THE OPTIMAL MANAGEMENT OF AN OCEAN FISHERY By TROND BJ0RNDAL Sivil^konom, The Norwegian School of Economics and Business Administration, 1975 Sivil^konom HAE, The Norwegian School of Economics and Business Administration, 1980 A THESIS 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 December 1984 ® Trond Bjcfrndal, 1984 In p r e s e n t i n g t h i s t h e s i s requirements f o r an British it freely available for Columbia, I agree that understood that for h i s or be her s h a l l not Economics The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 D a t e DE-6 (3/81) December 12, 1984 s h a l l make study. I the of further this Columbia thesis head o f this my It i s thesis a l l o w e d w i t h o u t my permission. Department O f University representatives. be the Library g r a n t e d by copying or p u b l i c a t i o n f i n a n c i a l gain the for extensive copying of s c h o l a r l y p u r p o s e s may by the f o r r e f e r e n c e and permission department or f u l f i l m e n t of advanced degree at of agree that in partial written - i i- ABSTRACT The objective of t h i s thesis i s to study the optimal management of North Sea herring. The analysis i s based on a dynamic bioeconomic model for a f i s h resource, consisting of a model of population dynamics and a net revenue function. The model of population dynamics i s described difference equation. by a delay- The model distinguishes between natural growth and mortality i n the existing stock as opposed to new stock, which takes place with a time l a g . recruitment to the The model i s estimated based on time series data for the period 1947-82. The net growth function i s shown to exhibit depensation, a phenomenon not uncommon for schooling f i s h l i k e herring. In f i s h e r i e s economics, the production function i s often treated i n a rather r e s t r i c t i v e manner. The approach of this thesis i s to spec- i f y a general production function, where output (harvest) i s a function of variable inputs, stock size and other fixed f a c t o r s . (1968, 1971 and 1975) Cross-sectional and aggregate time series (1963-77) data sets for the North Sea herring f i s h e r y are a v a i l a b l e . The cross-sectional data f a c i l i t a t e d i r e c t estimation of the production function (Cobb-Douglas). The time series data are used to estimate a harvest supply function (Cobb-Douglas), and by d u a l i t y theory the parameters of the corresponding production function are derived. A hypothesis of increasing returns to scale i n a l l inputs i s accepted in a l l model s p e c i f i c a t i o n s . - iii - The stock output e l a s t i c i t y generally varies between 0.1 and 0.5. Bio- nomic e q u i l i b r i u m — i . e . , the open access stock l e v e l — i s estimated to be close to zero. The l a s t two r e s u l t s are attributed to the fact that the resource i n question i s a schooling one. The model i s extended by introducing stock dynamics and the concept of a sole resource manager. An intertemporal p r o f i t function i s maximized and an expression for the optimal stock l e v e l i s derived. Some new a n a l y t i c a l results with regard to the relationship between the optimal stock l e v e l and the production technology are derived. The quantitative r e s u l t s show that the inclusion of costs i n the intertemporal p r o f i t function causes a considerable increase i n the optimal stock l e v e l . The assertion that a low stock output e l a s t i c i t y implies that costs have a negligible e f f e c t on the optimal stock l e v e l i s therefore not necessarily t r u e . nature of the production technology. This i s a r e s u l t of the nonlinear The optimal stock l e v e l i s shown to be not very sensitive to moderate changes i n the discount r a t e . It i s i l l u s t r a t e d that costs have a s t a b i l i z i n g influence on the stock level. The optimal harvest quantity i s quite i n s e n s i t i v e to changes i n the stock l e v e l , a r e s u l t caused by the properties of the estimated model of population dynamics. L a s t l y , the model i s found to be robust in the sense that the d i f f e r e n t s p e c i f i c a t i o n s of the model of populat i o n dynamics and the production technology give r i s e to the same qualitative r e s u l t s . - iv - TABLE OF CONTENTS ABSTRACT LIST OF TABLES i i v LIST OF FIGURES vi ACKNOWLEDGEMENT vii DEDICATION viii 1.0 INTRODUCTION 2.0 2.1 2.2 2.3 2.3.1 2.3.2 2.4 THE BIOECONOMIC MODEL The Model of Population Dynamics A Fishery P r o f i t Function The Intertemporal P r o f i t Function Dynamic Optimization The Marginal Stock Effect Summary 7 7 12 15 15 24 29 3.0 3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 THE EMPIRICAL MODEL The Model of Population Dynamics The Stock-Recruitment Function The Net Growth Function The Complete Model The Production Function The Nature of the Fishery Empirical Estimation 31 31 32 39 43 46 48 51 4.0 4.1 4.2 THE OPTIMAL STOCK LEVEL The Influence of Costs The Influence of the Marginal Stock Effect 75 75 87 4.3 5.0 Some Management Issues SUMMARY 1 90 99 BIBLIOGRAPHY 104 APPENDIX 1: TECHNICAL DERIVATIONS 109 115 APPENDIX 2: DATA - v- LIST OF TABLES 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Al A2 A3 A4 A5 A6 Estimated Stock-Recruitment Functions Maximum Recruitment Estimation of Net Natural Growth Functions 6 (S ) fc Characteristics of Function S t e The Carrying Capacity of North Sea Herring Estimates of S m s y and MSY Summary of Cross-Sectional Data Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data Estimated Production Function (Cobb-Douglas) f o r North Sea Herring. Cross-Sectional Data. Main Season (June to August) Catches Only Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. K^t Exogenous. Time Series Data 1963-77 Derived Production Function Parameters (Cobb-Douglas) for North Sea Herring Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. K j t Endogenous. Time Series Data 1963-77 Derived Production Function Parameters (Cobb-Douglas) for North Sea Herring. K i t Endogenous 2SLS Nonlinear Estimation of Production Function Parameters (Cobb-Douglas) for North Sea Herring. S 2 Value Imposed Optimal Stock Level (S*) and Corresponding Harvest (H*) The Optimal Stock Level for the Variable Cost Case S e n s i t i v i t y to a 20% Change i n Cost/Price Ratio. Base Case Cost Alternative S e n s i t i v i t y to Changes i n the E f f o r t Output E l a s t i c i t y (a) S e n s i t i v i t y to Changes i n the Stock Output E l a s t i c i t y (B Q ) Effects of Changes i n Fleet Size Simulation of Spawning Stock 1983-87 B i o l o g i c a l Data Norwegian P a r t i c i p a t i o n i n the Herring Fishery The Cost of E f f o r t Index Average Annual Operating Costs for Norwegian Purse Seiners 1975-81 Average Daily Operating Costs for Norwegian Purse Seiners Norwegian Price Data 37 37 40 40 44 44 54 58 58 66 66 70 70 71 81 85 86 87 88 90 92 115 117 119 122 122 124 - vi- LIST OF FIGURES 2.1 Stock dynamics 8 2.2 MSE as a function of p o 28 2.3 S* as a function of p Q 28 2.4 MSE as a function of p o 28 2.5 S* as a function of p o 28 3.1 The Stock Recruitment Relationship 36 3.2 The Net Growth Function 42 3.3 Stock Dynamics (Steady State Stock Levels) 47 4.1 Fleet P a r t i c i p a t i o n and Stock Size 1963-77 77 4.2 Number of Catches and Stock Size 1966-77 77 4.3 S* as a function of B „ 89 - vii- ACKNOWLEDGEMENT This thesis has benefitted greatly from comments and advice from the members of my supervisory committee, G. R. Munro (Chairman), C. W. Clark, P. A. Neher, W. E. Schworm and R. S. Uhler. In a d d i t i o n , I have learned much from discussions with D. V. Gordon, R. Hilborn and M. Eswaran. My wife, Marit, and son, Tord, deserve thanks for encouragement and endurance during the Ph.D. programme. Data for the empirical work were supplied by the Marine Research I n s t i t u t e , Norway, The Directorate of F i s h e r i e s , Norway and the Norwegian Herring Fishermen's Cooperative. My studies at The University of B r i t i s h Columbia were made possible by a scholarship from The Norwegian School of Economics and Business Administration/the Norwegian University of Fisheries and a Government of Canada Award. The thesis was expertly typewritten by M. R. Brown. - viii - DEDICATION This thesis i s dedicated to my parents, Judith and Thorvald Bj^rndal. - 1 - 1.0 INTRODUCTION The objective of this thesis i s to study the optimal management of North Sea h e r r i n g . This f i s h stock was severely depleted i n the 1960s and early 1970s due to o v e r f i s h i n g , but has since been permitted to recover. This leads to questions about the target for the rebuilding programme and the approach to this l e v e l . These questions, which con- s t i t u t e the essence of a management plan, w i l l be analyzed by means of a dynamic bioeconomic model. In the t h e s i s , a bioeconomic model for a f i s h resource w i l l be developed. An important objective of the research i s that the model should provide a good representation of real world phenomena and meet theoretical properties that seem appropriate for the relevant functions. The main contribution of the thesis w i l l be i n the empirical application of the model to data for the North Sea herring f i s h e r y . The thesis i s based on c a p i t a l theory. The f i s h resource i s con- sidered a c a p i t a l stock under the control of a sole resource manager, and changes i n the stock over time are viewed as investments. The objective of the sole resource manager i s assumed to be to maximize the present value of the flow of net revenues from the f i s h e r y . The properties of the dynamic bioeconomic models commonly used i n f i s h e r i e s economics, e.g. the Clark-Munro model^\ are well known. In a d d i t i o n , these models have been adapted to analyze a number of special Clark and Munro (1975). - 2 - issues. However, most of this work has been done i n a theoretical framework. Indeed, Munro and Scott (1984) i n their recent survey of the f i e l d of f i s h e r i e s economics point out that the development to date i n the area of empirical estimation of f i s h e r i e s models has been l i m i t e d , and that the scope for further research i s great. In many ways this f i e l d i s s t i l l in i t s infancy. To put the thesis i n perspective, some of the c h a r a c t e r i s t i c s of the herring fishery w i l l be explained, followed by a discussion of the bioeconomic model and the empirical work that w i l l be undertaken. The bioeconomic model consists of a model of population dynamics and a net revenue function for the f i s h e r y . An important behavioural c h a r a c t e r i s t i c of North Sea herring that i s l i k e l y to influence both the b i o l o g i c a l growth function and the production technology i s the schooling behaviour. Schooling f i s h contract their feeding and spawning range as the stock i s reduced, with the size of schools often remaining unchanged. However, with modern f i s h f i n d i n g equipment harvesting can be p r o f i t a b l e even at low stock l e v e l s . There- f o r e , herring and other clupeids are especially vulnerable to predation by man, as nature's brakes on stock depletion may not be very e f f e c t i v e . Many of the important f i s h e r i e s of the world are based on c l u peids (anchovies, sardines, c a p e l i n , h e r r i n g s ) . However, i n the empirical l i t e r a t u r e there i s apparently no bioeconomic study of the management of a clupeid fishery and of the special problems caused by the schooling behaviour. It i s hoped that this thesis can present a c o n t r i - bution also by drawing attention to some of these problems. - 3 - The model of population dynamics w i l l recognize that a f i s h popul a t i o n commonly can be divided into several subpopulations. In a bio- economic as opposed to a purely b i o l o g i c a l context, the primary concern i s to i d e n t i f y the harvestable population. Since for North Sea herring this coincides with the spawning stock, the model w i l l be formulated i n terms of this variable rather than t o t a l biomass, as i s commonly the case. The model of population dynamics w i l l be formulated i n terms of discrete time, which i s generally more r e a l i s t i c than the more common continuous time models. The model w i l l distinguish between natural growth and mortality i n the existing stock as opposed to new recruitment to the stock. Moreover, recruitment w i l l take place with a time l a g , which i s r e a l i s t i c for most species. Alternative functional forms for the model of population dynamics w i l l be specified and estimated based on time series data for 1947-82. In f i s h e r i e s economics, the production function i s often treated 2) in a rather r e s t r i c t i v e manner. The Schaefer function (Schaefer, 1957), which i s linear i n both e f f o r t and stock s i z e , i s commonly used. While this function may be useful for expositional purposes, i t i s presumably less appropriate i n empirical applications, as i t a p r i o r i imposes a number of questionable assumptions. The approach of this the- sis w i l l be to specify a general production function, where output H = qES, where H i s harvest, E i s e f f o r t , S i s stock size and q a constant c a t c h a b i l i t y c o e f f i c i e n t . To economists, this i s a special case of the Cobb-Douglas f u n c t i o n . - 4 - (harvest) i s a function of variable inputs, stock size and other fixed factors. Some of the c h a r a c t e r i s t i c s particular to this fishery w i l l be taken into account i n the modelling of the production technology. In the production f u n c t i o n , the presence of fixed f a c t o r s — i n the form of c a p i t a l — w i l l be acknowledged. It i s assumed that there i s excess capacity i n the f l e e t with no alternative use, causing the opportunity cost of c a p i t a l to be low. This presumably represents a r e a l i s - t i c description of many developed f i s h e r i e s in the short to medium 3) long run . The long run s i t u a t i o n , when c a p i t a l no longer i s redun- dant, w i l l not be considered. E x t e r n a l i t i e s are frequently encountered in f i s h e r i e s . Static e x t e r n a l i t i e s are associated with the interactions between boats on the fishing grounds. These can be negative, e.g. due to overcrowding and gear c o l l i s i o n s , or p o s i t i v e , e.g. due to sharing of information about locations of f i s h . The l a t t e r may p o t e n t i a l l y be important in the case presently under consideration, because of the importance of the search phase in a schooling f i s h e r y . The e x t e r n a l i t i e s need to be considered i n the estimation of the production function (Brown, 1974). For the North Sea herring f i s h e r y , cross-sectional data are available for the years 1968, 1971 and 1975. These data f a c i l i t a t e d i r e c t estimation of the production function (Cobb-Douglas). In addi- t i o n , aggregate time series data are available for the period 1963-77. In this period, the fishery was characterized by open access, so that C l a r k , Clarke and Munro (1979), i n an a n a l y t i c a l model, consider a case similar to this one. - 5 - myopic profit-maximizing behaviour w i l l be assumed. vest supply functon w i l l be estimated. A Cobb-Douglas har- By duality theory, the para- meters of the corresponding production function can be derived. The combination of the model of population dynamics and a net revenue function for the fishery represents the complete bioeconomic model. An intertemporal p r o f i t maximization w i l l be performed for the case of a sole owner, and an i m p l i c i t expression for the optimal stock l e v e l i s derived. In this way, the stock e x t e r n a l i t y — t h e effect of one period's harvest on the next period's stock l e v e l — w i l l be taken into account. Furthermore, the relationship between the optimal stock l e v e l and the production technology w i l l be analyzed. This m o d e l — i n conjunction with the empirical r e s u l t s — w i l l be used to analyze the optimal management of the North Sea herring f i s h ery. By this i s meant the optimal stock l e v e l with corresponding har- vest quantity for the case of a sole resource manager, which w i l l be estimated under different sets of assumptions. The thesis w i l l not deal with the p r a c t i c a l implementation of such a p o l i c y , which involves matters l i k e the adjustment phase, the regulation of e f f o r t and transboun- dary i s s u e s , although these matters w i l l be discussed. The analysis w i l l be performed i n the context of a deterministic model. The thesis i s organized as follows. In Chapter 2, the a n a l y t i c a l model i s developed, i . e . , the model of population dynamics and the production and p r o f i t functions. A dynamic optimization i s performed for the case of a sole owner, and the resulting equilibrium condition for the stock l e v e l i s contrasted to the free entry solution (bionomic - 6 - equilibrium). The relationship between the optimal stock l e v e l and the production technology i s analyzed. In Chapter 3, the empirical model i s specified and estimated. Various functional forms for the model of population dynamics are considered. The particulars of the fishery i n question—due to the school- ing behaviour of h e r r i n g — a r e discussed and taken into account i n the formulation of the production technology. A Cobb-Douglas production function i s estimated based on the cross-sectional data. For the aggre- gate f i s h e r y , alternative s p e c i f i c a t i o n s of the harvest supply function (Cobb-Douglas) are estimated, and the parameters of the corresponding production functions are derived. In Chapter 4, the dynamic optimization model of Chapter 2 i s combined with the empirical results of Chapter 3. Estimates of the bio- nomic equilibrium and the optimal stock l e v e l with corresponding harvest quantities are presented. The s e n s i t i v i t y of the optimal stock l e v e l to changes i n the discount r a t e , the cost/price r a t i o and the parameters of the production function i s i l l u s t r a t e d . Examples of approach paths to the steady state stock l e v e l are given, and matters concerning fishery regulations, transboundary issues and uncertainty are b r i e f l y discussed. The major results are summarized i n Chapter 5. Some avenues of future research, based on the material i n this t h e s i s , are o u t l i n e d . Appendix 1 contains the technical derivations of the dynamic optimization problem. in Appendix 2. The data for the empirical estimations are given - 7- 2.0 THE BIOECONOMIC MODEL The purpose of this chapter i s to develop the a n a l y t i c a l model on which this thesis i s based and to perform a dynamic optimization for the sole owner case. mics. Section 2.1 formulates the model of population dyna- The s t a t i c production and p r o f i t functions are treated i n Section 2.2. The concept of a sole owner and an intertemporal p r o f i t function are introduced i n Section 2.3. A dynamic optimization i s undertaken, and the solution i s contrasted to the bionomic equilibrium. The relationship between the sole owner stock l e v e l and the production technology i s analyzed. An attempt has been made to develop a general model, although i t was formulated with a s p e c i f i c application i n mind. The a p p l i c a b i l i t y of the model to species other than North Sea herring w i l l be discussed in the summary of the chapter. 2.1 The Model of Population Dynamics In i t s most simple form, changes i n the biomass of a f i s h stock over time w i l l come from additions to the stock due to recruitment and natural growth and deductions from the stock due to natural mortality and harvesting as i l l u s t r a t e d i n Figure 2.1 (adapted p. 25). from Ricker, 1975, - 8- Recruits Usable 1^^-Natural mortality stock Harvesting Natural growth Figure 2.1: Stock dynamics A fish population can generally be divided into several subpopulations. In a bioeconomic as opposed to a purely biological context, the primary concern is to identify the harvestable population. Since for North Sea herring this coincides with the spawning biomass, the model will be formulated in terms of this variable. The Interactions between recruitment, natural growth, natural mortality and harvesting have been fundamental in the development of the model of population dynamics. The following delay-difference equation w i l l be used to explain changes in the biomass over time: (2.1) G S t + i = (S t - H t )e " M + G(S t _ Y ) where St = spawning biomass in period t Ht = harvest quantity in period t G • mean instantaneous natural growth rate M = mean instantaneous natural mortality rate G(St_y) = recruitment to the stock, taking place with a delay of Y periods. The production function, H t , w i l l be discussed in Section 2.2. - 9 - Clark (1976a) has developed a model which i s similar to this one. However, Clark's model of population dynamics i s in terms of num- bers and not weight. Another delay-difference model i s the one devel- oped by Deriso (1980). The model can be formulated i n terms of t o t a l biomass, X, by replacing S with X i n equation (2.1). Recuitment w i l l then generally occur with a time lag of one period, giving a ^-value of zero. The reasoning behind the model requires some a m p l i f i c a t i o n . The f i r s t part of the right hand side of equation (2.1) denotes changes i n the biomass due to natural growth, natural mortality and harvesting. In the model, i t i s assumed that harvesting occurs i n a short season at the beginning of the period. The escapement, S t - Hfc, i s l e f t to grow at the net instantaneous growth rate G - M. Therefore, i n the absence of recruitment, changes i n the spawning biomass over time are given by (2.2) St+1 = (S t - H t ) e G_M If the fishery took place at the end of the period, we would have (2.2') St+1 = Ste G_M - Ht Yet another alternative would be the case of a continuous fishery throughout the period, (2.2") St+i = Ste G_M F " , where F i s instantaneous f i s h i n g m o r t a l i t y . Equation (2.2") i s known as the Ricker model (Ricker, 1975). Flaaten, 1983) A seasonal fishery (Bradley, 1970; can be formulated as a further development of this model. The q u a l i t a t i v e nature of the model i s unaffected by these specifications. Thus, the choice between them depends on what i s most - 10 - r e a l i s t i c for the fishery i n question. Since F i n a herring fishery may depend on stock size (Ulltang, 1980), i t i s appropriate to focus on harvest quantity rather than f i s h i n g m o r t a l i t y . The second part of the right hand side of equation (2.1) represents addition to the biomass due to recruitment, which i s assumed to occur at discrete time i n t e r v a l s . Moreover, r e c r u i t s w i l l normally j o i n the parent population several years after spawning. (2.3) Rt+1 = where Rt+l i s We postulate that g(S t ) t n e number of r e c r u i t s to the juvenile population as a function of the previous period's spawning biomass. A certain f r a c t i o n , X, w i l l survive the juvenile s t a g e ^ and j o i n the spawning biomass, so that (2.4) g(St_y)X i s the number of r e c r u i t s j o i n i n g the spawning biomass