2 0 . 8 3 1 1 9 E - 0 3 0 . 8 1 9 1 1 E - 0 3 0 . 7 6 7 1 4 E - 0 3 0 . 5 3 2 5 3 E - 0 3 302, 26. 2755. 291 . 378, 12. 1262, 138. 442. 30. 3383. 348. 299 . 35. 3991 . 357. 312. 25. 2530. 248. 375 . 39. 4039. 278 . 462. 24. 2949. 278. 500. 26. 3425. 305. 462. 22. 1975. 289. 461 . 16 . 1709. 183. 461 . 26. 2371 . 307. 314 . 18. 1875. 227. 461 . 25. 2260. 267. 299. 31 . 2616. 293. 331 . 29. 1922. 289. 461 . 29 . 2398. 307. 453. 29 . 2253. 294 . 451 . 28 . 2353. 268. 318. 34 . 1900. 301 . 487. 38. 3030. 317. 462. 28. 2473. 300. 723. 15. 3413. 205. 969. 20. 3509. 292. 969. 24 . 4034 . 355. 969. 23. 4449. 338. 742 . 20 . 2826. 275. 741 . 21 . 3088. 284. 742. 19 . 2757. 275. 731 . 15. 2534 . 212. 942 . 15. 2245. 4. ^ . £- • 941 . 25. 4529. 348. 941 . 24 . 4163. 332. 781 . 26. 4234 . 368. 781 . 26. 3829. 362. 488. 18. 1666. 226. 451 . 24. 2545. 317. 299. 19. 2149. 193. 451. 23. 2774 . 293. 451 . 18. 2323. 203, 373. 25. 3217. 288. 442 . 31 . 4128, 326. 299; 36. 4649, 350. 312. 32 . 3179. 265. 491 . 11 . 1462. 131 . 442. 30. 3740. 336. 375 . 3 7 . 3937 . 332 . 500 . 24 . 3558. 298. 462. 25 . 3005. 323. 461 . 22 . 2445. 264. 400. 10. 718. 130. 461 . 33. 2894 . 348. 299. 29. 2689. 234 . 331 . 28. 1966. 264. 451 . 30. 2773. 289. 328. 28. 1848. 262. 462 . 35. 2815 . 324 . 723. 9. 2288. 124. 726 . 15 . 4060. 969. 22. 4735. 312. 969. 20. 4278. 278. 463 3197. 0,8617 0.10083E- 02 1629 . 0.7747 0. 46189E- 03 3972. 0 . 8517 0. 12382E- 02 4030. 0.9903 0. 14607E- 02 2666. 0.9490 0.92598E- 03 3492. 1.1566 0. 14783E- 02 3492. 0.8445 0. 10793E- 02 3886. 0.8814 0. 12536E- 02 3376. 0.5850 0. 72285E- 03 2238. 0.7636 0. 62549E- 03 3941. 0.6016 0.86779E- 03 2889. 0.6490 0. 68625E- 03 3192, 0.7080 0. 82716E- 03 3909. 0.6692 0.95746E- 03 3638. 0.5283 0.70345E- 03 4093. 0,5859 0.87767E- 03 3826. 0.5889 0.82460E- 03 3536. 0.6654 0. 86120E- 03 3518. 0.5401 0. 69540E- 03 4231 . 0.7161 0. 11090E- 02 4067. 0.6081 0. 90512E- 03 2383. 1.4322 0. 12492E- 02 3341 , 1.0503 0. 12843E- 02 4342. 0.9291 0 , 14764E- 02 3833, 1.1607 0. 16283E- 02 3591 . 0.7870 0. 10343E- 02 3536, 0.8733 0.11302E- 02 3232. 0.8400 0. 10091E- 02 2536. 0.9992 0. 92744E- 03 2529. 0.8877 0.82167E- 03 4569. 0.9912 0. 16576E- 02 4436. 0.9385 0. 15237E- 02 4700. 0.9009 0. 15496E- 02 4299, 0.8907 0. 14014E- 02 3333. 0.4998 0. 52146E- 03 3726. 0,6830 0.79653E- 03 2384. 0.9014 0. 67264E- 03 3273. 0.8475 0. 86826E- 03 2431. 0.9556 0.72710E- 03 3276. 0.9820 0. 10069E- 02 3722. 1.1091 0. 12921E- 02 3985. 1.1666 0. 14551E- 02 2932. 1.0842 0. 99503E- 03 1278. 1.1440 o. 45761E- 03 4387. 0.8525 0. 11706E- 02 3735. 1.0541 0. 12323E- 02 3720. 0.9565 0. 11137E- 02 3785. 0.7939 0.94057E- 03 3187. 0.7672 0. 76528E- 03 1572. 0.4567 0. 22473E- 03 4377. 0.6612 0. 90582E- 03 3549. 0.7577 0 .84166E-03 3149. 0.6243 0. 61536E- 03 3703 . 0.7478 0.86795E- 03 2838. 0.6512 0. 57842E- 03 4127. 0.6821 0. 83110E- 03 1529. 1.4964 0.71614E- 03 2744. 1,4796 0 . 12708E- 02 4164 . 1.1371 0. 14821E- 02 3534. 1.2105 0 . 13390E- 02 464 9 6 9 . 2 5 . 7 4 2 . 1 4 . 7 4 1 . 2 0 . 7 4 2 . 1 4 . 7 3 1 . 11 . 9 4 2 . 1 4 . 9 4 1 . 2 6 . 9 4 1 . 2 6 . 7 8 1 . 2 5 . 7 8 1 . 1 7 . 4 8 8 . 1 9 . 4 5 1 . 2 4 . 2 9 9 . 2 6 . 4 5 1 . 1 8 . 3 7 8 , 3 1 . 4 4 2 . 3 5 , 2 9 9 . 3 9 . 3 1 2 . 3 2 . 4 0 7 . 2 8 . 4 9 3 . 1 5 , 4 4 2 . 3 4 . 3 7 5 . 4 4 . 5 0 0 . 2 2 . 4 6 2 . 2 7 . 4 6 1 . 2 6 . 3 0 0 . 2 3 . 4 9 9 . 1 9 , 3 0 0 . 2 9 . 4 6 1 . 2 9 . 2 9 9 . 2 9 . 3 3 1 . 2 6 . 4 6 1 . 3 0 . 