with a delay of Y periods. age. (2.5) The delay occurs while the juveniles mature to spawning Letting w denote the weight of new r e c r u i t s , we get G(S t _ T ) = g(S t _ Y )Xw where G(S -y) denotes recruitment i n weight to the spawning biomass. t North Sea herring spawn i n September every year. The following year a number of r e c r u i t s , called zero-group herring, j o i n the juvenile population as indicated by equation (2.3). survivors (equation 2.4) become sexually mature and j o i n the spawning or adult population (equation 2.5). The model may stock density. After another two years the Thus, i n this case, Y = 2, and the be generalized by l e t t i n g X depend on juvenile - 11 - delay between spawning and recruitment to the spawning stock i s three years. Time periods are defined so that the beginning of one period coincides with the point of time when a yearclass of new r e c r u i t s enters the spawning biomass. Before proceeding to the production function, the behaviour of the model under natural conditions w i l l be considered. In the absence of f i s h i n g , equation (2.1) i s reduced to (2.1•) St+1 = Ste G_M + G(S t _ Y ) = S t e 6 + G(St_y), 6 = G - M, where 6 i s the mean net natural growth r a t e . If there i s no f i s h i n g , changes i n the biomass over time are given by equation (2.1'). Under natural conditions, a f i s h population w i l l grow towards i t s carrying capacity, which i s the upper l i m i t of the stock size as determined by environmental conditions. However, i n equat i o n (2.1'), i t i s assumed that G and M are constants. In r e a l i t y , natural growth w i l l be density dependent because, ceteris paribus, there w i l l be r e l a t i v e l y more food available to a small stock than to a large one. Natural mortality may also be density dependent, e.g. i f the 2) effectiveness of predation depends on stock size or i f cannibalism occurs or becomes more frequent at high stock d e n s i t i e s . Hence one expects net natural growth to depend on stock density, with 6 = 6(S) and S(?) < 0, 2) This i s believed to be the case for schooling f i s h l i k e herring ( c f . Section 3.1). - 12 - where S i s the carrying capacity of the stock; otherwise an upper l i m i t to the size of the biomass would not exist under natural conditions. Equation (2.1) can now (2.6) St+1 be restated: = (S t - H t ) e 6(S t ) + G(S t _ y ) For the biomass to remain at i t s carrying capacity, one of the following relationships between recruitment and net natural growth i s required to hold: i. ii. G(S") = 0 * = 0 fi(?) G(S) > 0 + 6(<F) < 0 such that S = G(S)/(1 - e 6 ( S ) ). The second r e l a t i o n s h i p i s probably the more r e a l i s t i c one. When the model i s estimated empirically (Section 3.1), we check which one, i f any, of these two conditions 2.2 A Fishery P r o f i t will holds. Function It i s an objective of this analysis to treat the production cess i n a manner consistent with economic theory. the production pro- In fishery economics, function i s generally treated in a rather r e s t r i c t i v e manner. Much of the l i t e r a t u r e i s based on the Schaefer production function (Schaefer, 1957), H = qES, q constant. Harvest (H) i s l i n e a r i n both e f f o r t (E) and stock size (S) and i s thus homogeneous of degree two. These assumptions simplify the dynamic optimization problem considerably. The conditions under which this model holds are quite stringent (Clark, 1976). - 13 - The approach here i s to specify a general production function: (2.7) Ht = H(E t ;S t ,K t ) -> 3) Here, E t i s an n-dimensional vector of variable inputs, while fixed inputs consist of stock size (S t ) and an m-dimensional vector of other factors ( K t ) . The l a t t e r w i l l t y p i c a l l y describe technological aspects of the f l e e t (gear types, number of boats, size d i s t r i b u t i o n , f i s h finding equipment, e t c . ) . S and K are fixed i n any given time period, but can change over time. In cross-sectional analyses, factors that are constant and equal for a l l boats (e.g. stock size and aggregate number of boats) w i l l be suppressed i n the functional form. The production function i s assumed to be concave i n E and 4) increasing i n both variable and fixed inputs (Lau, 1978). P r o f i t s can be written as revenues minus variable costs: (2.8) P t = p t H(E t ;S t ,K t ) - i ^ 4 t E i t = p t [H(E t ;S t ,K t ) - ? c i t E ± t ] i where p t i s (nominal) price per unit output, c | t , i = l , . . . , n , i s (nominal) price per unit input i and c i t = c ; [ t /p t i s the normalized price per unit input i . Maximizing p r o f i t s with respect to variable inputs, expressions for the optimal l e v e l s of inputs as functions of prices and fixed factors are (2.9) E* derived: = f i ( c t ; S t , K t ) , i = l,...,n 3) An arrow denotes a vector. 4) Weak i n e q u a l i t y i s implied; s i m i l a r l y for other properties of the production and p r o f i t function throughout the t h e s i s . - 14 - Substituting (2.9) into (2.8) and dividing by output p r i c e , y i e l d s the normalized p r o f i t function (2.10) TT* = H(E*;S t ,K t ) - ^ c i t E ^ t = G(c t ; S t ,Kt) i This function gives maximized p r o f i t s for a given set of values {c t ;S t ,£(-}. The normalized p r o f i t function i s (Lau, 1978): 1. decreasing and convex i n normalized prices of variable inputs, 2. increasing i n the nominal price of output, 3. increasing i n fixed f a c t o r s , and 4. bounded, given S t and Kt. In this model formulation, stock size enters into the p r o f i t function as a fixed f a c t o r , while stock dynamics are disregarded. reason i s that the fishery to be considered was unregulated and terized by open access. i s assumed. The charac- Therefore, myopic profit-maximizing behaviour An intertemporal p r o f i t maximization, where stock dynamics are e x p l i c i t l y taken into account, w i l l be performed in Section 2.3. The data required for empirical analysis depend on which funct i o n s ) one desires to estimate. P r o f i t s , input demand and harvest sup- ply are a l l functions of normalized p r i c e s , stock size and fixed fact o r s , with p r o f i t s , input quantities and output quantity as the respect i v e dependent v a r i a b l e s . The a v a i l a b i l i t y of data w i l l determine which function i t i s feasible to estimate. The p r o f i t function i s often more amenable to empirical estimation than the production f u n c t i o n . The problems of simultaneity bias associated with d i r e c t estimation of the production function are - 15 - avoided. Estimation of a multi-input and/or multi-output system of equations i s also e a s i l y f a c i l i t a t e d . tions for North Sea herring w i l l be 2.3 In Section 3.2 production func- estimated. The Intertemporal P r o f i t Function In this section, stock dynamics and the concept of a sole owner w i l l be introduced. A dynamic optimization w i l l be undertaken and an i m p l i c i t expression for the optimal stock l e v e l derived. contrasted to the open access s o l u t i o n . This w i l l be In addition, the relationship between the optimal stock l e v e l and the production technology w i l l be analyzed. 2.3.1 Dynamic Optimization The production f u n c t i o n , as defined i n equation (2.7), contains inter a l i a a vector of variable inputs. As stated i n Section 2.2, this does not i n p r i n c i p l e represent any problems for empirical estimation of the production function or the dual, s t a t i c p r o f i t function. In a dynamic optimization, however, the situation i s somewhat different. Problems with multiple state variables are hard to solve analytically. Frequently, this can only be done after a severe simpli- f i c a t i o n of the problem (Clark, 1976). Therefore, i t w i l l henceforth be assumed that there i s only one control v a r i a b l e , f i s h i n g effort"*^ ( E t ) . Another reason for considering just one input i s that v a r i a b l e inputs commonly are combined i n fixed proportions, e.g. c a p i t a l and labour. Bjjzfrndal (1984) provides empirical evidence for this assert i o n . The concept of f i s h i n g e f f o r t i s discussed by Rothschild (1977). - 16 - In the empirical work to be undertaken, E t w i l l be defined as the number of boatdays. It i s assumed that the f i s h stock i s managed by a sole owner or resource manager whose objective i s to maximize the present value of net revenues from the f i s h e r y . (2.11) Normalized net revenues can be restated as 7rt = H ( E t ; S t , £ t ) - c t E t We assume that the manager of the fishery maximizes the present value of (2.11) subject to changes i n the population l e v e l given by (2.6) St+1 = (S t - H t ) e 5(S t ) + G(S t _ Y ) and the f e a s i b i l i t y constraint (2.12) 0<Et<E m a x This gives a discrete time, dynamic bioeconomic model with E t and S t as control and state variables r e s p e c t i v e l y . The upper l i m i t on e f f o r t i n a given time period ( E m a x ) w i l l be determined j o i n t l y by the size of the c a p i t a l stock (the number of boats) and nature (season length). In this formulation we have e x p l i c i t l y modelled stock changes over time, but not done so for changes i n the K-vector. The reason for not considering the dynamics of the K-vector i s that there i s great overcapacity i n the f l e e t and a lack of alternative employment opportunities ( B j 0 r n d a l , 1981). Consequently, opportunity cost for cap- i t a l i s low. One extension to the Clark-Munro model has been to analyze the investment problem i n physical c a p i t a l — i . e . , the K-vector in the present n o t a t i o n — i n addition to the investment problem i n the resource itself. Clark, Clarke and Munro (1979) consider several variations of this problem. In a case of special interest to this t h e s i s , i t i s - 17 - assumed that boats have no alternative use and thereby a n e g l i g i b l e opportunity cost, but depreciate over time. Therefore, gross investment in the f l e e t i s constrained to be non-negative. Assuming that the f i s h - ery previously has been open access and starts out at bionomic e q u i l i b rium, the authors show that optimal management i s d i f f e r e n t i n the short and i n the long run. In the short run, as c a p i t a l i s redundant, only operating costs are relevant i n the determination of the optimal stock level. In the long run, however, c a p i t a l i s no longer redundant and a l l costs are thus relevant. Therefore, the optimal stock l e v e l i n the long run i s higher than that i n the short run. In the present model, c a p i t a l depreciation i s not modelled as opposed to the Clark-Clarke-Munro model. If c a p i t a l through deprecia- tion should become scarce i n the long run, and the present model thus only considers the interim period, transversality conditions should have been s p e c i f i e d . It i s acknowledged that this may with the present model s p e c i f i c a t i o n . represent a problem However, the fishery to be con- sidered i s a minor one that i s complementary to other f i s h e r i e s ( c f . Section 3.2). Therefore, investment decisions are l i k e l y to be deter- mined mainly by the prospects i n the major f i s h e r i e s . This f l e e t w i l l then be available during the North Sea herring fishery season as i t may then have no alternative employment opportunity. Under such circum- stances, c a p i t a l w i l l always be available and i t s dynamics may be d i s - regarded. In other words, one can imagine the following circumstance: The s o c i a l manager i s faced with a cost/price r a t i o ( c t ) and a f l e e t with - 18 - some given attributes (the K-vector). He must then optimize the use of variable e f f o r t ( E t ) over time, taking stock dynamics into consideration. In the model, i t i s i m p l i c i t l y assumed that a l l boats are used i n any given year. This i s a way of reducing a two-control problem to a one-control problem. Whether a l l boats actually w i l l be used depends on the cost function and the returns to the number of boats i n the f i s h ery. If there are, for example, start-up or other fixed c o s t s , i t i s conceivable that the manager under certain circumstances would l e t some of the boats be i d l e . Thus, the manager would decide both on the number of boats to deploy and the number of days to u t i l i z e them. This two- control v a r i a b l e problem w i l l be considered i n future research^ \ Returning to the optimization problem, an intertemporal p r o f i t function can be derived by maximizing (2.11) with respect to E t to the stock dynamics (2.6). subject This would y i e l d expressions for E* and T T T as functions of {c t , S t , Kt> corresponding to those derived for the myopic case i n Section 2.2. However, we w i l l here change focus s l i g h t l y and derive expressions for the optimal stock l e v e l and the corresponding harvest. The method of Lagrange m u l t i p l i e r s may be used to derive optimal equilibrium conditions. We define the Lagrangean expression T L = E { d T T ( E t ; S t , t t ) - q t [ S t + 1 - (S t - H t ) e t=0 <S(S t ) - G(S t _ Y )]} A model of this kind i s given by Lewis and Schmalensee (1982). - 19 - where d = l/(l+r) i s the discount f a c t o r , r the rate of discount and q t the discounted value of the shadow price of the resource. F i r s t order necessary conditions f o r an optimum are: 1. f | « 0, t = 0,1,2, 2. | | « 0, t = 1,2,3,... Moreover, for a steady state equilibrium, the stock must be i n equilibrium: (2.13) 6 S t + 1 - St = S (S t - H t ) e * t ) + G(S t _ y ) - S t = 0 In Appendix 1, the shadow price of the resource i s shown to be: qt = d C The following i m p l i c i t expression for the optimal stock l e v e l S* i s derived (2.14) : e 6 Y ^ * ^ — + 1] + 6'(S*)[S* - G(S*)] + d G'(S*) = 1 + r Since this i s a steady-state stock l e v e l , the corresponding optimal harvest H* i s found by rewriting (2.6): (2.15) H* = H(S*) = S* + G ( S *( ) e S S *) * 6 ;s; ' Moreover, (2.16) 6 H'(S) - 1 + ' ( S ) [ S ' G ( S ) G e ( S ) " 1 The harvest i s maximized when this derivative i s set equal to zero, i . e . , Tfg i s the derivative of the net revenue function with respect to output. The cost/price r a t i o ( c t / p t ) i s assumed constant and equal to c. - 20 - (2.17) H'(S m s y ) = 0 where S m S y i s the stock l e v e l corresponding to Maximum Sustainable Y i e l d (MSY): (2.18) MSY = H ( S m s y ) The i m p l i c i t expression for the optimal stock l e v e l at f i r s t glance appears somewhat d i f f i c u l t to i n t e r p r e t . sons. This i s for two rea- F i r s t , i n t h i s model e f f o r t i s control variable rather than har- vest, as i s common i n most models. dynamics i s f a i r l y complex. Second, the model of population These two factors can be used to elucidate the interpretation of the equation. Let harvest (H) be control variable so that the net revenue function i s given by 7Tt = T r ( H t ; S t , f t ) Performing the dynamic optimization gives the following i m p l i c i t expression for the optimal stock l e v e l : (2.19) e C S Y h Here, the term -] + 6'(S*)[S* - G(S*)] + d G'(S*) = 1 + r (TTS + ^ R ) / T T h i s the standard expression for the Marginal Stock Effect i n a discrete time nonlinear model (Clark, 1976). The d i f - ference between this term and the corresponding one i n equation (2.14) i s thus caused by the use of e f f o r t rather than harvest as c o n t r o l . From equation (2.6) the following derivative can be obtained: (2.20) 3 S t + 1 / 3 S t = 6'(S t )[S t - H t ] e ,5(S t ) = 6"(S t )[S t - G(S t )] + + 6 ( S e 6 ( S t ) e t ) by equation (2.15) - 21 - This derivative must be understood i n terms of stock productivity: the change i n stock l e v e l i n period t+1 caused by a marginal stock change i n the previous period. Using this r e s u l t , equation (2.14) can be restated: ; (2.14') Y ^j-S- + 3 S t + 1 / 3 S t = 1 + r - d G'(S*) E This equation can be interpreted i n terms of stock adjustment at the margin. The l e f t hand side represents marginal b e n e f i t s . The term 8) C T T S / TTg i s the Marginal Stock Effect , which i s a measure of the impact of stock density on marginal sustainable resource rent (Clark and Munro, 6(S*) 1975). It i s multiplied by e , the growth factor that i s applied to stock escapement ( c f . equation 2.6). In the model, harvesting occurs at 6(S*) the beginning of the time period; hence e gone i n net growth due to harvesting. represents what i s fore- Therefore, at the optimum, this factor i s applied m u l t i p l i c a t i v e l y to the Marginal Stock E f f e c t . Together, the two terms represent the net benefits from a marginal stock adjustment. The terms on the right hand side represent the marginal tunity) cost of stock adjustment. (oppor- The stock adjustment w i l l cause a The Marginal Stock Effect may be explained with reference to the equilibrium condition for 1the optimal stock l e v e l (X*) i n the ClarkMunro (1975) model: F'(X*) - c (X*)F(X*)/[p - c(X*)] = r . F(X) i s the natural production function and c(X) the unit cost of harvesting. At the optimum, the own rate of return on the r e s o u r c e — c o n s i s t i n g of ( i ) the marginal physical product (F'(X*)) and ( i i ) c'(X*)F(X*)/[p - c ( X * ) ] , the M S E — i s equal to the discount r a t e . The MSE represents the impact of stock density on harvesting costs. I n t u i t i v e l y , i t can be understood by considering that an increase i n stock size w i l l increase catch per unit e f f o r t and hence reduce unit harvesting costs. The MSE i s analagous to the wealth effect i n modern c a p i t a l theory (Kurz, 1968). - 22 - marginal change i n recruitment of G'(S*), which can be p o s i t i v e , zero or negative. Since this change takes place with a time lag of Y T periods, the discounted value amounts to d G'(S*). If G'(S*) > 0, this term i s a net benefit (stock "appreciation") that i s subtracted from the rate of discount, i . e . , 1 + r - d^G'(S*) i s the "net" cost of stock adjustment, corresponding to a net user cost i n c a p i t a l theory. Y On the other hand, i f G'(S*) < 0, - d G'(S*) > 0, and the term enters as an additional cost (stock "depreciation"). Therefore, equation (2.14) states that at the margin the net benefits from the resource should equal the net cost of stock adjustment. If there i s no discounting (r = 0 ) , equation (2.14) can by use of (2.16) be written as (2.21) CTTS/TTe Assuming + H'(S*) = 0 ciTg/TT^, > 0, t h i s implies a ° ' msy Furthermore, i f harvesting i s costless (c = 0) or there i s no stock e f f e c t i n the production function (TT s (2.21') = 0 ) , (2.21) becomes H'(S*) = H ' ( S m s y ) = 0 by equation (2.17). The various cases discussed here w i l l a l l be i l l u s t r a t e d in the estimation of the optimal stock l e v e l (Chapter 4 ) . F i n a l l y , the sole owner solution w i l l be contrasted to bionomic equilibrium, i . e . , the free entry s o l u t i o n . In an open access f i s h e r y , e f f o r t , f l e e t p a r t i c i p a t i o n and stock size w i l l a l l be determined endogenously. So f a r , a net revenue function and stock dynamics have - 23 - been described. A complete analysis of the open access fishery would i n addition require an equation describing c a p i t a l dynamics, i . e . , entry and exit of boats to the industry. This problem w i l l not be pursued 9) further in this context, but represents an avenue for future research However, bionomic equilibrium may which furthermore may used here. be estimated for given f l e e t s i z e , be changed parametrically. This approach w i l l be Then, for a given f l e e t size (K-vector), an open access fishery w i l l be characterized by the d i s s i p a t i o n of rents: (2.22) TT T = H ( E t ; S t , £ t ) - c t E t = 0 An alternative statement of t h i s condition i s given by the equality between the average product and the r e a l cost of e f f o r t : (2.22') c t = H(E t ;S t ,K t )/E t This condition can be used to solve for the amount of e f f o r t (Ero) in the open access f i s h e r y : (2.23) E<x, = <t>(c,S,K) The second condition for the open access fishery i s given by equation (2.13), i . e . , the steady state condition for the stock. Com- bining these two equations gives the following i m p l i c i t expression for SOT, the bionomic equilibrium: (2.24) [S^ - H(c,S 0 0 ,K)]e S(Soo) + G(S,J - S«> = 0 It i s commonly asserted that an open access fishery i s characterized by an i n f i n i t e discount r a t e , because fishermen w i l l not be concerned with the e f f e c t of today's harvest on tomorrow's stock size (the stock In Chapter 4, some of the factors affecting entry and exit decisions to this industry w i l l be discussed. - 24 - externality). For this reason, i n f i n i t y subscripts have been used when denoting equilibrium values for e f f o r t and stock size i n the open access fishery. An inspection of equation (2.24) shows that the open access stock l e v e l , i n addition to the b i o l o g i c a l f a c t o r s , depends on the cost/price r a t i o with increasing i n c. The differences between the sole owner and open access solutions are quite s t r i k i n g . While there are discounting and marginal adjustments i n the former, this i s not so i n the l a t t e r . 