4 5 1 , 3 4 . 3 2 8 . 1 5 . 4 8 7 . 3 7 . 4 6 2 . 3 1 . 3 2 8 , 2 8 . 7 2 3 . 1 5 . 7 2 6 . 1 4 . 9 6 9 , 2 5 . 9 6 9 . 2 0 . 9 6 9 . 2 5 . 7 4 2 . 9 . 7 4 1 . 1 4 . 7 4 2 . 1 4 . 7 3 1 . 1 4 . 9 4 2 . 2 1 . 9 4 1 . 2 5 . 9 4 1 . 2 5 . 7 8 1 . 2 0 . 7 8 1 . 2 6 . 5 6 6 0 . 3 4 3 . 2 8 1 5 . 2 0 1 . 3 7 4 8 , 2 7 2 , 2 6 9 0 . 2 2 5 . 2 6 5 9 . 1 5 7 . 2 9 1 4 . 2 1 0 . 4 9 4 5 . 3 5 3 . 5 5 3 4 . 3 6 1 . 4 6 6 7 . 3 4 7 . 2 9 3 3 . 2 4 6 . 2 2 1 2 , 2 3 1 . 2 7 8 1 . 2 9 1 . 3 1 3 7 . 2 6 1 . 2 2 1 2 . 2 4 3 . 4 3 6 2 . 3 2 6 . 4 7 5 8 . 3 6 1 . 5 3 9 5 . 3 4 4 , 3 6 2 8 . 2 8 1 . 3 3 9 8 . 3 0 9 . 2 4 1 9 . 1 7 6 . 4 2 6 5 . 3 5 0 . 5 1 0 1 . 3 5 6 . 3 3 6 0 . 2 6 3 . 3 5 6 2 . 3 4 2 . 3 2 4 8 . 2 9 9 . 2 3 7 0 . 2 7 1 . 3 2 9 7 . 2 3 7 . 3 0 9 6 . 3 1 3 . 3 0 7 1 . 3 0 4 . 2 7 5 1 . 2 7 3 . 1 8 6 6 . 2 7 1 . 3 5 8 1 . 3 0 2 . 3 7 4 3 . 3 2 4 . 1 0 9 1 . 1 5 6 . 3 8 9 0 . 3 4 0 . 2 7 2 8 . 2 7 2 , 3 1 7 9 . 2 8 5 . 4 1 0 9 . 2 0 3 . 3 5 3 0 , 1 9 4 . 4 9 9 2 . 3 3 2 . 3 9 5 5 . 2 4 9 . 5 3 7 8 . 3 1 7 . 1 9 7 9 . 1 2 5 . 2 2 3 2 . 1 9 5 . 2 4 3 7 . 1 9 7 . 2 7 1 3 . 1 9 0 . 4 1 5 8 . 2 8 9 . 5 1 1 4 . 3 4 2 . 5 3 1 5 . 3 2 5 . 4 1 3 9 . 2 7 7 . 4 7 3 1 . 3 5 8 . 3 7 9 3 . 1 . 4 9 2 2 2 5 1 1 . 1 . 1 2 1 1 3 3 1 6 . 1 . 1 3 0 3 2 5 4 4 . 1 . 0 5 7 4 1 9 7 9 . 1 . 3 4 3 6 2 5 2 5 . 1 . 1 5 4 1 4 4 1 6 . 1 . 1 1 9 8 4 5 0 8 . 1 . 2 2 7 6 4 1 5 7 . 1 . 1 2 2 7 3 0 2 0 . 0 . 9 7 1 2 2 5 1 8 , 0 . 8 7 8 5 3 0 6 8 , 0 . 9 0 6 5 2 8 7 5 . 1 . 0 9 1 1 2 4 5 2 . 0 . 9 0 2 1 3 9 5 1 . 1 . 1 0 4 0 3 8 7 5 . 1 . 2 2 7 9 3 9 3 8 . 1 . 3 7 0 0 3 0 2 4 . 1 . 1 9 9 7 3 5 4 0 . 0 . 9 5 9 9 2 0 9 7 . 1 . 1 5 3 6 4 5 1 8 . 0 . 9 4 4 0 . 4 2 1 5 . 1 . 2 1 0 2 3 0 4 3 . 1 . 1 0 4 2 3 7 8 7 . 0 . 9 4 0 6 3 2 1 1 . 1 . 0 1 1 5 2 7 0 8 . 0 . 8 7 5 2 2 9 1 7 . 1 . 1 3 0 3 3 3 5 4 . 0 . 9 2 3 1 3 6 8 3 . 0 . 8 3 3 8 3 2 5 6 , 0 . 8 4 4 9 3 0 0 9 . 0 . 6 2 0 1 3 3 7 9 . 1 . 0 5 9 8 4 1 7 1 . 0 . 8 9 7 4 1 7 2 3 . 0 . 6 3 3 2 4 3 0 2 . 0 , 9 0 4 2 3 3 2 3 . 0 . 8 2 0 9 3 2 2 6 . 0 . 9 8 5 4 2 4 5 5 . 1 . 6 7 3 7 2 3 7 2 . 1 . 4 8 8 2 4 0 4 1 , 1 . 2 3 5 3 3 0 3 7 . 1 . 3 0 2 3 3 5 2 1 . 1 . 5 2 7 4 1 6 1 6 . 1 . 2 2 4 6 2 1 3 0 . 1 . 0 4 7 9 2 0 2 8 . 1 , 2 0 1 7 2 3 7 5 . 1 . 1 4 2 3 3 4 0 2 . 1 . 2 2 2 2 4 1 3 7 . 1 . 2 3 6 2 4 0 2 3 . 1 . 3 2 1 2 3 3 8 1 . 1 . 2 2 4 2 3 9 8 6 . 1 . 1 8 6 9 0 , 1 7 7 1 6 E - 0 2 0 . 8 8 1 1 0 E - 0 3 0 . 1 1 7 3 1 E - 0 2 0 . 8 4 1 9 7 E - 0 3 0 . 8 3 2 2 7 E - 0 3 0 . 9 1 2 0 8 E - 0 3 0 . 1 5 4 7 8 E - 0 2 0 . 1 7 3 2 1 E - 0 2 0 . 1 4 6 0 8 E - 0 2 0 . 9 1 8 0 3 E - 0 3 0 . 7 3 6 6 0 E - 0 3 0 . 9 2 6 0 7 E - 0 3 0 . 1 0 4 4 6 E - 0 2 0 . 7 3 6 6 0 E - 0 3 0 . 1 4 5 2 5 E - 0 2 0 . 1 5 8 4 4 E - 0 2 0 , 1 7 9 6 5 E - 0 2 0 . 1 2 0 8 1 E - 0 2 0 . 1 1 3 1 5 E - 0 2 0 . 8 0 5 5 3 E - 0 3 0 . 1 4 2 0 2 E - 0 2 0 . 1 6 9 8 6 E - 0 2 0 . 1 1 1 8 9 E - 0 2 0 . U 8 6 1 E - 0 2 0 . 1 0 8 1 6 E - 0 2 0 . 7 8 9 2 1 E - 0 3 0 . 1 0 9 7 9 E - 0 2 0 . 1 0 3 1 0 E - 0 2 0 . 1 0 2 2 6 E - 0 2 0 . 9 1 6 0 8 E - 0 3 0 . 6 2 1 3 8 E - 0 3 0 . 1 1 9 2 5 E - 0 2 0 . 1 2 4 6 4 E - 0 2 0 . 3 6 3 3 0 E - 0 3 0 . 1 2 9 5 4 E - 0 2 0 . 9 0 8 4 2 E - 0 3 0 , 1 0 5 8 6 E - 0 2 0 . 1 3 6 8 3 E - 0 2 0 . 1 1 7 5 5 E - 0 2 0 . 1 6 6 2 3 E - 0 2 0 . 1 3 1 7 0 E - 0 2 0 . 1 7 9 0 9 E - 0 2 0 . 6 5 9 0 1 E - 0 3 0 . 7 4 3 2 6 E - 0 3 0 . ~ 8 1 1 5 2 E - 0 3 0 . 9 0 3 4 3 E - 0 3 0 . 1 3 8 4 6 E - 0 2 0 . 1 7 0 3 0 E - 0 2 0 . 1 7 6 9 9 E - 0 2 0 . 1 3 7 8 3 E - 0 2 0 . 