2.3.2 The Marginal Stock Effect The Marginal Stock Effect (MSE) i s defined as (2.25) MSE = cir /TT HI Costs, output price and the production technology influence the optimal stock l e v e l through the MSE, as an inspection of the i m p l i c i t expression for S* (equation 2.14) w i l l r e v e a l . Therefore, the MSE plays an impor- tant role i n the optimization problem and accordingly deserves further study. Another reason for studying the MSE i s that the underlying pro- duction technology i s nonlinear as opposed to the more commonly used Schaefer function. For this reason some new and more general r e s u l t s about the relationship between the production technology and the optimal stock l e v e l may be obtained. Moreover, the effects of changes i n v a r i - ous parameter values on the optimal stock l e v e l may be predicted. In this way the present analysis may illuminate the empirical r e s u l t s that w i l l be forthcoming (Chapter 4 ) . - 25 - The analysis w i l l be performed i n the context of the Cobb-Douglas production function H = A^sP°7rK? j j Proofs of a l l results are given in Appendix 1. By the properties of the p r o f i t function, *s > 0 However, can be both negative and p o s i t i v e . For harvesting to occur, i t i s a necessary and s u f f i c i e n t condition that TT£ The MSE can also be defined as (2.26) MSE > 0 -* MSE >_ 0 = m(c,a,po,S,K) where K = 71 F i r s t , i t can be ascertained that the optimal stock l e v e l i s increasing i n the MSE. Second, note that from equation (2.25) the MSE i s increasing i n the cost/price r a t i o and i t vanishes i n the zero cost case. 1. MSE The following results can also be derived: i s decreasing i n a provided c > 0 and ir ^ 0 with lim 11 2. MSE i s decreasing i n K provided c > 0 and TT MSE = 0. a-'4-" oo ^ 0 with lim h K> MSE = 0. » The i n t u i t i o n behind the f i r s t r e s u l t i s that p r o f i t s are increasing i n the output e l a s t i c i t y of e f f o r t . For a s u f f i c i e n t l y high ot-value, costs become n e g l i g i b l e and the MSE vanishes. A consequence of this result i s that productivity improvements over time, reflected by increases i n the value of a , would imply a diminishing importance of costs. - 26 - The implication of the second result i s that for a s u f f i c i e n t l y large f l e e t , the resource should be managed as i f there are no c o s t s . The p r o f i t function i s increasing i n K. e f f o r t i s decreasing i n K. A l s o , the optimal l e v e l of Thus, costs w i l l also be decreasing i n K and become n e g l i g i b l e for a s u f f i c i e n t l y large K. This r e s u l t may be v i s u a l i z e d by considering the search e f f e c t . The area that can be searched and/or the i n t e n s i t y of the search w i l l increase with the number of boats i n the f i s h e r y * ^ . With modern communication equipment, information about locations of f i s h can be transferred to the rest of the f l e e t . Thus, the amount of variable e f f o r t w i l l be reduced. Another phenomenon can be cited for boat s i z e , the other fixed factor that w i l l be considered i n the empirical a n a l y s i s . It can be noted that the larger the size of the boat, assuming locations of f i s h to be known, the fewer the number of t r i p s (days) that are needed to take a certain catch. The r e l a t i o n s h i p between MSE and stock size w i l l not be the subject of special a n a l y s i s . This i s because stock size enters most of the other terms of equation (2.14), while the other parameters of the MSE only appear there. F i n a l l y , we turn to the stock output e l a s t i c i t y , which may be the most i n t r i g u i n g case. This i s done by considering the range of values this e l a s t i c i t y can take on. It should be recalled that there are no start-up c o s t s . - 27 - 1. po = 0 -> TTs = MSE = 0 In this case, the f i s h stock i s not an argument i n the production function and harvesting costs are independent of stock s i z e ^ ^ . the fishery i s p r o f i t a b l e , i t i s established by equation If (2.21') that = o -*• S* = S m s y , H* = r MSY. Moreover, Clark (1982) has shown that oo + s* = 0, r H* = 0. Accordingly, for 0 < r < °°, 0 < s* < s m s y and 0 < H* < This result may MSY. be modified somewhat, because S* may go to zero for f i n i t e discount r a t e s . On the other hand, i f the fishery i s unprofitable, the stock i s l e f t unexploited at i t s carrying capacity. For positive p o values, two cases may assuming TT > 0, MSE decreasing i n p o . emerge. In the f i r s t one, w i l l f i r s t be increasing and then This i s i l l u s t r a t e d i n Figure 2.2, with corre- sponding stock l e v e l s given i n Figure 2.3. Such a r e l a t i o n s h i p could emerge, for example, i n the model of Clark and Munro (1975). This case i s analyzed by Levhari, Michener and Mirman (1981). - 28 - However, a second case may also occur, which i s when MSE and thus S* are decreasing functions of p Q . This result is obtained when the fishery is unprofitable at a zero stock effect, i.e., TTg < 0 given p o = 0; see Figures 2.4 and 2.5. The occurrence of this circumstance depends in a nontrivial manner on the relationship between various parameters of the model. The consequence i s that profits are increasing in the stock output elasticity. An increase in p o may then Increase profits, which may reduce the optimal stock level and thereby increase the harvest. Figure 2.4: MSE as a function of p o . Figure 2.5: S* as a function of p Q . - 29 - In both cases, as p o tends to i n f i n i t y , the MSE goes to zero. The i n t u i t i o n behind this result i s the following: As the output e l a s t i c i t y of the stock increases, the "contribution" of the stock to i t s own harvest increases. stock improves. In a sense, the productivity of the Therefore, as p o increases, less variable e f f o r t w i l l be required to obtain a certain harvest quantity. In the l i m i t , as the stock output e l a s t i c i t y increases to i n f i n i t y , the amount of e f f o r t and hence costs become n e g l i g i b l e . Mathematically, the result i s obtained by l e t t i n g the stock output e l a s t i c i t y go to i n f i n i t y . In p r a c t i c e , p o w i l l be i n the range zero to one. Thus, the r e s u l t may be obtained for r e l a t i v e l y speaking "large" values of the stock output e l a s t i c i t y . 2.4 Summary The purpose of this chapter was to develop a bioeconomic model that on the one hand was general enough to be applicable to a number of d i f f e r e n t f i s h e r i e s while on the other hand being i d e a l l y suited for the fishery to be analyzed i n this t h e s i s . We w i l l now b r i e f l y address this matter. One of the main c h a r a c t e r i s t i c s of the model of population dynamics i s the time lag i n the stock recruitment function. As such a time lag i s common for many, i f not most, species, this feature greatly enhances the realism of the model. In addition, the survival rate of the juvenile stage i s assumed constant, although this may e a s i l y be r e c t i f i e d by introducing density dependence. - 30 - This model i s appropriate mainly for species with a f a i r l y short delay between spawning and recruitment to the adult stock, so that the assumption of a constant survival rate i s not u n r e a l i s t i c . North Sea herring i s an obvious example i n this respect. The model may ing be less appropriate where the delay between spawn- and recruitment i s f a i r l y long. Such species may tion of two b i o l o g i c a l l y interdependent c a l l for optimiza- f i s h e r i e s , one exploiting the juvenile stock and the other the adult stock. This would involve a reformulation of the model of population dynamics and represents a potential for future research. The p r o f i t function that has been developed i s quite general. The presence of fixed factors other than stock size i n the production function probably represents r e a l i t y for many developed f i s h e r i e s . i s presumably true even i n the somewhat long run, due alternative employment opportunities for c a p i t a l . This to the lack of The model formulation i s seen to be analogous to a case analyzed by Clark, Clarke and Munro (1979). Where appropriate, the production process may a function of variable inputs and stock size only. be redefined to be Another p o s s i b i l i t y i s to introduce a start-up cost in the p r o f i t f u n c t i o n . We believe that the bioeconomic model presented i n this paper i s well suited for an analysis of the management of North Sea herring. In a d d i t i o n , i t i s appropriate for the study of a number of other species. The model's general a p p l i c a b i l i t y may refinements of i t s various parts. be further enhanced by suitable - 31 - 3.0 THE EMPIRICAL MODEL In this chapter, alternative functional forms for the bioeconomic model w i l l be specified and estimated. Section 3.1 deals with the model of population dynamics, while the production function i s treated in Section 3.2. Appendix 2 contains data on which some of the empirical work i s based. 3.1 The Model of Population Dynamics North Sea autumn-spawning herring (Clupea harengus L.) consists of three spawning stocks, with spawning grounds east of Scotland, England and i n the English Channel. However, the three stocks mix on the feeding grounds i n the central and northern North Sea, rendering i t impossible to d i s t i n g u i s h between catches from the three stocks. It i s therefore customary to treat the three stocks as one u n i t , as indeed i s the approach i n this study. North Sea herring becomes sexually mature in agegroup two. Thus, the juvenile stock consists of agegroups zero and one, while the adult or spawning stock consists of agegroups two and older. e r a l l y l i v e s u n t i l the age of ten. The herring gen- On the biology of North Sea herring, see S a v i l l e and Bailey (1980). The model of population dynamics (equation 2.6), which i s defined in terms of the spawning stock, consists of two parts: the stockrecruitment function and the net growth function. We w i l l f i r s t look at these two parts separately and then combine them. - 32 - 3.1.1 The Stock-Recruitment Function According to Ricker (1975), three types of recruitment can be distinguished: 1. knife-edge recruitment, where a l l f i s h of a given age become vulnerable at a particular time i n a given year, and their v u l n e r a b i l i t y remains constant throughout their l i v e s ; 2. platoon recruitment, where the v u l n e r a b i l i t y of a yearclass increases gradually, but during any year each individual f i s h i s either f u l l y catchable or not catchable; and 3. continuous recruitment, where there i s a gradual increase i n vulnera b i l i t y of members of a yearclass over a period of two or more years. Continuous recruitment i s probably most common, while knife-edge recruitment i s least common. However, both platoon and continuous recruitment are often approximated by knife-edge recruitment i n analyti c a l models. The model we have formulated does assume knife-edge recruitment. It may also be noted that available data i n many instances presupposes t h i s type of model, as indeed i s the situation for North Sea herring. Ricker (1975, p.281) discusses some properties which are desirable i n a curve of r e c r u i t s against parent stock: 1. It should pass through the o r i g i n , so that when there i s no adult stock there i s no reproduction. - 33 - 2. It should not f a l l to the abscissa at higher stock l e v e l s , so that there i s no point at which reproduction i s completely eliminated at high d e n s i t i e s . 3. The rate of recruitment (recruits/parents) should decrease continuously with increases i n parental stock. 4. Recruitment must exceed parental stock over some range of parental stock values (when measured i n equivalent u n i t s ) . The following alternative functional forms w i l l be used to estimate the relationship between r e c r u i t s i n period t+1 ( R t + i ) and spawning biomass in period t ( S t ) : = Ste a(1 S - t 1. R 2. Rt+i = aS t /(b + 3. R t + i = as|? 4. R t+i = aS t - bsj t+1 /b > St) The f i r s t three functional forms are known as the Ricker, Beverton-Holt and Cushing stock recruitment functions respectively (Cushing, 1977). The fourth s p e c i f i c a t i o n i s a quadratic function. Ricker function i s the only one to s a t i s f y the four properties l i s t e d above. For the Ricker function, i. lx - R = S for S = b and 3R ^S=0 = 6 a For the Beverton-Holt function, i. lim R = a The - 34 - Moreover, the quadratic function i s only well behaved for 0 _< S _< a/b. It should be noted that while recruitment here i s dealt with i n deterministic manner, i t i s i n practice often a highly stochastic process. Ricker notes that "year-to-year differences i n environmental c h a r a c t e r i s t i c s cause fluctuations i n reproduction at least as great as those associated with v a r i a t i o n i n stock density over the range observed - sometimes much greater" (Ricker 1975, p. 274). For this rea son, i t i s often d i f f i c u l t to detect any clear relationship between the number of r e c r u i t s and parent stock size from observed data. Appendix 2 gives the data on which these regressions are based: annual spawning biomass and number of r e c r u i t s for the period 1947-81. The data, estimated by v i r t u a l population analysis, are given i n research reports published by the International Council for the Exploration of the Sea. The spawning biomass consists of agegroups two and o l d e r . Spawning takes place i n September each year. New r e c r u i t s enter the juvenile stock one year l a t e r as zero-group h e r r i n g . Figure 3.1 i s a scatter diagram r e l a t i n g the number of r e c r u i t s zero-group h e r r i n g — i n year t+1 ( R t + j ) to the spawning biomass i n the previous year ( S t ) . 1956, We note the high number of r e c r u i t s i n the years 1960 and 1981, which i s presumably due to exceptionally favourabl environmental conditions. This i s a phenomenon that i s known to occur for clupeids at certain i n t e r v a l s . Ulltang (1980) suggests that the stock-recruitment r e l a t i o n exhibits depensation at low stock l e v e l s . This means that the rate of - 35 - recruitment ( r e c r u i t s / p a r e n t s ) — o r net growth, i n a growth m o d e l — i s increasing over a range of stock d e n s i t i e s ^ . This i s contrary to the third property of stock-recruitment functions as outlined by Ricker, but i s not uncommon for schooling f i s h . As noted above, schools serve among other purposes as protection against predation. Predation may become less e f f e c t i v e at higher stock l e v e l s , which means that the r e l a t i v e mortality rate can be decreasing in stock s i z e . This phenomenon may give r i s e to a depensatory stock- recruitment curve (Clark, 1976, ch. 7 ) . If there i s depensation, this w i l l influence the speed of recovery i f actual stock l e v e l i s i n this range. Indeed, i t may cause the 2) recovery of overexploited clupeids to be very slow None of the stock-recruitment functions defined above exhibits depensation. For North Sea h e r r i n g , however, the present stock l e v e l i s probably outside the range where there may be depensation. Thus, we need not be concerned about this phenomenon i n our a n a l y s i s . The estimated regressions for the four stock-recruitment functions postulated above are given i n Table 3.1. The relations between R t+ 1 gives maximum r e c r u i t - and S t are drawn i n Figure 3.1. Table 3.2 ment for the Rt+i = g(S t ) regressions. Mathematically, depensation corresponds to an i n f l e c t i o n point: the second derivative i s equal to zero, while the f i r s t derivative changes sign from positive to negative. 2) An example i s probably given by Norwegian spring-spawning ring. her- - 36 i Figure 3.1: The Stock Recruitment Relationship - 37 - Table 3.1: Estimated Stock-Recruitment Functions Functional form Regression Ricker ln(R/S) = BevertonHolt (1/R) = Cushing InR = 3) Quadratic 1) 2) 3) R 3 2.87** 0.81** x 10~ S (16.68) (-6.90) 0.32 x 10 (0.43) -4 + 0.11** x (1/S) (5.17) 5.89** + 0.40**lnS (5.99) (2.75) R = 12.74**S (6.53) 2 2 F DW 0.59 47.62 0.65 26.77 — 0.36 7.54 - - 0.12 0.004**S (-4.57) 2) 1.64 1.78 Time series: 1947-81 (n = 35). R i s measured i n m i l l i o n s , while S i s measured i n 1,000 tonnes. The Beverton-Holt and Cushing models have been estimated with f i r s t order autocorrelation, using an i t e r a t i v e Cochrane-Orcutt procedure. In the ordinary least squares regressions of these two models, the Durbin Watson s t a t i s t i c s were 1.25 and 1.32 r e s p e c t i v e l y , t s t a t i s t i c s i n parentheses. ** denotes s i g n i f i c a n t at 95% l e v e l . To estimate the F s t a t i s t i c , the f i r s t observation had to be excluded i n the Beverton-Holt and Cushing functions. For the quadratic function, the regression i s forced through the o r i g i n . Therefore, the F test i s i n v a l i d and the R-square may be i n c o r r e c t . The function i s well defined f o r 0 < S < 3.07 m i l l i o n tonnes. Table 3.2: Maximum Recruitment S 1,000 tonnes Functional form Ricker: 2 R = Se ' Beverton-Holt: R = 87(1 " S/3 ' 551 1,240 > 31 450 S ? 0 < _——— 3,480 + S OO 0 Cushing: R = 361.6e^*^ ^ Quadratic: R = 12.74 S - 0.004 S CO 2 1,530 R millions 8,040 31,450 CO 9,800 - 38 - The Ricker and Beverton-Holt functions explain 59% and 65% respectively of the t o t a l variance of the dependent v a r i a b l e . Taking the time span, with possible changes i n the environmental variables into account, the degree of explanation i s f a i r l y good. However, this i s not so much the case for the other two functions. One of the parameters i n the Beverton-Holt function i s insignificant. The inverse of this parameter gives maximum recruitment (Table 3.2), which i s considerably higher than what has actually been observed during the data period. For this reason, we must be cautious in using this function for p r e d i c t i o n s . The Cushing function has been estimated for a number of A t l a n t i c and P a c i f i c herrings with point estimates of b i n the range 0.2 to 0.7 (Cushing, 1971). The present estimate i s also i n this range. For this function, recruitment i s continously increasing i n stock s i z e . The above regressions relate zero-group herring i n year t+1 to the previous year's spawning biomass. to agegroup two. However, not a l l r e c r u i t s survive The survival rate of equation (2.4) i s defined as follows: X = e^V-Mi) where MQ and respectively. i s the natural mortality rate of yearclass zero and one Estimates of M Q and are 0.4 and 0.3 3) . The mean These estimates include f i s h i n g mortality for j u v e n i l e h e r r i n g . M and are the sums of average f i s h i n g mortality f o r the period 1976-80 (Anon., 1982) and natural mortality, estimated to be 0.10 ( S a v i l l e and B a i l e y , 1980). Fishing mortality i s mainly due to bycatches of juvenile herring. If these could be reduced, the y i e l d from the spawning stock would increase. Q 39 - - weight of agegroup two herring i s estimated to be 126 grams (Anon., 1977). We then have estimates of a l l parameter values of the stock- recruitment equation (2.5): (2.5') (M +M G(S t _ 2 ) = g ( S t _ 2 ) e - 0 l > w where w i s the mean weight of new r e c r u i t s . If the Ricker function i s used, this becomes: 2 G(S t _ 2 ) = S t _ 2 e ' 87(1 S - t-2/3,551)e-0.70>126 = 1.10St_2e-°- 81 x 10 " 3s t-2 where G(S t _ 2 ) i s measured i n 1,000 tonnes. The stock-recruitment functions for North Sea herring have been estimated on the basis of data for 1947-81. This period i s probably not long enough for any major long-run changes i n the aquatic environment to 4) have taken place . Moreover, data that can be used to test the effects of possible changes i n the aquatic environment on recruitment are not available. 3.1.2 The Net Growth Function The basic postulate about the net natural growth r a t e , 6, i s that i t i s related to the size of the biomass. The value of 6 has been estimated annually for the time period 1947-82 (Appendix 2 ) . Two funct i o n a l forms for the relationship between <5 and S are s p e c i f i e d : Posthuma (1971) shows that temperature conditions on the spawning grounds during the incubation time seem to affect yearclass strength, possibly through d i f f e r e n t i a l egg mortality at d i f f e r e n t temperatures . - 40 - i: S(S) = a + bS ii: 6(S) = a + bS + cS 2 The results of the regressions are given i n Table 3.3. Table 3.3: Estimation of Net Natural Growth Functions 2 Functional form 6(S) = 0.15** 0.43** x 10" (7.96) (-3.63) 6(S) = 0.11** + 0.46 x 10" (4.52) (1.25) 1) 4 4 S S - 0.27** x 10" (-2.51) 7 S 2 R F DW 0.37 20.31 1.44 0.47 14.87 1.82 Time series: 1947-82 (n = 36). Stock size as of January 1 has been used. The Durbin Watson s t a t i s t i c for the f i r s t regression i s i n the indeterminate range, but autocorrelation has not been corrected f o r . The estimate of the f i r s t order rho i s only s i g n i f i c a n t at the 90% confidence l e v e l . In a d d i t i o n , correcting for autocorrelation causes only marginal changes i n parameter estimates, t s t a t i s t i c s i n parentheses. ** denotes s i g n i f i c a n t at 95% l e v e l . Table 3.4: C h a r a c t e r i s t i c s of Function S^e Maximum net growth Functional form a Ste " St;e bS t a+bSt-cS Stock l e v e l M i l l i o n tonnes Net growth M i l l i o n tonnes Level of zero growth M i l l i o n tonnes 1.71 0.14 3.49 1.84 0.20 3.05 2 The linear regression of 6 on S explains 37% of the t o t a l v a r i ance of the dependent v a r i a b l e . Moreover, 6'(S) < 0 for a l l stock levels. The quadratic regression on S explains 47% of the t o t a l - 41 - variance of the dependent v a r i a b l e . The degree of explanation i s r e l a - t i v e l y good, considering the time-span and effects of possible changes i n the aquatic environment. Furthermore, the net natural growth r a t e , while positive at low stock l e v e l s , i s continuously decreasing with increases i n the biomass l e v e l . In the absence of recruitment and harvesting, changes i n the stock l e v e l are given by ( c f . equation 2.2): s t+l - s t e This function i s graphed i n Figure 3.2 for the two cases under considera t i o n , and some of i t s c h a r a c t e r i s t i c s are given i n Table 3.4. There are two main differences between the two functions. First, although maximum net growth occurs at roughly the same stock l e v e l s , net growth i s considerably higher i n the quadratic model. The second d i f - ference i s that net growth remains positive u n t i l a considerably higher stock l e v e l i n the linear than i n the quadratic model. These r e s u l t s w i l l have consequences f o r the model of population dynamics. For the quadratic net growth function, an interesting feature of the combined function i s that i t exhibits depensation at stock l e v e l s lower than 0.58 m i l l i o n tonnes. This effect i s not present when the l i n e a r net growth function i s used"^. Thus, there i s a q u a l i t a t i v e d i f ference between the two models. Both parts of the model of population dynamics have now been estimated and we turn to the combined model. ~^Both the f i r s t and second derivatives of S e ^ ^ are s t r i c t l y p o s i t i v e f o r this function; hence there can be no depensation. - 42 - Ste fi(S) Million tonnes (i) (ii) a bS Ste " t Ste t t Figure 3.2: The Net Growth Function - 43 - 3.1.3 The Complete Model In the absence of f i s h i n g , the f i s h stock w i l l develop according to the delay difference equation St+1 6 = S t - e ^ ^ + G(S t _ 2 ) Various functional forms both for the stock recruitment and the net growth function have been estimated. The d i f f e r e n t alternatives w i l l now be combined i n the model of population dynamics. Two points are of special i n t e r e s t : 1. 