1 5 7 5 4 E - 0 2 465 T a b l e 2 Multi-purpose Fishing Vessels. Apart from the estimated fi s h i n g m o r t a l i t i e s , the data are the same as used to estimate the harvesting functions for the multi-purpose fishing vessels above. These data were l i s t e d in appendix 5.3-A and the estimated f i s h i n g m o r t a l i t i e s l i s t e d below correspnd exactly to them. Data period: 1974-77 Source: Marine Research Insti t u t e . (Unpublished data on fis h i n g m o r t a l i t i e s ) . Variable code: ( 1 ) Yearly operating days. (2) Estimated vessel s p e c i f i c demersal f i s h stocks (tons caught per day at sea). (3) Estimated fi s h i n g m o r t a l i t i e s . (1) (2) (3) (1) (2) (3) (1) (2) (3) 2 7 5 . 0 . 9 2 0 0 2 9 0 . 1 . 1 069 2 2 8 . 2 . 8 9 4 7 2 7 4 . 3 . 3 6 1 3 1 7 3 . 2 . 3 6 4 2 2 0 6 . 2 . 2 4 2 7 1 7 9 . 1 . 3017 2 8 5 . 1 . 1 509 1 5 1 . 1 . 8874 2 1 3 . 1 . 7277 2 1 1 . 3 . 4 9 7 6 2 1 8 . 1 . 3899 2 2 6 . 3 . 2 7 8 8 2 3 8 . 1 . 1219 3 0 3 . 3 . 4 8 1 8 2 5 7 . 5 . 0 9 3 4 3 3 5 . 3 . 6 0 6 0 3 1 8 . 3 . 4 6 8 6 3 0 2 . 2 . 7 0 5 3 2 9 4 . 3 . 6 6 3 3 2 6 8 . 3 . 8 1 3 4 2 9 2 . 3 . 1 5 0 7 1 8 9 . 5 . 4 3 9 2 2 4 8 . 4 . 1 4 1 1 3 0 9 . 3 . 6 1 8 1 2 5 1 . 2 , 1 1 1 6 2 5 6 . 1 . 3477 2 4 7 . 0 . 8 9 0 7 5 2 . 0 . 9 4 2 3 2 8 9 . 1 . 9135 2 1 3 . 0 . 8 2 6 3 197 . 1 .7462 2 4 6 . 0 . 6 6 6 7 2 8 8 . 0 . 6 5 2 8 2 7 5 . 1 . 8727 2 2 3 . 1 . 7489 3 1 6 . 1 . 6329 0 . 1 5 4 8 4 E - 0 3 0 . 1 9 6 4 5 E - 0 3 0 . 4 0 3 9 2 E - 0 3 0 . 5 6 3 6 5 E - 0 3 0 . 2 5 0 3 1 E - 0 3 0 . 2 8 2 7 4 E - 0 3 0 . 1 4 2 6 0 E - 0 3 0 . 2 0 0 7 4 E - 0 3 0 . 1 7 4 4 2 E - 0 3 0 . 2 2 5 2 2 E - 0 3 0 . 4 5 1 6 6 E - 0 3 0 . 1 8 5 4 4 E - 0 3 0 . 4 5 3 4 9 E - 0 3 0 . 1 6 3 4 0 E - 0 3 0 . 6 4 5 6 6 E - 0 3 0 . 8 0 1 1 1 E - 0 3 0 . 7 3 9 3 0 E - 0 3 0 . 6 7 5 0 4 E - 0 3 0 . 5 0 0 0 0 E - 0 3 0 . 6 5 9 1 2 E - 0 3 0 . 6 2 5 4 6 E - 0 3 0 . 5 6 3 0 4 E - 0 3 0 . 6 2 9 1 4 E - 0 3 0 . 6 2 8 5 2 E - 0 3 0 . 6 8 4 2 2 E - 0 3 0 . 2 9 6 2 7 E - 0 3 0 . 1 9 2 8 6 E - 0 3 0 . 1 2 2 9 8 E - 0 3 0 . 2 7 3 9 1 E - 0 4 0 . 3 0 9 1 3 E - 0 3 0 . 9 8 3 8 4 E - 0 4 0 . 1 9 2 3 0 E - 0 3 0 . 9 1 6 7 6 E - 0 4 0 . 1 0 5 0 9 E - 0 3 0 . 2 8 7 8 8 E - 0 3 0 . 2 1 8 0 1 E - 0 3 0 . 2 8 8 4 4 E - 0 3 1 8 3 . 1 . 1 803 3 2 7 . 1 . 1988 3 4 5 . 1 . 6 522 2 5 4 , 1 , 8 622 2 4 4 . 2 . 2 9 5 1 3 2 1 . 0 . 8 8 1 6 3 0 8 . 0 . 8 5 7 1 3 1 2 . 1 . 4 776 3 1 1 . 0 . 8 3 9 2 2 3 8 . 2 . 0 0 8 4 1 8 3 . 1 . 8 142 2 2 9 . 3 . 4 5 8 5 1 8 9 . 2 . 7 4 6 0 2 7 1 . 1 . 3 137 3 2 0 . 2 . 2 2 1 9 2 9 2 . 1 . 2 466 1 9 0 . 2 . 7 7 8 9 3 0 2 . 1 . 9 636 2 3 2 . 2 . 3 1 9 0 1 9 9 . 2 . 4 7 2 4 2 1 0 . 1 , 9 476 3 3 7 . 2 . 6 9 7 3 9 3 . 4 . 0 4 3 0 2 8 2 . 3 . 3 4 4 0 2 0 1 . 2 . 1 4 4 3 1 6 9 . 1 . 5 3 2 5 2 1 3 . 2 . 5 7 7 5 2 2 8 . 3 . 4 7 8 1 2 6 4 . 1 . 4 015 2 4 0 , 2 . 1 9 5 8 2 1 3 . 3 . 3 7 0 9 3 2 9 , 3 . 3 2 5 2 3 2 2 . 2 . 5 7 1 4 2 3 6 . 1 . 8644 2 1 0 . 3 . 8 1 9 0 6 4 . 4 , 2 9 6 9 2 2 7 . 4 . 0 5 2 9 0 . 1 2 0 7 4 E - 0 3 0 . 2 1 9 1 3 E - 0 3 0 . 3 1 8 6 3 E - 0 3 0 . 2 6 4 4 1 E - 0 3 0 . 3 1 3 0 4 E - 0 3 0 . 1 5 8 2 0 E - 0 3 0 . 1 4 7 5 8 E - 0 3 0 . 2 5 7 7 0 E - 0 3 0 . 1 4 5 9 0 E - 0 3 0 . 2 6 7 2 0 E - 0 3 0 . 1 8 5 5 9 E - 0 3 0 . 4 4 2 7 3 E - 0 3 0 . 2 9 0 1 2 E - 0 3 0 . 1 9 9 0 0 E - 0 3 0 . 3 9 7 4 5 E - 0 3 0 . 2 0 3 4 8 E - 0 3 0 . 2 9 5 1 5 E - 0 3 0 . 3 3 1 4 9 E - 0 3 0 . 3 0 0 7 4 E - 0 3 0 . 2 7 5 0 3 E - 0 3 0 . 2 2 8 6 3 E - 0 3 0 . 5 0 8 1 3 E - 0 3 0 . 