2 * S, the carrying capacity of the stock; and S m S y , the stock l e v e l corresponding to Maximum Sustainable Yield (MSY). S m g y i s found by solving equation (2.17) and MSY i s subsequently given by (2.18). Estimates of the carrying capacity and S m S y are given i n Tables 3.5 and 3.6 r e s p e c t i v e l y . In Section 2.1, three alternative specifications of the harvesting process were given (equations 2.2, 2.2' and 2.2"). While the development of the stock under natural conditions and thus S w i l l be unaffected by the choice of harvest function, i t w i l l have some effect on the estimates of S m S y and MSY. An implication of this i s that the timing of the seasonal harvest may be of importance for the harvest quantity (see also Flaaten, 1983). - 44 - Table 3.5: The Carrying Capacity of North Sea Herring. M i l l i o n tonnes. Net growth function Stock-recruitment function Linear Quadratic Ricker 4.29 3.55 Beverton-Holt 7.87 4.73 Cushing 6.43 4.18 Quadratic 3.49 3.06 Table 3.6: Estimates of and MSY. S „ m<s Million tonnes. Net growth function Linear Quadratic Stock-recruitment function °msy MSY °msy MSY Ricker 1.42 0.58 1.57 0.61 Beverton-Holt 3.75 0.99 2.80 0.93 Cushing 2.92 0.62 2.39 0.64 Quadratic 1.61 0.69 1.67 0.72 The Beverton-Holt stock-recruitment function gives higher e s t i mates of S than any of the others. This i s due to the one insignificant parameter for this function, which gave a maximum recruitment that was u n r e a l i s t i c a l l y high. S i m i l a r l y , recruitment i s continuously increasing i n stock size for the Cushing function, which accounts for the high carrying capacities for this case. - 45 - The carrying capacity i s higher for the l i n e a r than for the quad r a t i c net growth function, given stock-recruitment function. The reason i s that net growth i s positive up to a higher stock l e v e l for the linear than for the quadratic net growth function ( c f . Figure 3.2). At the carrying capacity, net growth i s i n a l l instances nonposit i v e , while recruitment i s nonnegative. The stock of North Sea herring may i n recent times have been closest to i t s carrying capacity i n the early post World War II years, due to low f i s h i n g pressure during the war. Stock estimates f o r this period range between three and four m i l l i o n tonnes. According to Table 3.5, this indicates that the Ricker and the quadratic stock recruitment functions may y i e l d the more r e a l i s t i c r e s u l t s . The r e s u l t s i n Table 3.6 show that S m s y i s somewhat sensitive to choice of growth function. However, this i s less so for the MSY. In addition, the MSY i s r e l a t i v e l y stable across stock-recruitment funct i o n s , except for the Beverton-Holt case. The model of population dynamics that w i l l be used i n the estimations of the optimal stock l e v e l (Chapter 4) w i l l be the combination of the Ricker stock-recruitment function and the quadratic net growth function. There are a number of reasons for that. F i r s t , the Ricker func- tion i s preferred on a p r i o r i grounds to i t s a l t e r n a t i v e s ^ . Second, from an empirical point of view, these two functions perform as well as The fact that this function does not exhibit depensation i s not important, as fishery management presumably w i l l ensure that this phenomenon w i l l not be of p r a c t i c a l importance i n the future. - 46 - or better than their a l t e r n a t i v e s . T h i r d , the predictions of this model (Tables 3.5 and 3.6) appear to be quite reasonable. of population dynamics i s i l l u s t r a t e d i n Figure 3.3. The preferred model To f a c i l i t a t e graphical representation of a function involving a time l a g , a steady state stock l e v e l i s assumed. the difference between S i t+ The harvest quantity i s represented by and the 4 5 ° l i n e . It i s noteworthy that the steady state harvest quantity i s f a i r l y constant over a wide range of stock values. It should be stressed that the bioeconomic model w i l l not be q u a l i t a t i v e l y affected by the choice of a particular model of population dynamics. There could be some quantitative d i f f e r e n c e s , however, depending on the combination of stock recruitment and net growth function i n question. Moreover, these differences would be greater for stock l e v e l s than for harvest q u a n t i t i e s . 3.2 The Production Function This section w i l l start out with a discussion of the nature of the North Sea herring fishery: the technology employed, the uses that are made of herring and the factors that are important determinants of f i s h i n g power. The purpose of t h i s exercise i s to derive a conceptual production function. This w i l l be formalized, and the corresponding p r o f i t function w i l l be derived. F i n a l l y , results from the estimated production functions w i l l be presented. - 47 - St+1 Million » tonnes 3 2 1 Million tonnes Figure 3.3: Stock Dynamics (Steady State Stock Levels) - 48 - 3.2.1 The Nature of the Fishery The North Sea herring fishery takes place i n the central and northern North Sea. There has t r a d i t i o n a l l y been a prespawning fishery 7 from May to J u l y , and a spawning fishery i n August and September ^. Herring i s used for reduction into f i s h meal and f i s h o i l and f o r human consumption. The former has t r a d i t i o n a l l y been the more important use i n terms of quantity. Usually, more than one catch w i l l be needed to f i l l up the hold of a boat. Once the catch i s on board the boat, the quality w i l l start d e t e r i o r a t i n g . This problem can be aggravated because of warm weather, as the f i s h e r y takes place i n summer. This circumstance could cause short t r i p s with r e l a t i v e l y low catches, so that hold capacity i s r a r e l y f u l l y u t i l i z e d . A very important behavioural c h a r a c t e r i s t i c of North Sea herring, as of other clupeids, i s the schooling behaviour. Schooling takes place both to make the search for food more e f f e c t i v e and to reduce the effectiveness of predators (Partridge, 1982). It i s also important to note that schooling f i s h contract their feeding and spawning range as the stock i s reduced, with the size of schools often remaining unchanged. The schooling behaviour has also permitted the development of very e f f e c t i v e means of harvesting, e s p e c i a l l y the purse seine. With modern f i s h f i n d i n g equipment, this means that harvesting can be 'in 1968, 1971 and 1975, respectively 78%, 77% and 62% of Norwegian catches were taken i n the months June to August (Bjorndal, 1984). - 49 - p r o f i t a b l e even at low stock l e v e l s . harvesting Murphy notes that the e f f e c t i v e techniques, "when combined with the a b i l i t y to locate schools over wide areas and converge on them, means that high yields can be obtained from declining stocks v i r t u a l l y u n t i l the end; thus there are no economic brakes on exploitation u n t i l collapse" (Murphy, 1977, p. 285). For these reasons, the stock output e l a s t i c i t y i s expected to be 8) low; i n the extreme, i t may even be equal to zero . Moreover, i t has been asserted that a low stock output e l a s t i c i t y implies that costs have a n e g l i g i b l e effect on the optimal stock l e v e l . Both these assertions w i l l be tested e m p i r i c a l l y . If there i s no stock effect in the production function, extinct i o n of the stock i s "optimal" i n the sole owner case for a high f i n i t e or an i n f i n i t e discount rate ( c f . Section 2.3). Although this i s a spe- c i a l case for the sole owner, i t i s asserted that an open access f i s h e r y i s characterized by an i n f i n i t e discount r a t e . Therefore, the combina- tion of a zero stock output e l a s t i c i t y and an open access fishery would lead to the extinction of the stock. but low, open access may the reasons why If the stock effect i s positive s t i l l cause severe stock depletion. These are clupeids pose some special management problems. As a r e a l world phenomenon, a number of clupeids are in a severely depleted state (Murphy, 1977). In the empirical work to be undertaken, data for the Norwegian purse seine f l e e t w i l l be used. The purse seiners i n question are of For this reason, the Schaefer production function i s e s p e c i a l l y inappropriate. - 50 - 120-180 foot length, with a crew size of 9 to 12. The boats are equipped with powerblock, modern f i s h f i n d i n g and communications equipment. The l a t t e r f a c i l i t a t e s the sharing of information about locations of herring, which commonly takes place. technology, started i n 1963. The f i s h e r y , u t i l i z i n g this In the middle of the 1970s, however, the stock was nearly depleted and the fishery was 1977. prohibited at the end of Severe regulations have been i n effect ever since so as to allow the stock to recover ( c f . Table A l , Appendix 2 ) . The following factors should be considered for inclusion i n the production function: 1. E^: The number of boat-days. 2. St: Stock s i z e . 3. Kit: The number of boats i n the f i s h e r y . This factor i s important because we are dealing with a search f i s h e r y , where information about locations of f i s h tends to be shared. fit. This phenomenon may give r i s e to an external bene- On the other hand, too many boats may on the f i s h i n g grounds, a negative 4. K2t : cause overcrowding externality. Size of boats (measured i n gross registered tonnes). This variable may be of importance because catch per t r i p may depend on boat s i z e . 5. K3 t : Fish-finding equipment such as echo-sounder and sonar. This type of equipment contributes to the quality of search e f f o r t and may function. thus be an important variable in the However, already i n 1963 production a l l boats had echosounders - 51 - and two thirds had sonars. A l l boats were equipped with sonar and powerblock by 1967 (Bakken and Dragesund, 1971). For these reasons, this variable w i l l generally be equal for a l l boats. 6. K4t: It has accordingly been omitted from the a n a l y s i s . Gear type. This variable would be important i n many analyses but i s redundant i n the present one, because there i s only one gear type. In this formulation, one variable input has been s p e c i f i e d . can e a s i l y v i s u a l i z e others, e.g. f u e l expenditures. One However, this var- iable has been omitted because cross-sectional analyses show i t to be highly correlated with boat-days (BjjzCrndal, 1984). Similarly, a vari- able l i k e the number of man-days has been omitted because i t would be nearly perfectly correlated with boat-days, as crew size per boat i s commonly f i x e d . In the period under consideration, the fishery was characterized by open access, and myopic profit-maximizing behaviour i s therefore assumed. 3.2.2 Empirical Estimation Two data sets are available for estimation of production/profit functions: 1. Cross-sectional data for 1968, 1971 and 1975. As prices and stock size are fixed i n the short run, harvest i s estimated as a function of variable inputs and fixed factors that vary across boats. - 52 - 2. Aggregate time series data for 1963-77. While the number of observations i n the cross-sectional data sets permits some f l e x i b i l i t y i n choice of functional forms, the limited number of observations i n the time series c a l l s for a functional form that i s parsimonious i n parameters. For this reason a Cobb-Douglas 9) function i s chosen , of which the Schaefer model happens to be a special case. The production function i n logarithmic form i s : (3.1) l n H t = InA + -'.ja^nE^ + polnSt + |pjlnKjt Assuming profit-maximizing and price-taking behaviour, the dual p r o f i t function can be derived (Lau and Yotopoulos, 1972): (3.2) lnir* = InA* + Z a * l n c l t + p o l n S t + .Zp*lnKj t where c^ t i s the normalized price of input i ( c f . Chapter 2 ) . The harvest supply function i s given by (3.3) l n H t = l n A Q + E a * l n c i t + p*,lnSt + z p * l n K j t Since the production and p r o f i t functions are s e l f - d u a l , the parameters of one can be derived exactly from the parameters of the other. The r e s t r i c t i o n s on the parameters are: A* = f(A,a) A Q = g(A,a) (3.4) u a i = Za ± < 1 i = a ~ i / d ~ v ) , i = l,...,n 9) The complex model of population dynamics that was introduced i n Section 2.1 may be d i f f i c u l t to combine with a f l e x i b l e functional form for the p r o f i t function. This would represent another independent reason for choosing the Cobb-Douglas function. - 53 - f j = p j / ( l - y ) , j = 0,1,...,m It can be shown (Lau and Yotopoulos, 1972) that constant returns i n a l l inputs implies Ep* = 1, which may be tested for s t a t i s t i c a l l y . The empirical work w i l l proceed by f i r s t giving results from the cross-sectional estimations and subsequently from the time series regressions. Furthermore, a parameter estimate from the cross-sectional analysis w i l l be imposed on the time s e r i e s . Cross-Sectional Results Cross-sectional data for the Norwegian purse seine f l e e t are available for three years: 1968 (44 boats), 1971 (23 boats) and 1975 (102 boats). While the f i r s t two years constitute samples, complete information about the 1975 f i s h e r y i s a v a i l a b l e . The data set includes information per boat on harvest quantity, variable inputs and fixed factors^\ With this type of data, direct estimation of the production function (equation 3.1) i s f e a s i b l e . Table 3.7 gives a summary of the cross-sectional data. Over the years i n question, there has been a decrease i n mean harvest per boat and mean number of trips per boat. Average catch per t r i p increased by more than 50% from 1968 to 1971, but was then more than halved from 1971 to 1975. The 1975 l e v e l was therefore only 24% lower than i n 1968. There has also been a substantial increase i n mean tonnage, e s p e c i a l l y from 1971 to 1975. ^ ^ D a t a on harvest quantities and variable inputs were made available by Noregs S i l d e s a l s l a g (the Norwegian Herring Fishermen's Cooperative). Data on vessel attributes (fixed factors) were supplied by the Directorate of F i s h e r i e s , Norway. - 54 - Table 3.7: Summary of Cross-Sectional Data Year 1968 44 44,350 1971 23 1975 102 Mean 1968 Mean Trips per boat St .dev. Mean St.dev. 1,008 726 9.86 4.45 21,345 930 874 5.87 3.32 36,199 355 338 4.70 3.59 Catch per t r i p Tonnes Year Harvest per boat Aggregate harvest Tonnes Boats i n sample Days per t r i p 1^ Gross register tonnes St.dev. Mean St.dev. 94.7 45.5 6.45 2.24 279.1 113.8 1971 151.3 83.6 6.85 2.22 315.2 112.0 1975 71.7 38.8 * 423.2 146.5 1) Mean Based on sample sizes of 41 and 21 r e s p e c t i v e l y . 1975 are not yet a v a i l a b l e . St.dev. Figures f o r An important change i n i n s t i t u t i o n a l setting also took place i n the period under investigation: While the fishery was characterized by open access i n a l l three years, t r i p quotas were introduced i n 1975. This regulation stated that catch per t r i p could be a maximum of 100 tonnes plus 40% of the boat's cargo c a p a c i t y * ^ up to 380 tonnes. The e f f e c t this may have had on the production function w i l l be investigated. ^The smallest size of the purse seiners i n this fishery i s more than 200 tonnes, giving t r i p quotas of at least 180 tonnes. - 55 - When i t comes to variable inputs per boat, data are available on 1. the number of t r i p s , 2. the number of boat-days, and 3. estimated f u e l expenditure (1968 and 1971 o n l y ) . In the regressions to be performed, boat-days w i l l be selected as the variable input. mations. The same variable w i l l be used i n the time series e s t i - Bj0rndal (1984) estimates production functions with the number of t r i p s and f u e l expenditure as variable inputs (1968 and 1971 d a t a ) . The r e s u l t s show that the output e l a s t i c i t y of fuel expenditure i s i n s i g n i f i c a n t and that the exclusion of this input has very l i t t l e influence on the remaining parameter estimates. This j u s t i f i e s the omission of f u e l expenditure from the present estimations. With regard to fixed factors per boat, data are available on the boat's length, width, depth, tonnage, material type (wood or s t e e l ) , the year b u i l t , engine horsepower, i t s make and the year i t was b u i l t . Many of these vessel attributes are highly correlated. The following factors w i l l be included as arguments of the production function: K 2 = tonnage (gross register tonnes) K 3 = engine horsepower K 4 = year the boat was b u i l t The choice of factors needs some a m p l i f i c a t i o n . Tonnage i s included because i t represents a measure of potential cargo capacity. Engine horsepower i s included because i t influences the speed of boats. The t h i r d f a c t o r , year b u i l t , introduces an element of vintage c a p i t a l as i t serves to d i s t i n g u i s h between old and new boats. - 56 - Length, width and depth have been excluded because they can hardly make any independent contribution to the catching power of a boat. Moreover, tonnage i s a product of these three v a r i a b l e s . Mate- 12) r i a l type vest. w i l l also hardly make any independent contribution to har- In a d d i t i o n , i t i s assumed that a l l relevant information about the engine i s given by i t s horsepower. Two factors that w i l l influence production per boat have been excluded from the analysis: stock size and aggregate number of boats i n the f i s h e r y . The reason i s the cross-sectional nature of the estima- t i o n s , which implies that stock size and f l e e t p a r t i c i p a t i o n w i l l be 13) constant and equal for a l l boats. Therefore, these two fixed factors can be suppressed i n the functional form. The following Cobb-Douglas production function i s thus established*^ : where a P2 P3K P4e £ h = AET$ K 3 V h = production (harvest) per boat i n a given year 2 and i t i s assumed that e ~ N ( 0 , a ) . One would on a p r i o r i grounds expect the regulations i n 1 9 7 5 — i f b i n d i n g — t o affect the parameter estimate of tonnage ( p 2 ) > since this i s 12) In 1968, 1971 and 1975 there were f i v e , two and two wooden^ boats r e s p e c t i v e l y . 13) In r e a l i t y , stock size w i l l decline over the year, and f l e e t p a r t i c i p a t i o n may vary over the season. Below, r e s u l t s from estimations of whole year and main season f i s h e r i e s w i l l be presented. The assumptions of constant S t and K j t are presumably more r e a l i s t i c i n the l a t t e r than i n the former case. 14) Subscripts for boat number and year have been suppressed f o r a l l variables. - 57 - the variable that represents hold capacity. However, Table 3.7 shows that mean harvest per t r i p i n 1975 was l e s t t r i p quotas ( c f . footnote 11). regulations somewhat questionable. expect p 2 to be less than one. considerably lower than the smal- This makes the effectiveness of the Furthermore, one would a p r i o r i This i s because more than one catch typ- i c a l l y w i l l be needed to f i l l up the hold of a boat, and warm weather may cause short t r i p s with low catches. The r e s u l t s of the regressions are given in Table 3.8. thesis that a l l parameter values i n 1968 rejected according to a Chow t e s t . years have also been pooled. and 1971 A hypo- are equal i s not Therefore, the data for the two The result indicates that although stock size and f l e e t p a r t i c i p a t i o n changed considerably from 1968 to 1971 ( c f . Tables Al and A2), the parameters of the production function remained unchanged. 1971 and 1975 The hypothesis that a l l parameter values in the regressions are equal i s , however, r e j e c t e d . The output e l a s t i c i t y for the number of boat-days i s s i g n i f i cantly greater than one i n a l l regressions, i . e . , there are increasing returns to the variable input. This somewhat surprising result may be explained i n the following manner: The herring i s not randomly d i s t r i buted i n the ocean. At the beginning of the fishing season, time w i l l be spent searching for h e r r i n g . ered, the boat may Once a catch has been made and d e l i v - return to the location of the f i r s t catch i n the hope that the herring i s s t i l l there. The data appear to confirm that aver- age catch on the f i r s t t r i p i s lower than average catch per t r i p for the whole season. In a d d i t i o n , average number of days on the f i r s t appears to be higher than average days per t r i p for the season. trip - 58 - Table 3.8: Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data*^ Year ln A 1968 -8.98 (-0.44) 1971 81.22* (1.46) 1.42** 0.45 (6.74) (0.83) r P4 2 2) n F RSS 0.01 (0.47) 0.73 24.86 41 18.24 1.08* -0.04* (1.53) (-1.47) 0.78 14.22 21 6.25 0.09 -0.0002 (0.30) (-0.02) 0.73 38.00 62 26.76 0.71 41.78 73 17.74 1.31** 0.72** -0.20 (9.39) (1.70) (-0.52) 36.22* 1.16** 0.33 (-1.44) (12.06) (1.24) 1975 1) P3 P2 18.34 1.25** 0.57** (0.05) (11.66) (1.74) 1968 & 1971 2) a -0.23 (1.01) 0.02* (1.53) t s t a t i s t i c s are given i n parentheses. * denotes s i g n i f i c a n t at 90% l e v e l . ** denotes s i g n i f i c a n t at 95% l e v e l . The boats for which there are no records of the number of boat-days have been excluded from the regressions. Table 3.9: Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data. Main Season (June to August) Catches O n l y ^ Year 1968 1971 1975 1) In A a P2 -18.94 1.09** 0.42 (-0.87) (5.68) (0.84) 35.04 1.33** 0.95** (0.66) (8.16) (1.83) -35.20 0.93** 0.12 (-1.29) (8.36) (0.41) P3 P* r 2 F n 0.01 (0.99) 0.60 11.39 36 -0.02 0.61 (0.96) (-0.73) 0.88 19.34 16 0.60 24.31 69 -0.19 (-0.45) -0.04 (-0.15) 0.02* (1.40) t s t a t i s t i c s are given i n parentheses. * denotes s i g n i f i c a n t at 90% l e v e l . ** denotes s i g n i f i c a n t at 95% l e v e l . - 59 - The phenomenon described here may also be the cause of positive externality among boats: once one boat finds herring, other boats w i l l be informed of i t s l o c a t i o n . One can therefore imagine a fishery with a main season and preand post-season f i s h e r i e s , as indeed i s the case for North Sea herring. Catch per day may then increase during the pre-season, remain somewhat constant during the main season and decrease during the post-season fishery. Towards the end of the season, a thinning of the stock w i l l also occur. This could give r i s e to increasing returns for the year as a whole, with constant returns during the main season. For this reason, the production function was also estimated using data for the main season (June to August) only (Table 3.9). These results show that the output e l a s t i c i t y of days for the main season fishery i s not s i g n i f i c a n t l y different from one for 1968 and 1975. Although the c o e f f i c i e n t i s s t i l l s i g n i f i c a n t l y larger than one for 1971, the point estimate i s lower than i n Table 3.7. These r e s u l t s appear to confirm the hypothesis about the interactions between pre-, main and post-season fisheries. For the whole year f i s h e r i e s (Table 3.8), a l l point estimates of the output e l a s t i c i t y of tonnage ( p 2 ) are less than one. For the com- bined 1968 and 1971 regression, i t i s s i g n f i c a n t l y less than one. The low point estimate i n the 1975 regression could be due to the regulations that were i n e f f e c t during that year. However, since mean harvest per t r i p i s considerably less than the smallest t r i p quotas, this explanation does not appear to be very l i k e l y . - 60 - An important change that took place between 1971 and 1975 was i n the r e l a t i v e shares of the catch going to reduction and human consumption. While about 10% of t o t a l catch was delivered for human consump- t i o n i n 1971, the r e l a t i v e share increased to almost 90% i n 1975*"*^. Quality considerations when delivering for human consumption t y p i c a l l y cause short t r i p s with r e l a t i v e l y low c a t c h e s * ^ . It appears l i k e l y that this structural change, rather than the system of t r i p quotas, i s the reason for the low point estimate of p 2 for 1975. The other two parameters—p3 and p 4 — a r e i n some instances of the wrong sign and/or i n s i g n i f i c a n t . This phenomenon may be attributed to the high degree of c o r r e l a t i o n among the fixed f a c t o r s . However, i t could also be that these two factors i n one or more years were r e l a t i v e l y unimportant. 