2 1 0 1 8 E - 0 3 0 . 5 2 7 1 4 E - 0 3 0 . 2 4 0 9 3 E - 0 3 0 . 1 4 4 7 8 E - 0 3 0 . 3 0 6 8 9 E - 0 3 0 . 4 4 3 2 9 E - 0 3 0 . 2 0 6 8 3 E - 0 3 0 . 2 9 4 5 9 E - 0 3 0 . 4 0 1 3 6 E - 0 3 0 . 6 1 1 5 5 E - 0 3 0 . 4 6 2 8 5 E - 0 3 0 . 2 4 5 9 6 E - 0 3 0 . 4 4 8 3 2 E - 0 3 0 . 1 5 3 7 3 E - 0 3 0 . 5 1 4 2 8 E - 0 3 2 5 8 . 3 . 5 7 7 5 2 7 6 . 3 . 2 1 0 1 2 0 9 . 4 , 1 5 7 9 2 9 2 . 4 . 1 2 3 3 3 0 0 . 2 . 5 3 6 7 2 6 7 . 3 . 0 6 3 7 2 6 3 . 1 . 7 7 9 5 2 9 2 . 3 . 8 3 2 2 2 5 4 . 3 . 4 6 0 6 1 4 2 . 2 . 5 0 0 0 3 6 . 2 . 6 1 1 1 2 9 9 . 1 . 7 8 2 6 2 3 8 . 1 . 7 3 5 3 1 9 7 . 0 . 8 0 2 0 1 8 3 . 1 . 2 459 2 4 6 . 1 . 4 8 3 7 2 7 3 . 1 . 8 7 1 8 3 1 5 . 1 . 2 2 2 2 2 8 8 . 1 . 4 306 3 1 6 . 1 . 1361 1 3 7 . 1 . 2 409 2 9 7 . 1 . 7 003 2 7 4 . 0 . 7 4 4 5 2 6 6 . 1 . 1 9 9 2 1 9 6 . 1 . 4 949 2 6 0 . 1 . 1 4 6 2 1 9 5 . 0 . 9 2 3 1 1 5 6 . 0 . 7 4 3 6 2 6 5 . 1 . 4 039 2 7 0 . 0 . 8 1 1 1 2 2 5 . 1 . 1 556 2 7 7 . 1 . 4982 2 0 0 . 1 . 8 400 2 5 4 . 2 . 6 5 7 5 1 8 0 . 4 . 6 8 8 9 2 8 2 . 2 . 5 9 5 7 8 4 . 7 . 2 9 7 6 0 . 5 1 5 9 6 E - 0 3 0 . 4 9 5 2 7 E - 0 3 0 . 4 8 5 7 7 E - 0 3 0 . 6 7 3 0 4 E - 0 3 0 . 4 2 5 4 0 E - 0 3 0 . 4 5 7 2 6 E - 0 3 0 . 2 6 1 6 1 E - 0 3 0 . 6 2 5 5 2 E - 0 3 0 . 4 9 1 3 6 E - 0 3 0 . 1 9 8 4 5 E - 0 3 0 . 4 0 7 0 2 E - 0 4 0 . 2 3 0 7 9 E - 0 3 0 . 1 7 8 8 3 E - 0 3 0 . 6 8 4 1 4 E - 0 4 0 . 9 8 7 2 4 E - 0 4 0 . 1 5 8 0 5 E - 0 3 0 . 2 2 1 2 6 E - 0 3 0 . 1 6 6 7 0 E - 0 3 0 . 1 7 8 4 0 E - 0 3 0 . 1 5 5 4 5 E - 0 3 0 . 7 3 6 1 0 E - 0 4 0 . 2 1 8 6 6 E - 0 3 0 . 8 8 3 3 2 E - 0 4 0 . 1 3 8 1 3 E - 0 3 0 . 1 2 6 8 7 E - 0 3 0 . 1 2 9 0 3 E - 0 3 0 . 7 7 9 4 0 E - 0 4 0 . 5 0 2 2 8 E - 0 4 0 . 1 6 1 0 8 E - 0 3 0 . 9 4 8 2 7 E - 0 4 0 . 1 1 2 5 8 E - 0 3 0 . 1 7 9 6 9 E - 0 3 0 . 1 5 9 3 4 E - 0 3 0 . 2 9 2 2 8 E - 0 3 0 . 3 6 5 4 5 E - 0 3 0 . 3 1 6 9 6 E - 0 3 0 . 2 6 5 4 3 E - 0 3 466 2 5 2 . 0 . 7 1 4 3 0 . 7 7 9 4 0 E - 0 4 2 1 0 . 4 , 6 7 6 2 0 . 4 2 5 2 1 E - 0 3 2 7 9 . 1 . 0932 0 . 1 3 2 0 7 E - 0 3 2 8 2 . 3 , 6 7 3 8 0 . 4 4 8 5 9 E - 0 3 2 5 4 . 3 . 7 3 6 2 0 . 4 1 0 9 2 E - 0 3 2 7 9 . 1 . 6667 0 . 2 0 1 3 4 E - 0 3 1 6 0 . 2 . 2 1 2 5 0 . 1 5 3 2 8 E - 0 3 2 7 4 . 1 , 7226 0 . 2 0 4 3 8 E - 0 3 1 2 2 . 1 . 6721 0 . 8 8 3 3 2 E - 0 4 2 6 7 . 4 . 9 1 0 1 0 . 5 6 7 6 6 E - 0 3 2 9 7 . 2 . 3 7 7 1 0 . 3 0 5 7 0 E - 0 3 3 1 2 . 3 . 6 8 2 7 0 . 4 9 7 5 2 E - 0 3 2 3 8 . 2 . 0 0 4 2 0 . 2 0 6 5 4 E - 0 3 3 1 4 . 2 . 7 8 3 4 0 . 3 7 8 4 4 E - 0 3 2 8 2 . 3 . 7 8 0 1 0 . 4 6 1 5 8 E - 0 3 2 1 1 . 1 . 4692 0 . 1 3 4 2 3 E - 0 3 2 4 1 . 2 . 2 0 7 5 0 . 2 3 0 3 6 E - 0 3 3 2 3 . 3 . 2 3 2 2 0 . 4 5 2 0 5 E - 0 3 2 6 8 . 3 . 0 4 8 5 0 . 3 5 3 7 6 E - 0 3 2 8 2 . 3 . 0 6 0 3 0 . 3 7 3 6 8 E - 0 3 3 4 2 . 2 . 1 5 5 0 0 . 3 1 9 1 2 E - 0 3 2 2 6 . 1 . 1549 0 . 1 1 3 0 1 E - 0 3 3 2 3 . 3 . 0 0 6 2 0 . 4 2 0 4 4 E - 0 3 2 8 1 . 3 . 8 9 3 2 0 . 4 7 3 7 0 E - 0 3 2 9 8 . 1 . 7 953 0 . 2 3 1 6 5 E - 0 3 2 6 9 . 1 . 9071 0 . 2 2 2 1 3 E - 0 3 2 5 9 , 2 . 1 1 2 0 0 . 2 3 6 8 5 E - 0 3 1 9 7 . 2 . 6 2 4 4 0 . 2 2 3 8 6 E - 0 3 2 0 3 . 4 . 4 5 3 2 0 . 3 9 1 4 3 E - 0 3 2 2 0 . 3 . 6 5 0 0 0 . 3 4 7 7 0 E - 0 3 2 7 0 . 3 . 8 3 3 3 0 . 4 4 8 1 6 E - 0 3 2 7 8 . 4 . 2 2 3 0 0 . 5 0 8 3 4 E - 0 3 2 7 5 . 4 , 5 7 4 5 0 . 5 4 4 7 1 E - 0 3 2 6 9 . 3 . 2 7 1 4 0 . 3 8 1 0 4 E - 0 3 3 3 2 . 2 . 3 4 3 4 0 . 3 3 6 8 7 E - 0 3 2 7 2 . 4 . 3 1 9 9 0 . 5 0 8 7 8 E - 0 3 3 1 6 . 2 . 2 3 7 3 0 . 3 0 6 1 3 E - 0 3 3 1 4 . 4 . 0 7 6 4 0 . 