7 The main results from this s e c t i o n * ^ can be summarized as follows: 1. There appear to be constant or increasing returns to variable inputs, i . e . , the number of days per boat in the f i s h e r y . 2. It i s postulated that the interactions between pre-, main and postseason f i s h e r i e s can give r i s e to increasing returns for the year as *"^Source: Annual reports for 1971 Herring Fishermen's Cooperative. and 1975, the Norwegian 16) If the d e l i v e r y i s intended for human consumption, the boat w i l l usually have to leave the f i s h i n g grounds within 12 hours of making the f i r s t catch i n order to ensure high q u a l i t y . ^ ^ B j ^ r n d a l (1984) contains further estimations of production functions based on these cross-sectional data. This includes estimation of a translog f u n c t i o n . - 61 - a whole, but constant or decreasing returns during the main season. The empirical evidence appears to confirm t h i s . 3. A hypothesis that a l l parameter estimates i n the production function are equal was accepted for the 1968 rejected for the 1971 4. and 1971 regressions, but and 1975 regressions. There appear to be decreasing returns to fixed f a c t o r s , as was expected. 5. The effectiveness of t r i p quotas that were introduced i n 1975 i s found to be questionable. Time Series Results The aggregate time series data for the North Sea herring fishery contain information about fixed f a c t o r s , including stock s i z e , prices and harvest quantity. estimated. With these data, a harvest supply function can be This i s done for the following alternative s p e c i f i c a t i o n s of the production function: Al. H t A2. Bl. A^sjSf^e B2. where D.. = { 1 for 1963 and 1974-77 and by assumption g t ~ 0 otherwise N(0,o^). - 62 - There are two main models. Model A assumes that both the number 18) of boats and boat size matter of boats. , while model B only includes the number In the l a t t e r formulation, i t i s i m p l i c i t l y assumed that catch per t r i p i s unaffected by boat s i z e . While the cross-sectional regressions estimate production per boat, the present functions relate to aggregate or industry production. Due to the time series data, stock size (S t ) and fleet p a r t i c i p a t i o n (K^ t ) are included as arguments, because they do not remain constant over time. However, boat-days (E t ) and boat size (K.2t) are arguments i n both the cross-sectional and the time series functions, although the d i s t i n c t i o n between boat-days at the micro and at the aggregate l e v e l should be kept i n mind. Both models are specified with and without a dummy v a r i a b l e . The dummy has been introduced because the t r i p quotas that were i n effect during the year 1974-77, might affect our profit-maximizing hypothe19) sis . In addition, the dummy has been given a value of one for 1963. Since this i s the year the fishery s t a r t e d , the fishermen may for a number of reasons not have exploited the opportunities f u l l y . This may have caused a deviation from profit-maximizing behaviour. For these reasons, s p e c i f i c a t i o n s with the dummy variable are a p r i o r i preferred to those without. In the regressions, i s represented by average boat size per year. 19) In the l a s t few years of the data period, a r e l a t i v e l y larger proportion of catches was delivered for human consumption as opposed to reduction ( c f . cross-sectional regressions). This might be an independent reason for specifying a dummy v a r i a b l e . - 63 - I d e a l l y , the cost of e f f o r t should be represented by the opportun i t y cost of this factor of production. It can be argued that the deci- sion to participate in the North Sea herring fishery i s a marginal one. Norwegian purse seiners participate in up to f i v e different seasonal f i s h e r i e s (Bj^rndal, 1981). Of these, the capelin fishery i s the major one revenues. i n terms of quantity and Only the mackerel f i s h e r y , starting i n the f i r s t half of J u l y , overlaps to a s i g n i f i c a n t extent with this f i s h e r y . Therefore, the North Sea herring sidered a minor one compared to the capelin fishery or to the other f i s h e r i e s combined. the other f i s h e r i e s . Furthermore, i t may fishery may be viewed as complementary to Thus investment decisions w i l l mainly depend on the prospects in the major f i s h e r i e s and be made with an eye tility. ery. we be con- to versa- However, no special equipment w i l l be required for this f i s h - Once the investment decisions have been made, the situation that primarily are concerned with, a decision to participate i n this f i s h 20) ery w i l l be based on (expected) marginal revenues and costs . This also corresponds to the open access nature of the fishery during the data period. Variable costs consist of wages, f u e l and materials. While i t i s f a i r to assume that market prices represent opportunity costs for f u e l and materials, this i s not necessarily so for labour. A particular phe- nomenon of t h i s industry i s that the returns to labour vary from year to year due to yearly fluctuations in prices and stock s i z e . Therefore, an If applicable, t h i s should include opportunity cost from the alternative f i s h e r y . - 64 - opportunity cost i n the form of the wage i n manufacturing industry i s used for labour. An index for the cost of e f f o r t i s constructed i n Appendix 2, which also gives a further analysis of costs. Two types of e x t e r n a l i t i e s are associated with open access f i s h eries. Static e x t e r n a l i t i e s refer to interactions among boats. The stock externality refers to the effects of this period's harvest on the next period's stock l e v e l . The s t a t i c e x t e r n a l i t i e s can be positive or negative. On the p o s i t i v e side, the search e f f e c t i s l i k e l y to be important i n a schooling f i s h e r y . Examples of negative e x t e r n a l i t i e s are gear c o l l i s i o n s and overcrowding on the f i s h i n g grounds. For this fishery i n p a r t i c u l a r , i t should be recalled that i t i s an ocean f i s h e r y . Overcrowding i s there- fore less l i k e l y than e.g. Moreover, only one i n a coastal f i s h e r y . type of gear i s used, which makes gear c o l l i s i o n s less l i k e l y than i n some other f i s h e r i e s . For the time period i n question, i t i s believed that there might have been some negative interactions between the boats 21) in the l a t t e r half of the 1960s, but otherwise not . However, a posi- t i v e externality through the search e f f e c t i s believed to have been present throughout the data period. The production technology i s estimated on the assumption that the s t a t i c externality cannot be controlled f o r . however, can be controlled The stock e x t e r n a l i t y , i n the case of a sole resource manager (Chapter 4 ) . Source: Olav Dragesund, professor of f i s h e r i e s biology, University of Bergen (private communication). - 65 - It i s arguable whether the number of p a r t i c i p a t i n g boats should be considered an endogenous or an exogenous v a r i a b l e . (K^ t ) As d i s - cussed above, the boats have no alternative employment opportunities for parts of the season (except being i d l e ) , while at the end of the season there may exist an a l t e r n a t i v e . purely by reasoning. Hence, t h i s question cannot be resolved Therefore, the production function w i l l be e s t i - mated with K^t as a l t e r n a t i v e l y exogenous and endogenous v a r i a b l e . The regression r e s u l t s , with K^t exogenous, are given i n Table 3.10. The parameters of the production function, derived from the supply function, are given i n Table 3.11. The method for estimating the 22) variances of the derived parameters i s given by Kmenta (1971) From the supply function, the following results can be derived: 1. The hypothesis of increasing returns in a l l f a c t o r s — v a r i a b l e and f i x e d — i s accepted i n a l l models. The imputed degree of homogeneity i s given i n Table 3.8. 2. The supply e l a s t i c i t y of e f f o r t , a*, i s i n s i g n i f i c a n t i n model Al and barely s i g n i f i c a n t i n model B l . the dummy v a r i a b l e . The coefficent i s affected by Point estimates are of roughly the same magni- tude i n specifications with dummy; the same holds for models without dummy. 3. The supply e l a s t i c i t y of the stock, p*>, i s s i g n i f i c a n t l y less than one i n model B, while not s i g n i f i c a n t l y d i f f e r e n t from one i n model A. The estimate i s affected by the dummy v a r i a b l e . This i s r e a l l y a large sample method. - 66 - Table 3.10: Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. Exogenous. Time Series Data 1963-77^ t Model InA* * Po a a* p2 p! r ft 2 Al -20.01** (-2.10) -0.33 (-0.91) 1.11** (2.64) 1.41** (11.22) 1.54 (1.22) - 0.97 A2 -18.94** (-2.86) -0.73** (-2.49) 0.91** (3.03) 1.29** (12.29) 1.43* (1.62) -0.76** (-3.05) 0.98 Bl -8.70** (-3.75) -0.46 (-1.31) 0.61** (6.04) 1.47** (12.33) B2 -8.48** (-4.86) -0.87** (-2.85) 0.45** (5.12) 1.34** (12.24) 1) - - - -0.75** (-2.77) 0.97 0.98 Models A2 and B2 have been corrected for f i r s t order autocorrelation. Model A l has a DW s t a t i s t i c of 1.81 and an F s t a t i s t i c of 91.64, while Bl has a DW s t a t i s t i c of 1.79 and an F s t a t i s t i c of 116.50. t s t a t i s t i c s are given i n parentheses. * denotes s i g n i f i c a n t at 90% l e v e l . ** denotes s i g n i f i c a n t at 95% l e v e l . Table 3.11: Derived Production Function Parameters (Cobb-Douglas) for North Sea H e r r i n g ^ Model A a 2 Po Pi P2 Al 0.12 6 x 10" 0.25 (0.20) 0.84 (0.43) 1.06 (0.23) 1.16 (1.09) A2 0.26 4 x 10" 0.42 (0.10) 0.53 (0.22) 0.75 (0.11) 0.82 (0.57) Bl 0.0037 0.31 (0.17) 0.42 (0.10) 1.01 (0.19) B2 0.0151 0.46 (0.09) 0.24 (0.12) 0.72 (0.11) 1) Standard errors are given i n parentheses. been estimated. - P5 -0.44 (0.19) -0.40 (0.13) j=o' 3.31 2.22 1.74 1.42 The variance of A has not - 67 - 4. The estimated supply e l a s t i c i t y of the number of boats, p^, i s f a i r l y stable across d i f f e r e n t model specifications and i s s i g n i f i c a n t l y larger than one i n a l l models. 5. The supply e l a s t i c i t y of boat s i z e , p 2 , i s s i g n i f i c a n t at the 90% l e v e l i n model A2. 6. Judged by t s t a t i s t i c s , specifications with the dummy variable appear to perform better than those without, as was expected. On a p r i o r i grounds, specifications with the dummy v a r i a b l e — models A2 and B2—were preferred to those without. borne out by the empirical r e s u l t s . This expectation was Furthermore, due to the s i g n i f i - cance of p 2 , model A2 w i l l be preferred to B2. The other main d i f f e r - ence between these two models i s the estimate of the p* parameter. Variables S t and K 2 t are correlated. When K 2 t i s dropped from the regression, some of i t s e f f e c t w i l l accordingly be caught up by p * . When considering the derived output e l a s t i c i t i e s of the product i o n function (Table 3.11), one notes that there are s t r i c t l y decreasing returns to e f f o r t (boat-days), with point estimates i n the range 0.25 to 0.46. This i s i n contrast to the cross-sectional r e s u l t s , which gave constant or increasing returns to e f f o r t . The difference may be caused by the d i f f e r e n t nature of cross-sectional and time series regressions (Stapleton, 1981). The Schaefer production function imposes a stock output e l a s t i c i t y that i s equal to one. An interesting result for herring i s the r e l a t i v e l y low stock output e l a s t i c i t y (except for model A l , where the estimate i n c i d e n t a l l y has the highest standard e r r o r ) . The result must be attributed to the fact that we are dealing with a clupeid f i s h e r y . - 68 - Other r e s u l t s give point estimates of p^ larger than one i n models Al and B l , which indicates increasing returns to the number of boats. This may be due to sharing of information during the search phase of the f i s h e r y . The estimates of p 2 are somewhat higher than those of the cross-sectional regressions. It has hitherto been assumed that f l e e t p a r t i c i p a t i o n i s exogenous. I f , on the other hand, of simultaneity b i a s . i s endogenous, there w i l l be a problem However, even i f this i s the case, i t can be argued that the bias i s small. This i s because the models are well specified with several t r u l y exogenous variables and the variances of the error terms are small ( c f . Maddala, 1977, ch. 11). sons, the simultaneity problem may For these rea- not be serious and this procedure may be preferred to a more a r b i t r a r i l y defined simultaneous equation system. The consequence of l e t t i n g K^t be endogenous w i l l , however, s t i l l be investigated, which can be done by two stage least squares estimation. This procedure requires extra instrumental variables i n order to ensure consistent estimates. These are the dependent, endoge- nous and exogenous variables lagged once and the cost/price r a t i o i n the 23) mackerel fishery . The inclusion of the l a t t e r variable means that the e f f e c t of changes i n r e l a t i v e prices i n the herring and mackerel f i s h e r i e s are taken into account when determining p a r t i c i p a t i o n in the herring f i s h e r y . The r e s u l t s from the estimations of the supply 23) It i s assumed that the costs i n the mackerel fishery are the same as i n the herring f i s h e r y . This i s reasonable, because the technology and the fishing grounds are to a large extent the same. S t a t i s t i c a l Yearbook (Oslo: Central Bureau of S t a t i s t i c s of Norway) gives a price index for mackerel. - 69 - function are given i n Table 3.12 with derived production function parameters i n Table 3.13. Treating f l e e t p a r t i c i p a t i o n as an endogenous variable affects a number of parameter estimates. In p a r t i c u l a r both the supply e l a s t i c i t y of the number of boats and the absolute value of the supply e l a s t i c i t y of e f f o r t increase. When comparing r e s u l t s , however, i t should be borne in mind that the two stage least squares estimations contain one less observation than when i s endogenous. Since sample size i s small, this may a f f e c t estimated parameter values. The interpretation of r e s u l t s i s to a large extent similar to the case when ]sqt was exogenous. Although numerical results vary somewhat, q u a l i t a t i v e r e s u l t s remain unchanged. A hypothesis of increasing returns i n a l l inputs i s accepted i n a l l models. When considering the parameters of the production function (Table 3.13), treating K]^t as endogenous causes an increase i n the output e l a s t i c i t y of e f f o r t and a decrease i n the output e l a s t i c i t i e s of stock size and f l e e t s i z e . Model A l indicates constant returns to f l e e t s i z e , while point estimates are lower than one i n the three other models. As noted above, boat size (K 2 ) i s an argument i n both the crosssectional and the time series (model A) production functions. This f a c i l i t a t e s the imposition of an estimate of p 2 from the cross-sectional analysis on the time s e r i e s . This i s done because the cross-sectional estimations may give a better estimate of p 2 than the time s e r i e s . The reason i s that there i s much more v a r i a t i o n i n boat size i n the crosssectional than i n the time series data, where average boat size has been - 70 - Table 3.12: Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. K j t Endogenous. Time Series Data 1963-77^ * Model InA * a Po -20.77** -0.62 (-1.96) (-0.99) Al * Pi * P5 * P2 2) — 1.01** 1.62** 1.38 (2.11) (4.03) (1.00) 3 A2 > -21.38** -1.26** 0.78** 1.57** 1.38* -0.98** (-3.21) (-2.46) (2.45) (5.43) (1.65) (-3.45) Bl -14.98** -1.37* (-2.66) (-1.65) B2 -11.50** -1.43** 0.32** 1.64** (-3.65) (-2.52) (1.87) (5.05) 1) 2) 0.46** 2.14** (2.39) (3.84) — — - r Method -0.98** (-3.07) 2 2SLS 0.96 2SLS + AUTO 0.98 2SLS 0.93 2SLS + AUTO 0.97 t s t a t i s t i c s are given i n parentheses. AUTO means corrected for f i r s t order autocorrelation. 2SLS involves the loss of one degree of freedom (the f i r s t observation) due to the i n c l u s i o n of lagged variables as extra instrumental variables. In this model, the instrumental variable K.2t lagged once was dropped because of nearly perfect c o l l i n e a r i t y with K2 t . 3) Table 3.13: Derived Production Function Parameters (Cobb-Douglas) 1) for North Sea Herring. K^t Endogenous 2 Model A a Po Pi P2 Al 0.4 x 10-5 0.38 (0.24) 0.62 (0.44) 1.00 (0.20) 0.85 (0.99) A2 0.0001 0.56 (0.10) 0.34 (0.19) 0.69 (0.08) 0.61 (0.42) Bl 0.0025 0.58 (0.15) 0.19 (0.13) 0.90 (0.14) — B2 0.012 0.59 (0.10) 0.13 (0.09) 0.68 (0.08) — 1) Standard errrors are given i n parentheses. P5 — -0.43 (0.20) — -0.40 (0.23) j=o' 2.85 2.20 1.67 1.40 - 71 - used. In a d d i t i o n , the cross-sectional data contain r e l a t i v e l y many observations over several years. In this case, the following estimating equation for the supply function corresponding to model A2 i s used: l n H t = InA* - a l n c t / ( l - a ) + p 0 l n S t / ( l - a ) + p ^ n K ^ / d - a ) + p 2 l n K 2 t / ( 1-a) + p 5 D t / ( l - a ) + et Here, the r e l a t i o n s h i p between the parameters of the supply and the production functions (equation 3.4) has been taken into account. Nonlinear techniques are required for the estimation of this equation. An advantage of this approach i s that i t gives d i r e c t estimates of the parameters of the production function with corresponding variances. The results of the e s t i m a t i o n s — f o r p 2 values of 0.57 and 0.75—are given i n Table 3.14. Table 3.14: 2SLS Nonlinear Estimation of Production Function Parameters (Cobb-Douglas) f o r North Sea Herring. Model a Po Pi P2 P5 B? Value Imposed^ + •U 1 r 2 Al 0.56** (3.89) 0.39** (3.13) 0.92** (6.89) 0.57 - 2.44 0.94 Al 0.54** (3.71) 0.46** (3.61) 0.93** (6.85) 0.75 — 2.68 0.95 A2 0.55** (6.01) 0.32** (3.73) 0.76** (8.24) 0.57 -0.37** (-3.37) 2.20 0.98 A2 0.53** (5.63) 0.39** (4.40) 0.77** (8.10) 0.75 -0.38** (-3.34) 2.44 0.98 1) Asymptotic t s t a t i s t i c s are given i n parentheses. Due to the nonlinear estimation technique, model A2 could not be corrected for autocorrelation. - 72 - When comparing these r e s u l t s to the production function parameters derived from the supply function (Table 3.13), the very close correspondence for model A2 i s noticeable. The results are not that close for model A l . However, the estimates of a and p o i n Table 3.13 have somewhat large standard e r r o r s , which presumably explains this d i f ference i n r e s u l t s . When p 2 i s increased from 0.57 to 0.75, a l l para- meter estimates remain stable except for p o » This i s caused by the corr e l a t i o n between variables S t and K 2 t . These r e s u l t s give added c r e d i b i l i t y to the parameter estimates that were derived from the supply function. In addition, a high degree of s t a b i l i t y i s indicated for a number of parameter estimates. In the estimations of the harvest supply function, the cost of e f f o r t variable was approximated by an index. The index i n question i s a weighted average of indices for wages, f u e l and materials, the main components of the cost of e f f o r t ( c f . Appendix 2 ) . If one a l t e r n a t i v e l y used a wage index, this would cause n e g l i g i b l e changes i n the r e s u l t s . This i s because wages constitute the largest share i n the cost of effort. I f a f u e l price index i s used, the supply e l a s t i c i t y of e f f o r t in model Al increases i n absolute value and becomes s i g n i f i c a n t . Other parameter estimates would also be a f f e c t e d , but there would be no q u a l i t a t i v e changes i n the interpretation of the r e s u l t s . Therefore, the results presented i n Tables 3.12 and 3.13 are f a i r l y robust to choice of index for the cost of e f f o r t . The question of e x t e r n a l i t i e s was discussed above, and i t was noted that crowding e x t e r n a l i t i e s might have been present during the l a t t e r part of the 1960s. An inspection of f l e e t p a r t i c i p a t i o n i n the - 73 - f i s h e r y reveals that i t was at i t s highest during the years 1965-69 (Table A2, Appendix 2 ) . One might imagine the following s i t u a t i o n . low f l e e t s i z e s , there w i l l be no crowding e x t e r n a l i t i e s . At However, i f f l e e t p a r t i c i p a t i o n exceeds some threshold value, crowding might result. On the other hand, one would expect the search externality to be present at a l l f l e e t l e v e l s , although to a decreasing extent. The hypothesis about the crowding externality may be tested as follows. Define a new dummy—T>2 —that i s set equal to one f o r 1965-69 t K and zero otherwise, and introduce the variable D 2 t l t i * supply functions. tions. cant. 1 t n e harvest This has been done for the four model s p e c i f i c a - The estimated c o e f f i c i e n t was i n a l l cases negative and s i g n i f i This result indicates that there are crowding e x t e r n a l i t i e s at high f l e e t l e v e l s . The introduction of t h i s new variable also caused an increase i n the output e l a s t i c i t y of f l e e t size ( p i ) . This i s p l a u s i - b l e , because p^ may now represent the "pure" search e f f e c t , while the c o e f f i c i e n t of the new variable represents the crowding effect at high l e v e l s of f l e e t p a r t i c i p a t i o n . The empirical l i t e r a t u r e i s not abundant with studies of f i s h e r ies p r o f i t and production functions. However, some empirical work i s 24) based on models that are similar to the present one In a study of Norwegian spring-spawning herring, Ulltang (1976) found a stock output e l a s t i c i t y close to zero. However, this r e s u l t 24) Other empirical studies of production functions than those referred to here are given by Young (1979), Morey (1983) and Arnason (1984). - 74 - must be viewed with caution, as i t was estimated on the assumption that e f f o r t was constant during the period of investigation (1950-60). Hannesson (1983) and Schrank et a l . (1984) both estimated CobbDouglas production functions with e f f o r t and stock size as independent variables. Hannesson, i n applying this model to Norwegian cod and saithe f i s h e r i e s , estimated e f f o r t output e l a s t i c i t i e s t h a t — w i t h the exception of one gear t y p e — v a r i e d between 0.81 was s i g n i f i c a n t l y different from one. were i n the range 0.74 to 0.90. and 1.35, none of which The stock output e l a s t i c i t i e s Many of these c o e f f i c i e n t s were not s i g n i f i c a n t l y different from one. Schrank et a l . estimated production functions for Newfoundland cod, f l a t f i s h and redfish f i s h e r i e s . Con- stant returns to e f f o r t were obtained in a l l f i s h e r i e s , with point e s t i mates of the e f f o r t output e l a s t i c i t y varying between 0.88 and 1.09. The stock output e l a s t i c i t y could only be estimated for the cod f i s h eries with point estimates of 0.66 cantly less than one. and 0.88, which were not s i g n i f i - The results obtained by Hannesson and Schrank et a l . are therefore quite analogous: constant returns to e f f o r t , but with evidence that the stock output e l a s t i c i t y may be somewhat less than one. The difference i n stock output e l a s t i c i t i e s for the clupeid and the demersal f i s h e r i e s can be attributed fisheries. to the different nature of the While there may be some empirical evidence supporting the use of the Schaefer function i n demersal f i s h e r i e s , this i s not the case for clupeid f i s h e r i e s . - 75 - 4.0 THE OPTIMAL STOCK LEVEL In this chapter, the empirical results of Chapter 3 w i l l be combined with the dynamic optimization model for the sole owner that was developed i n Chapter 2. Estimates of bionomic equilibrium and the o p t i - mal stock l e v e l with corresponding harvest quantities w i l l be presented. Section 4.1 w i l l investigate the consequences of various assumptions about costs, while the s e n s i t i v i t y to changes i n other parameters of the Marginal Stock E f f e c t w i l l be considered i n Section 4.2. F i n a l l y , Section 4.3 w i l l b r i e f l y discuss a number of issues of r e l e vance to the management of North Sea h e r r i n g . 