5 5 4 2 4 E - 0 3 3 0 3 . 3 . 6 0 4 0 0 . 4 7 2 8 4 E - 0 3 3 0 0 . 4 . 2 0 0 0 0 . 5 4 5 5 8 E - 0 3 2 9 3 . 3 . 0 8 8 7 0 . 3 9 1 8 7 E - 0 3 2 7 4 , 3 . 5 8 0 3 0 . 4 2 4 7 7 E - 0 3 2 1 4 . 4 , 7 2 4 3 0 . 4 3 7 7 6 E - 0 3 2 5 1 . 2 . 0 6 3 7 0 . 2 2 4 2 9 E - 0 3 2 5 2 , 1 , 9149 0 . 2 3 3 8 2 E - 0 3 2 3 3 . 2 . 0 6 0 1 0 . 2 0 7 8 4 E - 0 3 2 9 7 , 4 . 3 7 7 1 0 . 5 6 2 9 0 E - 0 3 3 2 8 , 6 . 4 0 8 5 0 . 9 1 0 1 7 E - 0 3 2 4 8 . 1 . 1129 0 . 1 2 0 6 1 E - 0 3 1 9 7 . 0 . 7 8 6 8 0 . 6 7 7 3 5 E - 0 4 1 8 7 . 1 . 3048 0 . 1 0 6 6 3 E - 0 3 2 9 0 . 2 , 1 5 8 6 0 . 2 7 3 5 6 E - 0 3 2 6 3 . 2 . 3 6 1 2 0 . 2 7 1 3 8 E - 0 3 2 9 5 , 1 .3966 0 . 1 B 0 0 4 E - 0 3 3 1 4 . 2 . 2 9 3 0 0 . 3 1 4 6 4 E - 0 3 1 9 8 . 0 . 8 0 8 1 0 . 6 9 9 2 0 E - 0 4 3 0 3 . 1 . 6238 0 . 2 1 5 0 0 E - 0 3 2 0 9 . 2 . 8 5 6 5 0 . 2 6 0 8 9 E - 0 3 2 1 0 . 1 . 9429 0 . 1 7 8 3 0 E - 0 3 2 3 9 . 2 . 4 5 1 9 0 . 2 5 6 0 8 E - 0 3 2 1 5 . 2 . 0 2 3 3 0 . 1 9 0 1 0 E - 0 3 2 6 1 . 1 . 9 5 0 2 0 . 2 2 2 4 3 E - 0 3 2 5 3 . 1 . 7 5 8 9 0 . 1 9 4 4 6 E - 0 3 3 0 8 . 1 . 9 253 0 . 2 5 9 1 4 E - 0 3 2 4 4 . 1 . 4 7 9 5 0 . 1 5 7 7 6 E - 0 3 3 2 2 . 1 . 8 8 8 2 0 . 2 6 5 7 0 E - 0 3 2 7 3 . 1 , 8 4 6 2 0 . 2 2 0 2 5 E - 0 3 2 2 3 . 1 . 4 305 0 . 1 3 9 4 0 E - 0 3 3 3 0 , 2 . 2 6 9 7 0 . 3 2 7 3 1 E - 0 3 3 0 4 . 1 . 6 447 0 . 2 1 8 5 0 E - 0 3 2 4 4 . 1 , 8934 0 . 2 0 1 8 9 E - 0 3 3 0 3 . 1 . 1 3 8 6 0 . 1 5 0 7 7 E - 0 3 3 2 1 . 2 . 2 9 6 0 0 . 3 2 2 0 7 E - 0 3 3 2 3 . 0 . 7 5 8 5 0 . 1 0 7 0 7 E - 0 3 2 3 7 . 1 . 0 6 3 3 0 . 1 1 0 1 2 E - 0 3 2 4 5 . 0 . 8 5 3 1 0 . 9 1 3 3 3 E - 0 4 1 1 8 . 5 . 5 9 3 2 O . 2 8 8 4 2 E - 0 3 1 9 0 . 3 . 3 3 1 6 0 . 2 7 6 6 2 E - 0 3 2 8 0 . 1 . 7 6 7 9 0 . 2 1 6 3 2 E - 0 3 2 4 7 . 1 . 6316 0 . 1 7 6 1 1 E - 0 3 2 6 2 . 1 , 7 290 0 . 1 9 7 9 6 E - 0 3 2 6 0 . 2 . 4 2 6 9 0 . 2 7 5 7 5 E - O 3 8 4 . 5 . 0 7 1 4 0 . 1 8 6 1 6 E - 0 3 1 8 3 . 0 . 8 1 9 7 0 . 6 5 5 5 0 E - 0 4 2 2 4 . 2 . 2 2 7 7 0 . 2 1 8 0 6 E - 0 3 2 4 4 . 1 . 9 5 9 0 0 . 2 0 8 8 9 E - 0 3 1 4 1 . 2 . 3 1 9 1 0 . 1 4 2 9 0 E - 0 3 1 9 7 . 2 . 7 8 1 7 0 . 2 3 9 4 8 E - 0 3 3 0 6 . 1 . 4 183 0 . 1 8 9 6 6 E - 0 3 3 0 9 . 2 . 3 2 6 9 0 . 3 1 4 2 0 E - 0 3 2 1 6 . 1 . 1 3 8 9 0 . 1 0 7 5 0 E - 0 3 2 0 0 . 3 . 1 4 0 0 0 . 2 7 4 4 4 E - 0 3 3 0 2 . 0 . 6 3 9 1 0 . 8 4 3 4 1 E - 0 4 2 5 5 , 1 . 3 333 0 . 1 4 8 5 8 E - 0 3 2 8 6 , 2 . 9 8 2 5 0 . 3 7 2 7 6 E - 0 3 2 1 3 . 1 . 5 305 0 . 1 4 2 4 6 E - 0 3 1 5 4 . 2 . 6 4 9 4 0 . 1 7 8 3 0 E - 0 3 1 9 1 . 2 . 0 5 2 4 0 . 1 7 1 3 0 E - 0 3 2 1 1 . 2 . 7 7 7 3 0 . 2 5 6 0 8 E - 0 3 2 4 4 . 1 . 1 107 0 . 1 1 8 4 3 E - 0 3 3 2 0 . 1 . 5406 0 . 2 1 5 4 4 E - 0 3 2 1 8 . 1 . 6514 0 . 1 5 7 3 2 E - 0 3 3 2 5 . 2 . 3 9 6 9 0 . 3 4 0 4 2 E - 0 3 2 6 5 . 3 . 2 9 4 3 0 . 3 8 1 5 0 E - 0 3 3 2 4 . 2 . 5 7 4 1 0 . 3 6 4 4 6 E - 0 3 2 7 1 . 3 . 2 4 3 5 0 . 3 8 4 1 2 E - 0 3 2 1 1 . 3 . 8 3 8 9 0 . 3 5 3 9 7 E - 0 3 2 4 3 . 3 . 9 3 4 2 0 . 4 1 7 7 7 E - 0 3 3 0 3 . 2 . 8 7 4 6 0 . 3 8 0 6 3 E - 0 3 2 3 2 . 3 . 6 7 2 4 0 . 3 7 2 3 2 E - 0 3 2 6 8 . 3 . 5 6 7 2 0 . 4 1 7 7 7 E - 0 3 2 9 4 . 2 . 8 5 0 3 0 . 3 6 6 2 1 E - 0 3 Chapter 7.1: Data. (Simulation Data, see also tables A. F i shinq mortal i t i es. COD: AGE 1960 1961 1962 1 0.000 0.000 0.000 2 0.008 0.013 0.006 3 0. 1 03 0.089 0.110 4 0. 1 57 0.218 0.253 5 0.326 0. 144 0.347 6 0.353 0.365 0. 1 64 7 0. 187 0.394 0.438 8 0.372 0.208 0.398 9 0.488 0.407 0.416 10 0.487 0.430 0.581 1 1 0.815 0.494 0.532 1 2 1 .359 0.728 0.478 1 3 0.475 0.437 0.530 1 4 0.600 0.550 0.700 AGE 1968 1 969 1970 1 0.000 0.000 0.000 2 0.003 0.001 0.002 3 0.077 0.044 0.059 4 0.256 0.218 0.293 5 0. 184 0.371 0.346 6 0.243 0.208 0.400 7 0.282 0.311 0.298 8 0.882 0.503 0.576 9 0.900 0.856 1 .033 1 0 1 .506 0.905 0.685 1 1 1 .248 1.130 0.873 1 2 1 .799 0.881 1.418 1 3 0.704 1 .067 0.561 1 4 0.700 1 .000 0.900 AGE 1976 1 977 1978 1 0.000 0.000 0.000 2 0.002 0.000 0.000 3 0.063 0.021 0.031 4 0.265 0. 1 55 0.173 5 0.370 0.359 0.236 6 0.624 0.378 0.337 7 0.610 0.827 0.514 8 0.939 0.685 0.690 9 0.769 0.996 0.544 1 0 1 .303 0.616 0.532 1 1 1 .462 0.597 0.350 12 1 .267 0.654 0.810 13 3.027 0.131 1 .200 1 4 1 .340 1 .400 1 .400 467 4.1, 4.4 and 4.7) YEAR 1963 1964 1 965 1 966 1967 0.000 0.000 0.000 0.000 0.000 0.004 0.008 0.010 0.011 0.027 0. 1 07 0.064 0.101 0.071 0.098 0.267 0.305 0. 1 64 0. 1 94 0. 1 49 0.318 0.466 0.398 0. 1 52 0.234 0.353 0.252 0.498 0.417 0.129 0.277 0.406 0.343 0. 585 0.511 0.593 0.477 0.707 0.498 0.589 0.464 0.947 0.705 0.627 0.612 0.662 0.654 0.997 0.982 0.887 0.730 0.978 0.968 1 . 043 1 .201 0.571 1 .235 0.449 0.858 0.526 0.428 0.616 0.717 0.580 0.273 0.600 0.600 0.800 0.750 0.500 YEAR 1 971 1 972 1 973 1 974 1 975 0.000 0.000 0.000 0.000 0.000 0.006 0.011 0.013 0.008 0.003 0.085 0.076 0. 144 0. 1 03 0.131 0.323 0.280 0.321 0.385 0.303 0.532 0.513 0.453 0.500 0.524 0.624 0.562 0.598 0.460 0.576 0.510 0.809 0.715 0.745 0.643 0.355 0.612 0.705 0.971 0.811 0.600 0.693 0.687 0.815 1 .078 1 .068 1 .063 1.104 1.031 1 . 1 99 0.706 1 . 1 63 1 .097 1 .366 1 .475 0. 988 0.983 0.947 1 .260 1 .802 3.205 2.372 0. 130 0.555 - 0.816 1 .300 1 .500 0. 500 1.100 1.100 YEAR 1 979 1 980 1 981 1 982 1983 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.028 0.014 0.049 0.030 0.200 0. 153 0. 1 32 0. 187 0. 1 90 0.217 0.368 0.358 0.314 0.320 0.294 0.391 0.524 0.431 0.440 0.498 0.397 0.668 0.656 0.570 0.537 0.574 0.802 1 . 1 93 1 .020 0.550 0.575 0.778 1.146 1 . 1 30 0.381 0.518 1 .082 0.943 1 .030 0. 535 0.506 1 . 176 0.695 0.810 0.205 0.709 1 . 1 50 0.646 0.720 1 .408 0. 176 1.313 0.777 0.780 1 .400 1 .000 1 .000 1 .000 1 .000 ) 468 HADDOCK: AGE 1960 1961 1 0.004 0.002 2 0.054 0.203 3 0.162 0.222 4 0.481 0.323 5 0.914 0.761 6 0.761 1.196 7 0.707 2.229 8 0.612 0.708 9 0.800 0.800 AGE 1968 1969 1 0.000 0.002 2 0.047 0.060 3 0.292 0.236 4 0.405 0.384 5 0.804 0.617 6 1.020 0.566 7 0.799 0.749 8 0.676 0.421 9 0.800 0.670 AGE 1976 1977 1 0.000 0.000 2 0.025 0.002 3 0.125 0.049 4 0.348 0 . 1 9 8 5 0.563 0.476 6 0.767 0.587 7 0.682 0.921 8 0.525 0.906 9 0.800 1.150 YEAR 1 962 1 963 1 964 0. 004 0.001 0.000 0. 042 0.047 0.060 0. 294 0.250 0.547 0. 504 0.416 0.317 0. 635 0.809 0.646 0. 651 0.889 0.913 0. 562 0.898 1 .088 0. 646 0.709 1.319 0. 800 0.900 1 .000 YEAR 1970 1 971 1972 0. 000 0.000 0.000 0. 025 0.018 0.041 0. 185 0. 1 72 0.221 0. 467 0.415 0.620 0. 509 0.886 0.670 0. 794 1 .244 0.598 1 . 039 1.210 0.414 1 . 445 1 .087 0.370 0. 840 1 .000 0.400 YEAR 1978 1979 1 980 0. 000 0.000 0.000 0. 001 0.002 0.002 0. 018 0.019 0.021 0. 089 0. 175 0.114 0. 250 0.430 0.287 0. 719 0.633 0.494 1 . 105 0.778 0.633 1 . 900 1 .028 0.557 1.. 1 50 1 . 1 50 0.900 1 965 1966 1 967 0.000 0.002 0.001 0.043 0.020 0.094 0.616 0.350 0.403 0.856 0.754 0.885 0.856 0.607 0.693 0.688 0.672 0.321 0.933 0.605 0.443 0.975 1.113 0.269 1.100 0.600 0.300 1 973 1 974 1975 0.001 0.000 0.000 0.060 0.020 0.007 0.238 0.116 0. 