4.1 The Influence of Costs For the sole owner, three alternatives with regard to costs are considered: 1. zero costs; 2. variable costs, i . e . , f u e l and materials; and 3. v a r i a b l e , insurance and maintenance c o s t s . Capital costs have been disregarded because i t has been argued that they may not be relevant for the present a n a l y s i s . Estimates and a further analysis of costs and output price are given in Appendix 2. The zero cost case i s included because i t serves as a useful reference. Moreover, i t has been asserted that a low stock output e l a s t i - c i t y implies that costs have a n e g l i g i b l e effect on the optimal stock level. This assertion w i l l be considered i n l i g h t of the empirical results that w i l l be obtained. - 76 - One would expect a sole owner to include a l l relevant costs i n his maximization problem. priate concept. This would make cost category 3 the appro- In the following, this w i l l be termed the base case. However, the special case of only variable costs w i l l also be considered. Before proceeding to give estimates of the optimal stock l e v e l , the open access fishery w i l l be considered. In this way the two types of f i s h e r i e s can be compared, and the importance of resource management can be assessed. The estimations are based on model A2 for the produc- tion function. The open access fishery was b r i e f l y discussed i n Chapter 2. In this f i s h e r y , e f f o r t (boat-days), f l e e t size and stock size should a l l be treated as endogenous v a r i a b l e s . This complete model formulation represents an avenue for future research. For now bionomic equilibrium w i l l be considered only for given f l e e t p a r t i c i p a t i o n . Two Figure 4.1 figures help to i l l u s t r a t e the dynamics of this f i s h e r y . i s a phase-space diagram showing combinations of f l e e t p a r t i - cipation and stock size for the period 1963-77. The early years of this fishery (1963-68) were characterized by increasing f l e e t p a r t i c i p a t i o n and decreasing stock s i z e . Since the stock i n i t i a l l y was at a f a i r l y high l e v e l , t h i s period may represent "mining" of the resource. How- ever, the s i t u a t i o n changed i n 1968, with generally decreasing f l e e t p a r t i c i p a t i o n and stock size i n the ensuing years. declining stock caused decreased the industry. Presumably the p r o f i t a b i l i t y , which led to exit from K lt - 77 - t I 2 S t M i l l i o n tonnes Figure 4.2: Number of Catches and Stock Size 1966-77 - 78 - Wilen (1976) studies the common property exploitation of North P a c i f i c fur s e a l . This industry went through a dynamic process similar i n part to the one i l l u s t r a t e d i n Figure 4.1. However, exit from the industry caused the stock to recover, and the data i l l u s t r a t e d that a stable bionomic equilibrium was being approached. For North Sea herring, on the other hand, exit from the industry does not appear to have caused the stock to recover. Thus one can only speculate on the course of the stock development, had the fishery not been closed i n 1977 when the stock l e v e l was as low as 0.12 tonnes. An open access fishery may depletion. million have resulted i n further stock In other words, bionomic equilibrium may be very close to zero. In this context i t should be realized that the equilibrium i n question was a moving one, since costs and prices were changing over time. In p a r t i c u l a r , the cost/price r a t i o was decreasing i n the 1970s ( c f . Appendix 2 ) . This would a t t r a c t entry and decrease bionomic equi- librium. Further c o l l e c t i o n of cross-sectional data would f a c i l i t a t e drawing a phase-space diagram for e f f o r t (boat-days) and stock s i z e . How- ever, Figure 4.2, which shows combinations of annual number of catches*^ and stock size for the period 1966-77, may relationship. throw some l i g h t on this Comparing the two f i g u r e s , one finds that they essen- t i a l l y t e l l the same story, e s p e c i a l l y for post-1968 observations. Source: The Norwegian Herring Fishermen's Cooperative - 79 - One of the empirical results of Chapter 3 was that there i s a stock effect i n the production function, which means that harvesting costs depend on stock s i z e . This implies that bionomic equilibrium w i l l not be zero, as harvesting costs w i l l go to i n f i n i t y as stock size approaches zero. However, bionomic e q u i l i b r i u m — S ^ — c o u l d s t i l l be very low. For the model developed i n this thesis (equation 2.24), S^, has been estimated for f l e e t sizes varying between 50 and 150 boats and for 2) the cost/price r a t i o s of the sole owner case Soo was . In a l l the estimations found to be very close to zero, i . e . , the model predicts the near extinction of the stock. This result must be viewed with some caution. i s not f u l l y s p e c i f i e d . Second, the results involve extrapolation out- side the range of observed values, and estimated fore not remain constant. F i r s t , the model c o e f f i c i e n t s may there- However, the evidence strongly suggests that an open access fishery w i l l cause severe stock depletion. The presence of density dependent harvesting costs in the net revenue function serves as a brake on stock depletion. The r e s u l t s here show that for the case i n question this brake i s not s u f f i c i e n t to prevent severe depletion or possibly even near extinction of the stock i n an open access f i s h e r y . The consequence i s that the u t i l i z a t i o n of a resource l i k e herring poses some serious management problems. The cost/price r a t i o s i n the middle of the 1970s were t y p i c a l l y lower than the present day r a t i o . This i s because fuel prices were lower and the output price was higher. - 80 - Fishery economic theory predicts that there w i l l be entry to the fishery as long as rents are earned, i . e . , u n t i l bionomic equilibrium i s attained (Scott Gordon, 1954; how S c o t t , 1955). The theory does not predict fast bionomic equilibrium w i l l be reached. Considering the stock development i n the l a s t decades (Table A l , Appendix 1), this process may take some time, although i t has been acknowledged that the equilibrium i t s e l f was changing. This question may be studied further when a com- plete model of the open access fishery i s s p e c i f i e d . We w i l l now proceed to analyze the sole owner case. tions are based on model A2 for the production function. The estima- It should be stressed that r e s u l t s show that the d i f f e r e n t production functions give the same q u a l i t a t i v e r e s u l t s . zero. The dummy variable w i l l be set equal to This i s because profit-maximizing behaviour i s assumed. In the estimations of the production technology, only the Norwegian f l e e t was considered. I d e a l l y , since cooperative management of the resource i s analyzed ( c f . Section 4.3), the f l e e t s of other European countries that may included. p a r t i c i p a t e i n this fishery should also be The reason for not doing so i s that data on other countries' f l e e t s are d i f f i c u l t to get. In the base case, the number of boats in the fishery has been set to 300, while boat size has been set at the 1977 level. This represents 3) a higher f l e e t particpation than would be forthcoming from Norway . In other words, the f l e e t may be composed of boats from different European In 1982, the Norwegian purse seine f l e e t consisted of 193 boats (Brochmann and Josefsen, 1984). - 81 - countries . In addition, s e n s i t i v i t y analyses w i l l be performed to i l l u s t r a t e the e f f e c t on the optimal stock l e v e l of changes i n f l e e t participation. It i s here assumed that the sole owner technology i s the same as the open access one. However, the sole owner can c o n t r o l — i . e . , i n t e r n a l i z e — t h e stock e x t e r n a l i t y . As equation (2.14) cannot be solved e x p l i c i t l y for the optimal stock l e v e l ( S * ) , a solution i s found by means of simulation. The optimal harvest quantity (H*) i s subsequently found from equation (2.15). Results for the zero cost case and the base case are given i n Table 4.1. Table 4.1: Optimal Stock Level (S*) and Corresponding Harvest (H*) Zero Cost Case Base Case Discount rate s* Million tonnes H* Million tonnes 0 1.57 0.61 2.26 0.54 44% 6% 1.40 0.60 2.21 0.55 58% 12% 1.23 0.59 2.17 0.55 76% 18% 1.06 0.56 2.13 0.56 101% s* Million tonnes H* Million tonnes Percentage increase i n S* This assumes that the technologies of other countries' f l e e t s are the same as the Norwegian. This i s probably r e a l i s t i c , provided only purse seiners are considered. - 82 - The r e s u l t s i n Table 4.1 put the management of this resource i n perspective: while an open access fishery would cause severe depletion or possibly even near extinction of the stock, a sole owner would aim at achieving a stable stock l e v e l with a sustained harvest flow ad i n f i n i tum. From the results i n Table 4.1, a number of conclusions can be drawn: 1. The i n c l u s i o n of costs i n the intertemporal p r o f i t function causes a considerable increase i n the optimal stock l e v e l . This result i s e s p e c i a l l y noteworthy because the stock output e l a s t i c i t y i s low. The assertion that a low stock output e l a s t i c i t y implies that costs have a negligible effect on the optimal stock l e v e l i s therefore not necessarily t r u e . 2. S* i s not very sensitive to changes i n the discount rate and less so i n the base case than i n the zero cost case. This shows that costs have a s t a b i l i z i n g influence on the optimal stock l e v e l . 3. H* i s quite i n s e n s i t i v e to changes i n the optimal stock l e v e l and consequently also to changes i n the discount r a t e . This i s because the growth curve i s f a i r l y f l a t over a wide range of stock values ( c f . Figure 3.3). 4. In the zero cost case, S* = S m s y and H* = MSY f o r r = 0 ( c f . Table 3.6). Moreover, the numerical results show that stock extinc- t i o n i s "optimal" for a discount rate of about 52%. A consequence of the third result i s that for a change i n the discount r a t e , the harvest quantity—and thus gross r e v e n u e s — w i l l - 83 - usually be almost unchanged. However, the optimal stock l e v e l w i l l be a f f e c t e d , causing an adjustment i n harvesting costs. Changes i n other parameter values w i l l frequently have the same consequences, with adjustment on the cost side rather than on the revenue s i d e . The optimal stock l e v e l i s increasing i n the cost/price r a t i o . Therefore, i f a rental price of c a p i t a l was included i n the cost of e f f o r t , S* would be higher than the estimates for the base case given i n Table 4.1. This conforms to the a n a l y t i c a l results of Clark, Clarke and Munro (1979). On the other hand, a secular downward trend i n the cost/price r a t i o would gradually diminish the importance of costs. Bionomic equilibrium for the base case cost/price r a t i o was e s t i mated to be near zero. However, costs were seen to cause a considerable increase i n the sole owner stock l e v e l . These results may seem contra- d i c t o r y , but can be attributed to the nonlinear nature of the production function. A necessary and s u f f i c i e n t condition for harvesting to occur i n the sole owner case i s : .,E > 0 This condition can a l t e r n a t i v e l y be stated as VMP^ = aVAP^ > A C „ For the fishery i n question, ct i s about 0.5, and the harvesting condi- tion becomes VAPE > 2ACE - 84 - Thus, harvesting w i l l only take place provided the value of the average product of e f f o r t i s at least twice the value of the average"*^ cost of effort. Bionomic equilibrium, on the other hand, i s characterized by V A P „ = ACL, ( c f . equation 2.20'). The values of the marginal and the average product of e f f o r t are both increasing i n stock s i z e . Thus, there may be a threshold l e v e l for the stock size i n order for the harvesting condition to be s a t i s f i e d . For the base case cost/price r a t i o , numerical results show that TTg < 0 for steady state stock l e v e l s less than about 2.0 m i l l i o n tonnes. Accordingly, a sole owner would never drive the stock below this l e v e l even at an i n f i n i t e discount r a t e . These r e s u l t s show that for a s u f f i c i e n t l y high cost/price r a t i o , severe stock depletion w i l l never be optimal for the sole owner. Bionomic equilibrium, however, may words, costs may s t i l l be close to zero. be an e f f e c t i v e brake on stock depletion i n the sole owner case but not i n the open access case. explain why In other Furthermore, the results the inclusion of costs caused such a considerable increase in the optimal stock l e v e l despite the fact that the stock output e l a s t i c i t y was low. L a s t l y , i t should be pointed out that these results would not have been forthcoming i n the context of a l i n e a r model. The optimal stock l e v e l for the variable cost case i s given Table 4.2. Even when only variable costs are considered, they cause a marked increase i n S*. In this model, average and marginal costs are equal. - 85 - Table 4.2: The Optimal Stock Level f o r the Variable Cost Case Variable Cost Case Di scount rate S* Million tonnes H* Million tonnes Increase i n S* compared to zero cost case 0 1.85 0.59 18% 6% 1.73 0.60 24% 12% 1.63 0.61 33% 18% 1.51 0.61 42% When comparing the base case and the variable cost case, i t i s worth noting that the optimal stock l e v e l i s somewhat less sensitive to changes i n the discount rate at the higher than at the lower cost/price ratio. This shows that the higher the optimal stock l e v e l , the less sensitive i t i s to changes i n the discount r a t e . The influence of costs i s the cause of t h i s phenomenon, which w i l l also be i l l u s t r a t e d i n some l a t e r cases. In the variable cost case, numerical r e s u l t s show that the o p t i mal stock l e v e l i s close to but does not reach zero for an i n f i n i t e d i s count r a t e . The harvesting c o n d i t i o n — - n „ > 0 — i s s a t i s f i e d f o r lower stock l e v e l s than i n the base case. Therefore, costs are less e f f e c t i v e as a brake on stock d e p l e t i o n . The base case and the variable cost case represent two discrete alternatives. Table 4.3 i l l u s t r a t e s the s e n s i t i v i t y of S* to a marginal change—plus/minus 2 0 % — i n the cost/price r a t i o for the base case. - 86 - Table 4.3: S e n s i t i v i t y to a 20% Change i n Cost/Price Ratio. Base Case Cost Alternative Cost/price r a t i o : -20% Cost/price r a t i o : +20% Discount rate S* Million tonnes H* Million tonnes S* Million tonnes H* Million tonnes 0 2.10 0.57 2.42 0.50 6% 2.03 0.58 2.38 0.51 12% 1.96 0.58 2.35 0.52 It can be observed that for a given discount r a t e , the r e l a t i v e change i n S* i s only 7-10% ratio. as compared to a 20% change in the cost/price Therefore, the optimal stock l e v e l would be only marginally affected by moderate changes i n the cost/price r a t i o . The difference between the two cost alternatives may be summarized as follows: the choice of cost concept has a substantial effect on the optimal stock l e v e l , as i l l u s t r a t e d by Tables 4.1 and 4.2. However, once the choice has been made, S*—and even less so H * — i s not very sens i t i v e to moderate changes i n the cost/price r a t i o . Gallastegui (1983), i n a study of a sardine fishery in the Gulf of Valencia, estimates bionomic equilibrium to be somewhat larger than Smsv. The sole owner stock l e v e l i s for reasonable discount rates more than twice the size of S m s y . Moreover, S* i s not very sensitive to changes i n the discount r a t e . The consequences of discounting i n the case of baleen whales are i l l u s t r a t e d i n a paper by Clark (1976a). An increase i n the discount - 87 - rate from zero to f i v e percent reduces the optimal stock l e v e l by almost 50%. The reason for this severe effect of discounting as compared to the results obtained i n this thesis i s the low growth rate i n the whale population. 4.2 The Influence of the Marginal Stock Effect The Marginal Stock Effect was defined i n equation (2.26) as i n t e r a l i a a function of the cost/price r a t i o , the output e l a s t i c i t i e s of e f f o r t and stock s i z e , and fixed f a c t o r s . costs, the relationship Having analyzed the effect of between the optimal stock l e v e l and the other arguments of the Marginal Stock Effect w i l l now be considered. 1. Changes i n the E f f o r t Output E l a s t i c i t y The prediction elasticity. i s that S* i s decreasing in the e f f o r t output S* has been estimated for a-values of 0.51 and 0.61 respec- tively. Table 4.4: S e n s i t i v i t y to Changes i n the E f f o r t Output E l a s t i c i t y ( a ) * ^ ct = 0.51 Discount rate 1) S* Million tonnes a = 0.61 H* Million tonnes s* Million tonnes H* Million tonnes 0 2.99 0.31 1.80 0.60 6% 2.98 0.32 1.68 0.60 12% 2.98 0.32 1.56 0.61 In the base case, a = 0.56. - 88 - The results i n Table 4.4 show that the optimal stock l e v e l i s sensitive to changes i n the e f f o r t output e l a s t i c i t y , as the r e l a t i v e change i n the former variable i s larger than the r e l a t i v e change i n the latter. This appears to be the case even more for downward than for upward changes i n a . increasing a . The results tend towards the zero cost case with Also, a s u f f i c i e n t l y low e f f o r t output e l a s t i c i t y would render the fishery unprofitable. One observes that at the low e f f o r t output e l a s t i c i t y , the optimal stock l e v e l i s almost completely insensi- t i v e to changes i n the discount r a t e . 2. Changes i n the Stock Output E l a s t i c i t y Table 4.5 gives the optimal stock l e v e l for p o -values of 0.29 and 0.39. 1) Table 4.5: S e n s i t i v i t y to Changes i n the Stock Output E l a s t i c i t y (3^) Po = 1) 0.29 Po = 0.39 Discount rate S* Million tonnes H* Million tonnes S* Million tonnes H* Million tonnes 0 2.86 0.37 1.89 0.59 6% 2.86 0.37 1.79 0.60 12% 2.85 0.37 1.69 0.60 In the base case, p o = 0.34. For a given discount r a t e , S* i s sensitive stock output e l a s t i c i t y and especially to changes i n the so for downward changes i n 8 - 89 - Figure 4.3 illustrates in more detail the estimated relationship between the optimal stock level and the stock output elasticity for a zero discount rate. The result is quite interesting, namely that S* is a decreasing function of p o . In other words, the second type of relation- ship between MSE and B0 has been obtained. S* 0 0!l 0^2 0J3 0J4 015 0.'6 Figure 4.3: S* as a function of What happens in the case under consideration is the following: As p o decreases, the profitability of the fishery also decreases, leading to an increase in the optimal stock level. For sufficiently low p Q values, the fishery becomes unprofitable. When this i s the case, S* = S, the carrying capacity of the stock. 3. Changes in Fleet Participation The fixed factors of this model are the number of boats in the fishery and boat size. The base case estimations were based on a fleet participation of 300 boats and boat size as of 1977. Results have also been estimated for fleet sizes of 250 and 350 respectively, while boat size has been kept unchanged (Table 4;6). This corresponds to changes 2 in K of -12% and +11% respectively, where K = K^K^ . - 90 - Table 4.6: E f f e c t s of Changes i n Fleet Size Kx = 250 boats Kx = 350 boats Discount rate S* Million tonnes H* Million tonnes S* Million tonnes H* Million tonnes 0 2.46 0.49 2.12 0.56 6% 2.43 0.50 2.05 0.57 12% 2.39 0.51 1.99 0.58 The r e l a t i v e change i n S* f o r a given discount rate i s considerably less than the r e l a t i v e change i n f l e e t s i z e . The same result would hold f o r changes i n K2 and the combined fixed f a c t o r s . shown to be decreasing i n the fixed f a c t o r s . Moreover, S* i s S e n s i t i v i t y analyses for changes i n p^ and p 2 would y i e l d results analogous to those presented here. 4.3 Some Management Issues This section w i l l raise some issues that are important i n a pract i c a l management plan for North Sea herring. 