172 0.407 0.502 0.433 0.700 0.848 0.720 1 .091 0.906 0.842 1 . 167 1 . 1 49 0.994 1 .433 1 .220 0.823 1 .400 1 .200 1 .000 1 981 1982 1983 0.000 0.000 0.000 0.000 0.001 0.001 0.018 0.015 0.012 0. 106 0. 1 06 0. 100 0.295 0.299 0.300 0.666 0.437 0.500 0.757 0.696 0.600 0.539 0.939 0.800 0.500 0.700 0.800 469 SAITHE: AGE 1960 1961 1 0.000 0.000 2 0.007 0.015 3 0.052 0.154 4 0.148 0.199 5 0.285 0.336 6 0.293 0.331 7 0.236 0.199 8 0.251 0.130 9 0.283 0.126 10 0.219 0.217 11 0.184 0.256 12 0.291 0.541 13 0.316 0.292 14 0.300 0.300 AGE 1968 1969 1 0.000 0.000 2 0.000 0.000 3 0.015 0.020 4 0.054 0.105 • 5 0.095 0.160 6 0.172 0.252 7 0.289 0.402 8 0.372 0.439 9 0.281 0.411 10 0.339 0.347 11 0.324 0.155 12 0.255 0.273 13 0.323 0.117 14 0.300 0.300 AGE 1976 1977 1 0.000 0.000 2 0.001 0.000 3 0.010 0.003 4 0.182 0.085 5 0.342 0.243 6 0.469 0.349 7 0.401 0.357 8 0.412 0.356 9 0.345 0.514 10 0.534 0.378 11 0.335 0.572 12 0.367 0.319 13 1.251 0.225 14 0.500 0.500 YEAR 1 962 1 963 1964 0. 000 0.000 0. 000 0. 002 0.007 0. 001 0. 056 0.084 0. 063 0. 272 0.112 0. 227 0. 305 0.213 0. 254 0. 468 0.396 0. 308 0. 287 0.452 0. 278 0. 213 0.381 0. 235 0. 1 66 0.258 0. 185 0. 183 0.240 0. 1 73 0. 189 0.293 0. 1 40 0. 259 0.418 0. 1 63 0. 239 0.393 0. 216 0. 300 0.300 0. 300 YEAR 1 970 1971 1972 0. 000 0.000 0. 000 0. 000 0.000 0. 002 0. 005 0.011 0. 024 0. 092 0.064 0. 1 09 0. 1 68 0.239 0. 1 92 0. 258 0.356 0. 327 0. 401 0.496 0. 391 0. 497 0.674 0. 493 0. 521 0.829 0. 670 0. 513 0.641 0. 744 0. 399 0.929 0. 534 0. 431 1.171 1 . 1 70 0. 386 0.460 0. 961 0. 400 0.500 0. 700 YEAR 1 978 1 979 1 980 0. 000 0.000 0. 000 0. 000 0.000 0. 000 0. 01 1 0.011 0. 006 0. 064 0.095 0. 068 0. 134 0.151 0. 1 62 0. 202 0.301 0. 293 0. 447 0.321 0. 337 0. 660 0.491 0. 642 0. 304 0.428 0. 728 0. 370 0.770 0. 786 0. 327 0.813 0. 331 0. 670 0.660 0. 31 1 0. 484 0.976 0. 1 50 0. 500 0.500 0. 500 1 965 1 966 1967 0. 000 0. 000 0. 000 0. 001 0. 000 0. 003 0. 024 0. 015 0. 018 0. 134 0. 031 0. 069 0. 226 0. 127 0. 109 0. 239 0. 184 0. 245 0. 293 0. 219 0. 349 0. 239 0. 264 0. 327 0. 229 0. 223 0. 320 0. 188 0. 234 0. 298 0. 1 77 0. 229 0. 264 0. 1 58 0. 1 74 0. 190 0. 212 0. 288 0. 430 0. 300 0. 300 0. 300 1 973 1 974 1 975 0. 000 0. 000 0. 000 0. 001 0. 004 0. 000 0. 01 1 0. 064 0. 022 0. 097 0. 232 0. 210 0. 212 0. 180 0. 264 0. 328 0. 195 0. 207 0. 514 0. 263 0. 343 0. 455 0. 474 0. 383 0. 484 0. 431 0. 377 0. 662 0. 483 0. 444 0. 499 0. 620 0. 356 0. 795 0. 316 0. 743 1 . 087 0. 326 0. 388 0. 800 0. 500 0. 500 1 981 1 982 0. 000 0. 000 0. 000 0. 000 0. 013 0. 009 0. 087 0. 080 0. 1 74 0. 200 0. 292 0. 300 0. 282 0. 400 0. 354 0. 450 0. 515 0. 500 0. 737 0. 650 0. 513 0. 600 0. 990 0. 600 0. 830 0. 600 0. 600 0. 600 470 B. Other Simulation Data. Number of Recruitment Vessels Catch volume Operating rents Year Cod Had Sai Traw Boat Cod Had Sai Harv. Proc. Total 1960 151 .7 81.1 30.9 43 171 465. 3 87. 5 48. 0 NA NA NA 1 961 189.4 43.0 32.9 43 1 50 374. 6 1 02. 1 49. 8 NA NA NA 1 962 1 29.9 114.0 31.0 42 1 50 376. 3 1 19. 6 50. 4 NA NA NA 1 963 1 62. 1 91 .2 84.1 38 1 60 402. 0 1 02. 4 48. 4 NA NA NA 1 964 289.2 79.5 55.2 36 180 429. 3 99. 0 60. 4 NA NA NA 1 965 253.4 65.4 94.2 35 1 90 393. 6 99. 0 60. 1 NA NA NA 1 966 270.8 76.7 70. 1 32 200 356. 8 60. 1 52. 2 NA NA NA 1967 325.4 41 .7 68.0 30 220 345. 2 60. 2 76. 4 NA NA NA 1968 171.8 64.4 60.0 26 240 381 . 1 51 . 2 78. 6 NA NA NA 1 969 252.4 35.7 88.0 22 257 406. 4 46. 6 116. 3 NA NA NA 1970 184.5 40.3 65.7 20 269 470. 8 44. 5 116. 8 NA NA NA 1 971 177.0 30.6 49.8 25 265 453. 0 46. 1 1 36. 5 1 .038 1 .030 1 .033 1972 1 35.0 63.6 26.4 30 290 398. 5 39. 3 111. 3 1 .015 1 .017 1 .016 1 973 300.8 46.6 22.4 44 295 383. 4 45. 5 110. 9 0 .997 1 .064 1 .032 1974 167.6 58.6 22.6 55 280 374. 8 42. 6 97. 5 1 .000 1 .000 1 .000 1975 261 .8 90.4 26.6 60 265 371 . 0 45. 7 87. 9 1 .029 1 .032 1 .031 1 976 425.8 44. 1 35.0 70 260 347. 8 42. 4 81 . 9 1 .021 1 .043 1 .033 1 977 1 50.7 42.6 25.0 75 265 340. 1 39. 7 62. 0 1 .054 1 .007 1 .026 1 978 240.4 151.4 56.5 79 270 330. 4 43. 5 49. 7 1 .076 1 .028 1 .049 1 979 270.0 94.0 47.9 82 270 368. 1 58. 4 63. 5 1 . 077 1 .065 1 .069 1980 1 74.5 42.9 25. 1 86 280 435. 2 50. 9 58. 4 1 .015 1 . 1 06 1 .061 NA = Not avai l a b l e . 471 Appendix 2: On Computer Programs. As should be clear from parts II and III above, manipulations of the empirical f i s h e r i e s model developed in th i s study are computationally rather involved. To perform this task, a set of computer programs has been developed. These programs es s e n t i a l l y represent the basic structure of the empirical model. The p a r t i c u l a r data, representing the circumstances of s p e c i f i c f i s h e r i e s , are supplied via certain input channels. Due to t h i s seperation between basic structure and pa r t i c u l a r data, the programs should be easily applicable to other f i s h e r i e s . An outline of the main groups of computer programs employed in the thesis follows. Individual programs, being quite numerous, are not l i s t e d . (1) A general program to perform VP-analysis. (2) A c o l l e c t i o n of equilibrium or steady state programs to calculate i . a . : (a) Sustainable (joint) y i e l d . (b) Sustainable (joint) biomass. (c) Biomass per r e c r u i t . (d) Sustainable fishing c a p i t a l l e v e l s . (e) Corresponding p l o t t i n g routines. (3) A c o l l e c t i o n of dynamic simulation programs including: (a) B i o l o g i c a l r e s u l t s . (b) Economic r e s u l t s . (c) A f a c i l i t y to systematically compare simulation results with user supplied "actual" outcomes. (d) Corresponding pl o t t i n g routines. (4) A comprehensive program for maximizing economic rents from f i s h e r i e s including i . a . the following features: (a) Up to 10 f i s h i n g f l e e t s . (b) Up to 10 species of f i s h . (c) General ecological relationships between the species influencing a l l the natural elements of biomass growth e.g. natural mortality, individual weights and r e c r u i t -ment . (d) Exogenous environmental influences a f f e c t i n g a l l the above-mentioned elements of species growth. (e) Stochastic variation in a l l the above-mentioned ele-ments of biomass growth as well as the generation of fi s h i n g m o r t a l i t i e s . (f) Exogenous economic influences through (i) prices and ( i i ) f i s h i n g and processing technology. 472 (g) User s p e c i f i c a t i o n of f i s h i n g c a p i t a l dynamics includ-ing the degree of c a p i t a l m a l l e a b i l i t y . (h) User s p e c i f i c a t i o n of the fishing mortality production function. These programs are written in Fortran 77 and implemented on the VAX 11/780 computer. Several computer l i b r a r y routines are referenced. Being f a i r l y standard, however, they or their equivalents should be available at most computer centres.