1. the optimal approach path, 2. the transboundary nature of the f i s h e r y , 3. the regulation of e f f o r t , and They are: 4. uncertainty. As mentioned i n the Introduction, these issues are not considered part of the thesis proper. Rather, they represent issues f o r future research and w i l l be discussed only b r i e f l y . - 91 - Because the underlying model i s nonlinear, the optimal approach to the steady state i s asymptotic (Clark and Munro, 1975). The presence of a fixed f a c t o r — c a p i t a l — i n a linear model would also give r i s e to an asymptotic approach (Clark, Clarke and Munro, 1979). The conditions characterizing the optimal approach path—the equations of motion—are given i n Appendix 1. as part of the dynamic optimization problem. These conditions come out They may also be i l l u s - trated graphically i n the form of phase-space diagrams in an a n a l y t i c a l model (Neher, 1974). The task of finding the optimal path in an empirical problem i s , however, arduous, and especially so i n a model with a time l a g . The reason i s the high number of possible paths between the present day the steady state stock l e v e l s , to be arrived at at some time i n the future. From a policy point of view, the appropriate approach may therefore be to simulate various paths and compare them according to c r i t e r i a that are deemed important. given i n Table 4.7. Examples of two such paths are and - 92 - Table 4.7: Simulation of Spawning Stock 1983-87 1) Year Ht = 0 Spawning stock M i l l i o n tonnes H t = 0.20 m i l l i o n tonnes Spawning stock M i l l i o n tonnes 1983 1.05 0.82 1984 1.50 1.01 1985 2.05 1.29 1986 2.74 1.70 1987 3.33 2.19 Sg4 =0.50 m i l l i o n tonnes. The rapid increase i n stock size from 1982 to 1983 i s influenced by the good 1981 yearclass of r e c r u i t s ( c f . Table A l , Appendix 2 ) . The f i r s t case simulates the development of the stock under natur a l conditions, which corresponds to the bang-bang approach. This path i s both inoptimal and inconceivable, as a complete f i s h i n g moratorium would be very d i f f i c u l t to achieve, even i f that should be attempted. The case i s given i n order to be able to consider the delay i n attaining the steady s t a t e , when an asymptotic approach i s used rather than the most rapid approach. Accordingly, i n the second simulation i t i s assumed there i s an annual harvest quantity of 0.20 m i l l i o n tonnes. Assuming a 6% discount r a t e , the optimal stock l e v e l i s 2.21 m i l l i o n tonnes ( c f . Table 4.1). With a most rapid approach, this stock l e v e l would be attained i n 1986, after a four-year f i s h i n g moratorium. An annual harvest quota of 0.20 m i l l i o n tonnes would cause a delay i n reaching the equilibrium of two years. - 93 - The two p o l i c i e s can be compared i n present value terms. Assume that once the steady state has been reached, the fishery continues on a sustained b a s i s , i . e . , with an annual harvest quantity of 0.55 tonnes ad i n f i n i t u m . million Calculating the present values of catches or gross 6) revenues for the two p o l i c i e s — s t i l l assuming r = 0.06—gives a d i f - ference of about 1.5 p e r c e n t ^ . One would expect that i f the present values of net revenues were calculated, the difference would be of the same magnitude. This r e s u l t suggests that the loss—measured i n terms of present v a l u e — o f deviating from the optimal approach path may not be great. 8) This observation i s supported by Clark (1976) and Ludwig (1980) . The consequence i s that the resource manager may have some degree of freedom i n determining the approach to the steady state stock l e v e l . North Sea herring i s a transboundary resource that i s harvested by several European nations. Previously, the North West A t l a n t i c Fisheries Commission was responsible for making policy recommendations, but had no power to implement these over the wishes of i t s member countries. This was a major reason for the severe stock depletion that took place in the 1960s and 1970s. The optimal stock l e v e l would t y p i c a l l y be "overshot" in the year i t was reached ( c f . Table 4.7), giving a harvest quantity for that year which i s higher than the sustained one. This effect has been neglected i n the following c a l c u l a t i o n . ^^The asymptotic approach has the highest present value. 8) Although Ludwig's r e s u l t s are i n terms of a stochastic model, they are l i k e l y to carry over to the deterministic case. - 94 - After the introduction of extended f i s h e r i e s j u r i s d i c t i o n , North Sea herring has been considered a common resource between Norway and the European Economic Community (EEC). Therefore, management decisions are 9) decided upon j o i n t l y by Norway and the EEC . These decisions are inter a l i a based on b i o l o g i c a l advice from the International Council for the Exploration of the Sea. North Sea herring migrates between the Norwegian and the EEC f i s h e r y zones. This movement takes place partly due to seasonal migra- tion and partly according to developmental stages. common for many species (Gulland, 1980). Such migration i s This means that the d i f f e r e n t countries can exploit the resource at different stages in i t s l i f e cycle. The approach taken, however, i s that of cooperative management. Indeed, this i s also the approach of t h i s t h e s i s , as the sole resource manager can be compared to the j o i n t Norwegian-EEC body regulating the resource. Munro (1979) used a game theoretic model to analyze the coop- erative management of a f i s h resource shared by two countries. However, discount r a t e s , costs or consumer preferences may be d i f f e r e n t between the two countries, giving r i s e to differences in what the two countries perceive to be the optimal stock l e v e l . this c o n f l i c t i s studied. The cooperative resolution of In this t h e s i s , such d i f f e r e n c e s — i f they e x i s t — a r e assumed away. It should be mentioned i n passing that i t has proved d i f f i c u l t for the member countries of the EEC to agree upon a common f i s h e r i e s policy. - 95 - Levhari and Mirman (1980) analyze the competitive u t i l i z a t i o n of a resource shared by two countries, also i n a game theoretic framework. Not s u r p r i s i n g l y , the steady state stock l e v e l turns out to be lower i n the competitive than i n the cooperative case. Crutchfield (1983) ana- lyzes bionomic equilibrium for a fishery where several nations p a r t i c i pate and consider d i f f e r e n t management alternatives under extended f i s h eries j u r i s d i c t i o n . Although the present day management of North Sea herring i s meant to be cooperative, there i s no guarantee that this w i l l be kept up i n the future. Therefore, both cooperative and competitive management schemes should be considered. It may be p a r t i c u l a r l y interesting to include the migratory pattern of the resource in a game theoretic model. This represents an interesting topic for future r e s e a r c h * ^ . The problems with regard to regulating the fishery have been assumed away i n this a n a l y s i s , to a large extent by the introduction of a sole resource manager. Should the analysis be extended to consider these problems, the assumption of a sole owner can be retained. However, the f i s h i n g f l e e t i s i n private ownership and some kind of fishery regulation w i l l be required One can imagine a situation where the sole owner implements a management p o l i c y , the objective of which i s to restore the stock to Here, the interactions between Norway and the EEC have been o u t l i n e d . This i m p l i c i t l y assumes away the process whereby the EECcountries arrive at a common f i s h e r i e s p o l i c y , which indeed has proved difficult. ^On this t o p i c , see Hannesson (1978), Pearse (1979), Scott (1979), Clark (1982a) and Munro and Scott (1984). - 96 - some equilibrium l e v e l over a period of time. This policy can be achieved by setting annual escapement targets with corresponding total 12) allowable catch quotas to be made. will result. . As the stock recovers, there w i l l be rents If the amount of e f f o r t i s not c o n t r o l l e d , excess capacity This can cause rent dissipation and also represents a potential threat for stock depletion. Excess capacity can i n theory be prevented by input controls l i k e limited entry programmes and regulations of boat s i z e . However, pro- blems w i l l arise i f a l l inputs are not controlled or controllable and a regulated input can be substituted by an uncontrolled one. change may cause separate Technical complications. Output controls consist of r o y a l t i e s on f i s h landings and vidual catch quotas. who The quotas may indi- either be distributed to fishermen, then would c o l l e c t the rents, or be auctioned o f f . Clark (1982a) shows that i f the quotas are transferable, landings taxes and the quota system w i l l have the same effect i n terms of e f f i c i e n c y . r e s u l t may However, this be model s p e c i f i c and i s thus not necessarily true i n general. A special case w i l l occur i f the stock effect in the function i s zero. production If a landings tax i s imposed, the fishery w i l l either remain p r o f i t a b l e or i t w i l l become unprofitable. In other words, the tax w i l l either have no behavioural effect or harvesting w i l l cease Monitoring and enforcing such quotas commonly represent considerable problems. - 97 - discontinuously ( c f . Section 2.3). Thus, landings taxes may not be e f f e c t i v e for some clupeid f i s h e r i e s . The present analysis i s carried out i n the context of a determini s t i c model. However, real world management must take uncertainty into account, both with regard to costs and p r i c e s , the production technology and the resource i t s e l f . The l a t t e r type of uncertainty i s v i v i d l y i l l u s t r a t e d i n the estimated stock recruitment and net growth functions ( c f . Figure 3.1). Walters and Hilborn (1978) c l a s s i f y major sources of uncertainty in resource management as ( i ) random disturbances, ( i i ) parameter uncertainty and ( i i i ) ignorance about the appropriate forms of the underlying models. So f a r , most work i n this f i e l d has been concerned with random disturbances, but a number of papers on parameter uncertainty have also been published. Andersen and Sutinen (1984) give a survey of the l i t e r - ature on uncertainty i n f i s h e r i e s management. When i t comes to the p o s s i b i l i t y of extending the present analy- s i s to consider the e f f e c t s of uncertainty, i t should be pointed out that the complexity of the underlying bioeconomic model w i l l make stochastic optimization very d i f f i c u l t . Markov processes (Lewis, 1982) and simulation represent alternative methods of a n a l y s i s . Two other points that may can be made. be s p e c i f i c to the resource i n question Since several countries u t i l i z e North Sea herring, moni- toring harvesting quantities becomes e s p e c i a l l y d i f f i c u l t . be an added source of uncertainty. This would On the other hand, concentrating harvesting on the spawning stock tends to reduce the effect of uncertainty i n the stock recruitment r e l a t i o n s h i p . This i s because more - 98 - information about the size of the new they mature to spawning yearclasses can be obtained while age. In real l i f e , the management issues that have been discussed here are c l o s e l y i n t e r r e l a t e d . Excess f l e e t capacity and uncertainty i n f l u - ence both the approach path and the optimal stock l e v e l . The regulation of e f f o r t and the monitoring of catches are made more d i f f i c u l t because several nations are involved i n the f i s h e r y . a n a l y t i c a l or empirical study, i t may separately. On the other hand, i n an be necessary to treat these issues However, that w i l l be the subject of future research. - 99 - 5.0 SUMMARY The objective of t h i s thesis has been to analyze the optimal management of a f i s h resource—North Sea herring. model was constructed. A dynamic bioeconomic In the development of the model of population dynamics, important 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 were taken into account. The production technology and the net revenue function were specified i n manners consistent with economic theory. An a n a l y t i c a l solution for the sole owner stock l e v e l was derived. This was contrasted to bionomic equilibrium, i . e . , the open access s o l u t i o n . Alternative forms of the b i o l o g i c a l and economic functions were specified and estimated. Esti- mates of the optimal stock l e v e l with corresponding harvest quantities were obtained under d i f f e r e n t sets of assumptions. The presence of fixed factors i n the production f u n c t i o n — i n the form of c a p i t a l equipment—was acknowledged. This presumably represents r e a l i t y i n many developed f i s h e r i e s , e.g. the one being analyzed i n this t h e s i s , although the s i t u a t i o n would be d i f f e r e n t i n a developing f i s h ery. It was argued that the fishery i n question i s a minor complemen- tary one, and that therefore excess capacity would always be present during the herring season. This j u s t i f i e s the exclusion of c a p i t a l dynamics from the dynamic optimization problem. Apart from i m p l i c i t l y assuming a zero depreciation r a t e , the model resembles a case modelled by Clark, Clarke and Munro (1979), who analyzed simultaneous investment in f i s h stock and c a p i t a l stock. The schooling behaviour of the resource i n question was expected to have some consequences for management. The empirical results showed - 100 - that the function describing growth i n the existing stock exhibited depensation at low stock l e v e l s , a phenomenon not uncommon for schooling fish. If the f i s h stock i s reduced to a low l e v e l , depensation cause the recovery to be slow. may This has been a case in point for a num- ber of c l u p e i d s . For North Sea herring, the stock size now appears to be outside the range of stock values where there i s depensation. Modern f i s h f i n d i n g and harvesting equipment make i t possible to locate and catch schools p r o f i t a b l y even at low stock l e v e l s . Schooling f i s h are therefore especially vulnerable to predation by man. It has been asserted that the stock output e l a s t i c i t y w i l l be low for this kind of f i s h e r y . This was confirmed by the empirical r e s u l t s , with most estimates of the stock output e l a s t i c i t y in the range 0.1 to 0.5. It can be shown a n a l y t i c a l l y that a zero stock output e l a s t i c i t y combined with an open access fishery w i l l lead to the extinction of the stock. In the present case, the stock e f f e c t i s low but p o s i t i v e , which implies that harvesting costs serve as some kind of a brake on stock depletion. However, the empirical results show that this brake i s not very e f f e c t i v e , as bionomic equilibrium—based on cost/price r a t i o s that were i n e f f e c t i n the middle of the 1970s—is near zero. Although v i r - tual extermination of the species i s u n l i k e l y , near extinction of the stock and the p r a c t i c a l disappearance of North Sea herring as a commer- c i a l species i s a very r e a l p o s s i b i l i t y . This r e s u l t puts the management of clupeids i n perspective, because the c h a r a c t e r i s t i c s of this fishery are l i k e l y to carry over to f i s h e r i e s on other schooling species. The depleted state of a number of clupeids i s further evidence of this r e s u l t . Although open access - 101 - f i s h e r i e s w i l l cause depletion also i n the case of ground f i s h e r i e s , the stock dependence of harvesting costs w i l l be greater. of The possibility stock extinction i s therefore less i n these f i s h e r i e s than for c l u - peids. Failure to manage a clupeid fishery can have i r r e v e r s i b l e conse- quences . When i t comes to the case of a sole owner, i t had been asserted that a low stock output e l a s t i c i t y implies that costs have a n e g l i g i b l e effect on the optimal stock l e v e l . The results of this thesis show that this assertion i s f a l s e , at least i n the case of North Sea herring. The i n c l u s i o n of costs i n the intertemporal p r o f i t function are shown to cause a considerable increase i n the optimal stock l e v e l . The results show that bionomic equilibrium i s near zero, while the same cost/price r a t i o causes a marked increase i n the sole owner stock l e v e l . This apparent contradiction could be attributed to the nonlinear nature of the production function. It was shown that for a s u f f i c i e n t l y high cost/price r a t i o , i t would never be optimal for the sole owner to drive the stock to e x t i n c t i o n . Thus, costs may be an e f f e c t i v e brake on stock depletion i n the sole owner case, but need not be so i n the open access case. Two cost alternatives were considered. The f i r s t included only variable c o s t s , while the second included v a r i a b l e , insurance and maintenance costs. As stock size i s increasing i n costs, equilibrium bio- mass i s larger for the second than for the f i r s t cost a l t e r n a t i v e . Inclusion of c a p i t a l costs would have caused a further increase i n the optimal stock l e v e l . It was also shown that while the optimal stock l e v e l i s s i g n i f i c a n t l y influenced by the choice between the two discrete - 102 - cost a l t e r n a t i v e s , once the choice was made i t was not very sensitive to moderate changes i n the cost/price r a t i o . The optimal stock l e v e l was shown to be not very sensitive to moderate changes i n the discount r a t e . It was i l l u s t r a t e d that costs have a s t a b i l i z i n g influence on the stock l e v e l . The optimal harvest quantity was shown to be insensitive to changes i n the optimal stock l e v e l , a r e s u l t caused by the properties of the estimated model of popul a t i o n dynamics. For a change i n the discount r a t e , the optimal harvest quantity—and thus gross r e v e n u e s — w i l l usually be almost unchanged. However, the optimal stock l e v e l w i l l be a f f e c t e d , causing an adjustment i n harvesting costs. Changes i n other parameter values w i l l frequently have the same consequences. The relationship between the optimal stock l e v e l and the production technology was analyzed. Some new results were forthcoming, because the underlying model was nonlinear. It was shown a n a l y t i c a l l y and i l l u s t r a t e d empirically that the optimal stock l e v e l i s decreasing i n the output e l a s t i c i t y of e f f o r t and the amount of fixed factors i n the industry. Furthermore, i t was i l l u s t r a t e d that the optimal stock l e v e l — u n d e r certain c o n d i t i o n s — c a n be decreasing i n the stock output elasticity. This result was also obtained e m p i r i c a l l y . The p r a c t i c a l a p p l i c a b i l i t y of this thesis would be enhanced i f the present research were extended i n a number of ways. F i r s t , the approach path(s) to the optimal stock l e v e l should be investigated more closely. Second, matters concerning the transboundary character of the fishery need to be considered. studied. T h i r d , f i s h e r y regulations need to be Fourth, the e f f e c t s of uncertainty on resource management - 103 - should be analyzed. Although i t may be desirable a n a l y t i c a l l y to ana- lyze these topics separately, i t should be stressed that from the point of view of f i s h e r i e s management, they are c l o s e l y i n t e r l i n k e d . The analysis. concept of a sole owner has played an important part i n the In such an i n s t i t u t i o n a l s e t t i n g , the assumption of price- taking behaviour may seem inconsistent. It should here be borne i n mind that there are two markets for herring, one for reducton purposes and the second for human consumption. While the assumption of price-taking behaviour presumably i s innocuous i n the f i r s t market, i t may be more appropriate to model the sole owner as a monopolist i n the consumer market. This represents another avenue for future research. A f i n a l word on the present t h e s i s . the model i s quite robust. The results indicate that Different alternatives were specified both for the model of population dynamics and the production function. How- ever, q u a l i t a t i v e l y speaking, the results are not sensitive to choice of functional form. Although the numerical results may vary somewhat between model s p e c i f i c a t i o n s , the same q u a l i t a t i v e results are forthcoming. This result i s reassuring. - 104 - BIBLIOGRAPHY Anon. (1977). "Assessment of Herring Stocks South of 62°N 1973-75." Cooperative Research Report of the International Council for the Exploration of the Sea 60. Anon. (1982). "Report of the Herring Assessment Working Group for the Area South of 62°N." International Council for the Exploration of the Sea. C M . 1982/Assess: 7. Anon. (1983). "Report of the Herring Assessment Working Group for the Area South of 62°N." 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"A Generalized Harvest Function for Fishing: A l l o c a t i n g E f f o r t Among Common Property Cod Stocks." Unpublished paper, Department of Economics, University of Colorado. Munro, G. R. (1979). "The Optimal Management of Transboundary Renewable Resources." Canadian Journal of Economics 12:355-376. (1982). " B i l a t e r a l Monopoly i n Fisheries and Optimal Management Policy" i n Essays i n the Economics of Renewable Resources. Edited by L. J . Mirman and D. F. Spulber. Amsterdam, New York and Oxford: North-Holland. Munro, G. R., and Scott, A. D. (1984). "The Economics of Fisheries Management." Vancouver: Department of Economics, University of B r i t i s h Columbia. Discussion Paper 84-09. Murphy, G. I . (1977). "Clupeids" i n F i s h Population Dynamics. Edited by J . A. Gulland. New York: Wiley. Neher, P. A. (1974). "Notes on the Volterra-Quadratic Fishery." Journal of Economic Theory 8:39-49. Partridge, B. L. (1982). "The Structure and Function of Fish Schools." S c i e n t i f i c American 246:90-99. Pearse, P. H. (1979). "Property Rights and the Regulation of Commercial Fisheries." Vancouver: Department of Economics, University of B r i t i s h Columbia. Resources Paper No. 42. Posthuma, K. H. (1971). "The E f f e c t of Temperature i n the Spawning and Nursery Areas on Recruitment of Autumn-Spawning Herring i n the North Sea." Rapports et Proces-verbaux des Reunions 160:175-183. - 108 - Ricker, W. E. Computation and Interpretation of B i o l o g i c a l of F i s h Populations. Ottawa: Environment Canada. Statistics Rothschild, B. J . (1977). "Fishing E f f o r t " i n F i s h Population Dynamics. Edited by J . A. Gulland. New York: Wiley. S a v i l l e , A., and B a i l e y , R. S. (1980). "The Assessment and Management of the Herring Stocks i n the North Sea and to the West of Scotland." Rapports et Proces-verbaux des Reunions 177:112-142. Schaefer, M. B. (1957). "Some Considerations of Population Dynamics and Economics i n Relation to the Management of Marine Fishes." Journal of the Fisheries Research Board of Canada 14:669-681. Schrank, W. E.; Tsoa, E.; and Roy, N. (1984). An Econometric Model of the Newfoundland Groundfishery: Estimation and Simulation. S t . John's: Department of Economics, Memorial University of Newfoundland. Schumacher, A. (1980). "Review of North A t l a n t i c Catch S t a t i s t i c s . " Rapports et Proces-verbaux des Reunions 177:8-22. Schworm, W. E. (1983). "Monopsonistic Control of a Common Property Renewable Resource." Canadian Journal of Economics 16:275-287. Scott, A. D. (1955). "The Fishery: The Objectives of Sole-Ownership." Journal of P o l i t i c a l Economy 63:116-124. (1979). "Development of Economic Theory on Fisheries Regulation." Journal of the Fisheries Research Board of Canada 36:725-741. Stapleton, D. C. (1981). "Inferring Long-Term Substitution P o s s i b i l i t i e s from Cross-Section and Time-Series Data" i n Modelling and Measuring Natural Resource S u b s t i t u t i o n . Edited by E. R. Berndt and B. C. F i e l d . Cambridge and London: The MIT Press. U l l t a n g , 0. (1976). "Catch per Unit E f f o r t i n the Norwegian Purse Seine Fishery for Atlanto-Scandian (Norwegian Spring Spawning) Herring." FAQ Fisheries Technical Papers 155:91-101. (1980). "Factors Affecting the Reaction of Pelagic Fish Stocks to Exploitation and Requiring a New Approach to Assessment and Management." Rapports et Proces-verbaux des Reunions 177:489-504. Walters, C. J . , and Hilborn, R. (1978). "Ecological Optimization and Adaptive Management." Annual Review of Ecological Systems 9:157-188. - 109 - Wilen, J . E. (1976). "Common Property Resources and the Dynamics of Overexploitation: the Case of North P a c i f i c Fur Seal." Vancouver: Department of Economics, University of B r i t i s h Columbia. Resources Paper No. 3. Young, R. D. (1979). "Some Aspects of Production from an Ocean Fishery. Ph.D. d i s s . , Department of Economics, University of Santa Barbara. - 110 - APPENDIX 1: TECHNICAL DERIVATIONS 1. The General Model The Lagrangean for the maximization problem i s : (1) L = I ' C d ^ E t ^ t . t t ) - q t [ S t + 1 " (S t - H t ) e t=0 6(S t ) - G(S t _ Y )]} where Ht = H t ( E t ; S t , K t ) , q t = shadow price of the resource, d = 1/(1 + r) = the discount f a c t o r , and r = the discount r a t e . F i r s t order necessary conditions for an optimum are (Clark, 1976): i. -||- = 0, t = 0,1,2,... ii. ||r- = 0, t = 1,2,3,... To derive an expression for the steady state stock l e v e l , one proceeds by taking derivatives of (1) for general t (t >^ 1): <> 2 (3) f^= S-«tHEe d 6 ( S t ) =° | | - = d^;s + qt6-'(S)(St - H t ) e 6(S) + qte 6 ( S ) ( l - Hg) - q,--! + q Y + t G * ( S t ) = 0 1 (3 ) d ^ s + q t 6'(S)(S t - H t ) e 6(S) + qte 6 ( S ) ( l - H s ) + q Y + t G ' ( S t ) = q t _! The equations of motion are given by (2) and ( 3 ) . The shadow price of the resource i s found from equation (2): - Ill - ( 4 ) "t • * ' - ^ f e r E The shadow price i s inserted i n ( 3 ' ) , which i s simplified to become: (5) e 6(S) [HE ^ E - H 8 + 1] + 6'(S t )(S t - H t ) e 6(S t ) Y + d G'(S t ) = ± The normalized net revenue function i s defined as i r t = H(E t ;S t ,K t ) - c E t The derivatives are given by TT E S = H E n - c S These results can be used to rewrite (5): (5') e 6 ( S ) [ ^ + 1] + 6'(S)(S t - H t ) e E 6(S t ) + dV(St) = 1 + r For the stock to be i n steady s t a t e , one requires: (6) S t + 1 - St = ( S t - H t ) e 6(S t ) + G(S t _ y ) - S t = 0 Equation (6) can be solved for the steady state harvest quantity: (7) H = H(S) = S S - G(S) fi(S) e U t i l i z i n g equation (7) i n ( 5 ' ) , the following i m p l i c i t expression f o r the optimal stock l e v e l S* i s derived: (8) 6 Y e ^ * ^ - ^ - + 1] + 6'(S*)[S* - G(S*)] + d G'(S*) = 1 + r E Define (9) MSE = CTTS/TTE where MSE i s the Marginal Stock E f f e c t . - 112 - 2. The Cobb-Douglas Case The Cobb-Douglas production function i s given by (10) a H = AE sP°K, K = TTK^ , j where time subscripts have been suppressed. The net revenue function becomes (11) a TT = AE Sp°K - cE and the derivatives are given by TT = ctH/E - c Ji TTS = pQH/S The MSE i s accordingly (9 .) M S E = _|fH/S_ The l e v e l of e f f o r t needed to obtain the harvest corresponding given steady state stock l e v e l i s found by use of equation (7): a H = AE Sp°K = S - (12) E = (AKr 1 / a S ~ ( ^ e S-po/«[S - It can be ascertained that i. ii. | | < 0 and lim E = 0. K > c o S ) S ] e 1 / a to a - 113 - 3. The Marginal Stock E f f e c t The MSE i s defined as (9) MSE = CTr s /Tr E = m(c,ct,p0,S,K). The MSE vanishes for c = 0. By the properties of the p r o f i t function, The interactions between the MSE and the different parameters w i l l now be investigated. The Cobb-Douglas production function i s assumed with A = 1. The E f f o r t Output E l a s t i c i t y Assume for s i m p l i c i t y that p o = 1 and insert the appropriate derivatives in (9): M S E = a- ^ « E *SK - c Then 2 a 2a 1 3MSE _ ... c E KlnE + cE ~ SK U ; 3° " a 1 [dE - SK - c ] 2 2 This derivative i s negative provided i. c > 0, ii. Hg ^ c, i . e . , TT E ^ 0. If E < 1, a third condition would need to be s a t i s f i e d i n order to make the derivative negative; this special case can safely be disregarded. Both the numerator and the denominator of the MSE tend to i n f i n i t y as 00 a -»• , provided E > 1. L'Hopital's r u l e : The l i m i t i n g value i s found by applying - 114 - i• ucD = i lim MSE l i•m a-»-oo a^-oo ,. = lim a->co a E a_1 cE KlnE a-1 SK + aE SKlnE cKlnE ; E SK(1 + alnE) -1 = n0 Fixed Factors Assume for s i m p l i c i t y that a = p Q = 1 and insert the appropriate derivatives i n (9): MSE = C E K SK - c Then (14) 3MSE 2 c. = [SK - c ] 2 This derivative i s unambiguously negative i. ii. provided c >0 TTE i= 0. Moreover, lim MSE = lim ^ K->°° K->°° = ~ lim E = 0. K->oo The Stock Output E l a s t i c i t y Assume for s i m p l i c i t y that a = 1. Then MSE cSJESpOK 3 P sr°K - c j3 0 = 0 ••» MSE = 0 n. This case can be resolved by L'Hopital's rule: - 115 - E SSr ° KlnS MSE = lim ^ ca3 n E ^ sP°KlnS Po-"" P o -+<*> lim = lim cgnE Po— = c - — lim polim E P o — ' po S By use of equation (12), i t can be shown that this l i m i t i s zero provided S > 1. iii. 0 < p „ < co 3MSE 3 Po = [cEsP°K + C p E s P ° K l n S ] [ S P ° K - c] - Sp°KlnScp EsP°K n n " [sPo K - ]2 C = c E s P ° K [ s P ° K - cp^lnS - c ] [SP°K - c ] 2 The sign of t h i s derivative i s ambiguous, depending on the term i n brackets i n the numerator. the MSE and p o Therefore, d i f f e r e n t relationships between may occur, as outlined i n Section 2.3. In p a r t i c u l a r , i t i s claimed i n Section 2.3 that the MSE can be a decreasing function of p . Consider the derivative when evaluated at p 9 MSE _ cEK po=0 ~ K - c o Po o = 0: A necessary and s u f f i c i e n t condition for this derivative to be negative is K - c <0 However, note that TTg = K - c when A = a = 1 and p o = 0. This says that i f the fishery i s unprofitable when there i s no stock e f f e c t , i . e . , i T g < 0 at p decreasing i n p . Q o = 0, then the MSE—at least l o c a l l y — w i l l be - 116 - APPENDIX 2: DATA Table A l : B i o l o g i c a l Data Recruits the following year Millions Year Spawning stock per September 1 1,000 tonnes 1947 1948 1949 2,945 2,581 2,618 4,720 4,100 5,680 587 502 509 -0.1159 -0.0380 -0.0531 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 2,428 2,169 1,908 1,707 1,745 1,821 1,741 1,593 1,236 2,063 6,900 7,690 9,100 8,070 7,700 4,768 21,429 5,641 7,555 1,954 492 600 664 699 763 806 675 683 671 785 0.0053 -0.0307 -0.0174 0.0807 0.0216 0.1193 0.0770 0.0051 0.1763 0.1454 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1,871 1,601 1,132 1,800 1,829 1,340 1,116 817 390 359 16,686 7,085 8,740 10,907 5,709 5,289 7,581 7,623 3,820 9,081 696 697 628 716 871 1,169 896 696 718 547 0.0230 0.0926 -0.0252 0.1397 0.0903 0.1000 0.0893 0.0669 0.1754 0.1470 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 318 219 269 228 166 117 141 123 154 208 7,146 4,975 2,398 5,583 773 720 1,064 899 2,582 3,423 564 520 498 484 275 313 175 46 11 25 0.1306 0.1514 0.1817 0.1593 0.1404 0.1382 0.2265 0.1607 0.0237 0.0385 1980 1981 1982 238 368 498 12,414 14,958 61 95 0.0891 0.0589 0.1137 * Harvestj v quantity 1,000 tonnes Instantaneous net naturals growth rate * - 117 - 1) 2) Harvest of juvenile herring i s included. The net growth rate i s estimated according to the formula — 6 F S t + 1 = Ste t t + 6 f c = F t + l n ( S t + 1 / S t ) ; F t = f i s h i n g m o r t a l i t y . For these estimates the stock size as of January 1 has been used. * Not a v a i l a b l e . Sources: St: Anon. (1977) for 1947-54 Anon. (1982) for 1955-74 Anon. (1983) for 1975-82 Rt and F t : Anon. (1977) for 1947-54 Anon. (1982) for 1955-72 Anon. (1983) for 1973-81 Ht: Schumacher (1980) for 1947-76 Anon. (1982) for 1977-81 - 118 - Table A2: Norwegian P a r t i c i p a t i o n i n the Herring Fishery Number of participating purse seiners Year Norwegian harvest Tonnes Without powerblock With powerblock 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 3,454 147,933 586,318 448,511 334,449 286,198 134,886 220,854 210,733 136,969 135,338 66,236 34,221 33,057 3,911 16 148 150 72 14 0 0 0 0 0 0 0 0 0 0 0 47 134 262 312 352 253 201 230 203 153 165 102 92 24 1) 2) Index of average size of 2) purse seiners 1.000 1.060 1.124 1.192 1.264 1.340 1.420 1.505 1.677 1.900 2.015 2.115 2.173 2.331 2.406 Column 3 i l l u s t r a t e s the rapid introduction of powerblock i n the f l e e t . Boats with and without powerblock need to be weighted. The low number of data points makes the estimation of weights by nonlinear regression methods unfeasible. Therefore, weights are imposed: two boats without powerblock are set equal to one boat with powerblock. In the regressions, actual boat size has been used rather than the index l i s t e d here. It i s immaterial which one of these series i s chosen. Sources: Number of p a r t i c i p a t i n g purse seiners for 1971, 1972 and 1977: Noregs S i l d e s e l s l a g (Norwegian Herring Fishermen's Cooperative) Index of average boat s i z e : Based on Melhus (1978) Other data: The Directorate of F i s h e r i e s , Norway - 119 - COSTS AND PRICES The Cost/Price Ratio for the Harvest Supply Function In the harvest supply f u n c t i o n , the cost/price r a t i o enters as a variable. While data on the output price are a v a i l a b l e , this i s not the case for data on the cost of e f f o r t . Therefore, a time series needs to be constructed for this v a r i a b l e . The cost per boatday, c^, includes labour, f u e l and material c o s t s . In other words, only variable costs are considered. A l l that i s needed i s an index of the change i n cj- over time and not the absolute cost f i g u r e s . Price indices for wages, f u e l and materials are r e a d i l y available. The index for cj. w i l l then be a weighted average of these i n d i c e s , which raises the question of the appropriate set of weights. The Norwegian Budget Committee for the Fishing Industry (Budsjettkomiteen for fiskerinaeringen) has collected cost data for f i s h i n g vessels since 1968. On the basis of their data for purse seiners, the following cost shares have been established: Period Share of wages Share of f u e l Share of materials 79.20% 72.15% 13.80% 21.75% 7.00% 6.10% 1969-73 1974-77 A simple inspection of the annual data shows that the e f f e c t s of the f i r s t shock i n o i l prices became noticeable i n 1974, which i s the reason for the d i v i s i o n i n the time period. By using shares that are averaged over a period rather than annual shares, random differences from year to year are avoided. The d i f f e r e n t indices are given in Table A3, while price data are presented below i n Table A6. - 120 - Table A3: The Cost of E f f o r t Index Year Wage i n d e x ^ Fuel 2 v price index 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 100.0 106.2 115.8 124.2 133.7 144.4 158.0 177.1 198.8 216.5 239.5 281.0 336.6 391.8 434.6 100.0 104.0 102.0 102.0 108.9 111.9 110.9 113.9 128.7 132.7 141.6 200.0 221.8 250.5 274.3 1) 2) 3) 4) Raw materials price^s, index 100 104 109 108 108 106 111 124 130 128 136 168 193 206 212 Cost of effort.N • j 4) index 100.0 105.7 113.4 120.0 128.5 137.2 148.2 164.7 184.3 198.7 218.7 256.5 302.9 349.7 386.2 Index of hourly earnings i n manufacturing companies a f f i l i a t e d with the Norwegian Employers' Confederation. Wholesale price index for fuel and e l e c t r i c i t y . Wholesale price index for raw materials (except f u e l ) . Materials include, i n order of magnitude based on 1977 figures: miscellaneous costs, telephone expenses, harbour fees, some items of hired labour, including s o c i a l costs, i c e , e t c . Considering the contents of this cost category, the choice of index that has been made may be questioned. However, due to the low share of these c o s t s , i t would not matter i f another index were used. The weights for the period 1969-73 were used for the years 1963-68 as w e l l . Source: S t a t i s t i c a l Yearbook, Central Bureau of S t a t i s t i c s of Norway The p a r t i c u l a r wage index that i s chosen i s average earnings i n the manufacturing industry. We have used this rather than e.g. ex post seasonal returns i n the f i s h i n g industry. This i s because the - 121 - manufacturing industry represents important alternative employment opportunities for fishermen. For this reason changes in wages i n the f i s h i n g industry w i l l over time have to correspond to those i n industry. The Cost/Price Ratio for the Sole Owner While an index i s s u f f i c i e n t for the estimation of the supply function, this i s not the case for the cost/price r a t i o that i s used i n the estimation of the optimal stock l e v e l ( S * ) . In this case, the absolute value of the cost/price r a t i o matters, although S* i s , of course, l i n e a r l y homogeneous to proportional changes i n the cost of e f f o r t and the output price. The supply function was estimated using data for 1963-77. The optimal stock l e v e l , however, i s the target one would aim at achieving at some time i n the future. Therefore, present day, or i d e a l l y , expec- ted future, price and cost data should be used, rather than h i s t o r i c data. The optimal stock l e v e l i s derived on the assumption that there i s a sole owner. While i t has been argued that only variable costs were relevant i n the estimation of the supply function, this may be d i f f e r e n t in the case under consideration here. The following costs need to be considered by the sole owner: 1. variable costs, 2. insurance costs, 3. maintenance costs, and 4. rental price of c a p i t a l . - 122 - Category 1 e s s e n t i a l l y corresponds to the cost of e f f o r t , c^, that was discussed above. In Section 2.3 i t i s argued that the opportunity cost of c a p i t a l i s low. In this a n a l y s i s , i t w i l l be set equal to zero. Two 1. cost alternatives w i l l be considered: Variable costs only. This e s s e n t i a l l y corresponds to the case that has been considered. 2. Variable costs plus insurance and maintenance c o s t s . The remuneration of fishermen in the Norwegian purse seine f l e e t i s based on the share system. According to this system, the crew receive a certain percentage of revenue with the remainder accruing to the owners of gear and boat. For this reason, the following approach w i l l be used: a price forecast for herring w i l l be given, of which a s h a r e — t o be determined s h o r t l y — w i l l accrue to the c a p i t a l owner, i . e . , the sole owner of this a n a l y s i s . This means that variable costs w i l l include only f u e l and m a t e r i a l s . Table A4 gives average annual operating costs for purse seiners for the period 1975-81, while d a i l y costs—expressed given i n Table A5. i n real p r i c e s — a r e - 123 - Table A4: Average Annual Operating Costs for Norwegian Purse Seiners 1975-81. Figures i n kroner Year 1975 1976 1977 1978 1979 1980 1981 Number of Wholesale price operating index Materials Insurance Maintenance days Fuel 327,405 468,228 478,629 501,840 642,034 752,668 1,010,641 86,892 110,914 148,922 141,754 157,552 175,181 224,894 156,780 201,189 191,702 221,974 249,371 202,023 207,871 646,427 896,663 844,734 799,659 779,323 734,086 835,897 244 237 221 243 225 198 206 60 65 69 72 78 90 100 Sources: Cost data: Annual Cost Investigations 1975-81, The Budget Committee for the Fishing Industry (Budsjettkomiteen for fiskerinaeringen) Price index: S t a t i s t i c a l Yearbook 1983, Central Bureau of S t a t i s t i c s of Norway Table A5: Average Daily Operating Costs for Norwegian Purse Seiners. Figures i n 1975 kroner Year Variable costs Insurance and maintenance Total costs 1975 1976 1977 1978 1979 1980 1981 1,698 2,256 2,470 2,208 2,734 3,124 3,599 3,292 4,276 4,078 3,503 3,517 3,152 3,040 4,990 6,532 6,548 5,711 6,251 6,276 6,639 1) Conversion into real prices by means of wholesale price index given i n Table A4. - 124 - A number of comments are required. F i r s t , as noted i n Section 2.3, the boats in question participate i n a number of fisheries. It i s here assumed that d a i l y operating costs are equal i n the d i f f e r e n t f i s h e r i e s . Second, insurance and to some extent maintenance costs are r e a l l y fixed c o s t s . The rationale for treating them as variable i s that they, i n a given year, need to be covered incomes earned during the operating days. T h i r d , expenditures by on maintenance are to some extent influenced by an incentive to reduce taxes. For this reason, and i n order to avoid random differences," the Budget Committee averages boats' maintenance costs over three years. Fourth, the Budget Committee attempts to take stock changes into account, so that expenditures i n a given year pertain to that year only. The figures i n Table A5 show that d a i l y variable costs have doubled i n r e a l terms over the period in question. due to the development i n o i l p r i c e s . This i s e s s e n t i a l l y On the other hand, there i s no particular trend for insurance and maintenance c o s t s . The following assumptions w i l l be used to establish costs: 1. variable: average of 1980 and 1981; and 2. insurance and maintenance: average for 1975-81. This gives the following d a i l y cost of e f f o r t f i g u r e s , expressed i n 1975 kroner: Variable: c l = k r . 3,362.00 V a r i a b l e , insurance and maintenance: c 2 - k r . 6,913.00 It i s noticeable that the inclusion of insurance and maintenance costs more than double d a i l y operating c o s t s . - 125 - Having analyzed the costs, output price w i l l now be considered. Table A6 gives price data for the period 1963-77. These figures should only be considered together with the quantity figures i n Table A2. The high prices i n the last years of the period are due to small quantities combined with a high proportion of d e l i v e r i e s for human consumption. Table A6: Norwegian Price Data Year Average price of herring Kroner/tonne 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 387 339 342 357 235 213 309 437 408 358 640 830 1,226 1,422 2,359 Source: The Directorate of Fisheries The crew's share of gross revenue declined somewhat during the period 1975-81, and amounted to an average of 35.9% 1 for 1979-81 \ The government assesses a fee of 3.1% of the f i r s t hand value of Based on the Budget Committee's cost investigations. Average crew share for 1975-78 was 37.1%. The decline may be due to a s h i f t towards more c a p i t a l intensive boats. Some crew members (skipper, engine chief and cook) receive fixed remuneration i n addition to their shares. The error that i s caused by treating these fixed payments as variable i s n e g l i g i b l e . - 126 - fish 2) . This means that the sole owner's share amounts to 61%. As discussed above, herring i s used for reduction into f i s h meal and f i s h o i l and for human consumption. The f i s h meal market and demand for f i s h meal i s analyzed by Hansen (1979). the With the harvest quantities corresponding to the optimal stock l e v e l s estimated i n Chapter 4, i t i s quite l i k e l y that the major part of the harvest w i l l be used for reduction purposes. Hansen et a l . (1978) e s t a b l i s h price forecasts for f i s h meal and f i s h o i l and use these to derive f i r s t hand prices for d i f f e r e n t species. I d e a l l y , a more recent forecast would be preferred, but apparently this work has not been updated. It i s judged to be beyond the context of this thesis to undertake such a task. The price forecast i s derived as follows: Price forecast, based on Hansen et a l . (1978) for the period 1981-85, per tonne North Sea herring (1975 kroner): 20% extra for consumption d e l i v e r i e s : k r . 380.00 k r . 76.00 Average f i r s t hand price: Crew share and product fee (39%): k r . 456.00 k r . 178.00 Net price: k r . 278.00 This gives r i s e to the following cost/price r a t i o s , where the price per 1,000 tonnes has been used: 1. ci/p = 0.0121 2. c 2 /p = 0.0249 This completes the analysis of costs and p r i c e s . 2) Source: Cost investigation for 1979, Budget Committee for the Fishing Industry. The fee i s supposed to cover employers' pension contributions on behalf of their employees.
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The optimal management of an ocean fishery Bjørndal, Trond 1984
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Title | The optimal management of an ocean fishery |
Creator |
Bjørndal, Trond |
Publisher | University of British Columbia |
Date Issued | 1984 |
Description | The objective of this thesis is to study the optimal management of North Sea herring. The analysis is based on a dynamic bioeconomic model for a fish resource, consisting of a model of population dynamics and a net revenue function. The model of population dynamics is described by a delay-difference equation. The model distinguishes between natural growth and mortality in the existing stock as opposed to new recruitment to the stock, which takes place with a time lag. The model is estimated based on time series data for the period 1947-82. The net growth function is shown to exhibit depensation, a phenomenon not uncommon for schooling fish like herring. In fisheries economics, the production function is often treated in a rather restrictive manner. The approach of this thesis is to specify a general production function, where output (harvest) is a function of variable inputs, stock size and other fixed factors. Cross-sectional (1968, 1971 and 1975) and aggregate time series (1963-77) data sets for the North Sea herring fishery are available. The cross-sectional data facilitate direct estimation of the production function (Cobb-Douglas). The time series data are used to estimate a harvest supply function (Cobb-Douglas), and by duality theory the parameters of the corresponding production function are derived. A hypothesis of increasing returns to scale in all inputs is accepted in all model specifications. The stock output elasticity generally varies between 0.1 and 0.5. Bionomic equilibrium--i.e., the open access stock level--is estimated to be close to zero. The last two results are attributed to the fact that the resource in question is a schooling one. The model is extended by introducing stock dynamics and the concept of a sole resource manager. An intertemporal profit function is maximized and an expression for the optimal stock level is derived. Some new analytical results with regard to the relationship between the optimal stock level and the production technology are derived. The quantitative results show that the inclusion of costs in the intertemporal profit function causes a considerable increase in the optimal stock level. The assertion that a low stock output elasticity implies that costs have a negligible effect on the optimal stock level is therefore not necessarily true. This is a result of the nonlinear nature of the production technology. The optimal stock level is shown to be not very sensitive to moderate changes in the discount rate. It is illustrated that costs have a stabilizing influence on the stock level. The optimal harvest quantity is quite insensitive to changes in the stock level, a result caused by the properties of the estimated model of population dynamics. Lastly, the model is found to be robust in the sense that the different specifications of the model of population dynamics and the production technology give rise to the same qualitative results. |
Subject |
Fishery management |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-06-11 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096532 |
URI | http://hdl.handle.net/2429/25553 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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