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The optimal management of an ocean fishery Bjørndal, Trond 1984

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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 i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department Of Economics The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e December 12, 1984 DE-6 (3/81) - i i -ABSTRACT The objective of this thesis i s 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 i s 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 fi s h like herring. In fisheries economics, the production function is often treated in a rather restrictive manner. The approach of this thesis i s to spec-i f y 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 f a c i l i t a t e 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 correspon-ding production function are derived. A hypothesis of increasing returns to scale in a l l inputs is accepted in a l l model specifications. - i i i -The stock output elas t i c i t y generally varies between 0.1 and 0.5. Bio-nomic equilibrium—i.e., the open access stock l e v e l — i s 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 i s extended by introducing stock dynamics and the con-cept 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 l e v e l . The assertion that a low stock output el a s t i c i t y implies that costs have a negligible effect on the optimal stock level is therefore not necessarily true. This i s 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 i s illustrated that costs have a stabilizing influence on the stock l e v e l . The optimal harvest quantity i s quite insensitive to changes in the stock l e v e l , 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 popula-tion dynamics and the production technology give rise to the same quali-tative results. - iv -TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENT v i i DEDICATION v i i i 1.0 INTRODUCTION 1 2.0 THE BIOECONOMIC MODEL 7 2.1 The Model of Population Dynamics 7 2.2 A Fishery Profit Function 12 2.3 The Intertemporal Profit Function 15 2.3.1 Dynamic Optimization 15 2.3.2 The Marginal Stock Effect 24 2.4 Summary 29 3.0 THE EMPIRICAL MODEL 31 3.1 The Model of Population Dynamics 31 3.1.1 The Stock-Recruitment Function 32 3.1.2 The Net Growth Function 39 3.1.3 The Complete Model 43 3.2 The Production Function 46 3.2.1 The Nature of the Fishery 48 3.2.2 Empirical Estimation 51 4.0 THE OPTIMAL STOCK LEVEL 75 4.1 The Influence of Costs 75 4.2 The Influence of the Marginal Stock Effect 87 4.3 Some Management Issues 90 5.0 SUMMARY 99 BIBLIOGRAPHY 104 APPENDIX 1: TECHNICAL DERIVATIONS 109 APPENDIX 2: DATA 115 - v -LIST OF TABLES 3.1 Estimated Stock-Recruitment Functions 37 3.2 Maximum Recruitment 37 3.3 Estimation of Net Natural Growth Functions 40 6 (S ) 3.4 Characteristics of Function Ste fc 40 3.5 The Carrying Capacity of North Sea Herring 44 3.6 Estimates of Sm s y and MSY 44 3.7 Summary of Cross-Sectional Data 54 3.8 Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data 58 3.9 Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data. Main Season (June to August) Catches Only 58 3.10 Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. K^t Exogenous. Time Series Data 1963-77 66 3.11 Derived Production Function Parameters (Cobb-Douglas) for North Sea Herring 66 3.12 Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. K jt Endogenous. Time Series Data 1963-77 70 3.13 Derived Production Function Parameters (Cobb-Douglas) for North Sea Herring. K it Endogenous 70 3.14 2SLS Nonlinear Estimation of Production Function Parameters (Cobb-Douglas) for North Sea Herring. S2 Value Imposed 71 4.1 Optimal Stock Level (S*) and Corresponding Harvest (H*) 81 4.2 The Optimal Stock Level for the Variable Cost Case 85 4.3 Sensitivity to a 20% Change in Cost/Price Ratio. Base Case Cost Alternative 86 4.4 Sensitivity to Changes in the Effort Output El a s t i c i t y (a) 87 4.5 Sensitivity to Changes in the Stock Output E l a s t i c i t y (BQ) 88 4.6 Effects of Changes in Fleet Size 90 4.7 Simulation of Spawning Stock 1983-87 92 Al Biological Data 115 A2 Norwegian Participation in the Herring Fishery 117 A3 The Cost of Effort Index 119 A4 Average Annual Operating Costs for Norwegian Purse Seiners 1975-81 122 A5 Average Daily Operating Costs for Norwegian Purse Seiners 122 A6 Norwegian Price Data 124 - v i -LIST OF FIGURES 2.1 Stock dynamics 8 2.2 MSE as a function of po 28 2.3 S* as a function of pQ 28 2.4 MSE as a function of po 28 2.5 S* as a function of po 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 Participation 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 - v i i -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 addition, 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 Institute, Norway, The Directorate of Fisheries, Norway and the Norwegian Herring Fishermen's Cooperative. My studies at The University of British 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. - v i i i -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 herring. This fish stock was severely depleted in the 1960s and early 1970s due to overfishing, 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-stitute the essence of a management plan, w i l l be analyzed by means of a dynamic bioeconomic model. In the thesis, a bioeconomic model for a fish resource w i l l be developed. An important objective of the research is that the model should provide a good representation of real world phenomena and meet theoretical properties that seem appropriate for the relevant func-tions. The main contribution of the thesis w i l l be in the empirical application of the model to data for the North Sea herring fishery. The thesis i s based on capital theory. The fish resource is con-sidered a capital stock under the control of a sole resource manager, and changes in the stock over time are viewed as investments. The objective of the sole resource manager is assumed to be to maximize the present value of the flow of net revenues from the fishery. The properties of the dynamic bioeconomic models commonly used in fisheries economics, e.g. the Clark-Munro model^\ are well known. In addition, 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 in a theoretical framework. Indeed, Munro and Scott (1984) in their recent survey of the f i e l d of fisheries economics point out that the development to date in the area of empirical estimation of fisheries models has been limited, and that the scope for further research is great. In many ways this f i e l d is s t i l l in i t s infancy. To put the thesis in perspective, some of the characteristics 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 fishery. An important behavioural characteristic of North Sea herring that i s l i k e l y to influence both the biological growth function and the pro-duction technology is the schooling behaviour. Schooling fish contract their feeding and spawning range as the stock is reduced, with the size of schools often remaining unchanged. However, with modern fishfinding equipment harvesting can be profitable even at low stock levels. There-fore, herring and other clupeids are especially vulnerable to predation by man, as nature's brakes on stock depletion may not be very effective. Many of the important fisheries of the world are based on clu-peids (anchovies, sardines, capelin, herrings). However, in the empiri-cal literature there i s apparently no bioeconomic study of the manage-ment of a clupeid fishery and of the special problems caused by the schooling behaviour. It is hoped that this thesis can present a contri-bution also by drawing attention to some of these problems. - 3 -The model of population dynamics w i l l recognize that a fish popu-lation commonly can be divided into several subpopulations. In a bio-economic 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 stock, the model w i l l be formulated in terms of this variable rather than total biomass, as is commonly the case. The model of population dynamics w i l l be formulated in 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 in the existing stock as opposed to new recruitment to the stock. Moreover, recruitment w i l l take place with a time lag, 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 fisheries economics, the production function is often treated 2) in a rather restrictive manner. The Schaefer function (Schaefer, 1957), which i s linear in both effort and stock size, i s commonly used. While this function may be useful for expositional purposes, i t is pre-sumably less appropriate in empirical applications, as i t a priori 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 is harvest, E is effort, S i s stock size and q a constant catchability coefficient. To economists, this i s a special case of the Cobb-Douglas function. - 4 -(harvest) is a function of variable inputs, stock size and other fixed factors. Some of the characteristics particular to this fishery w i l l be taken into account in the modelling of the production technology. In the production function, 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 is assumed that there is excess capacity in the fleet with no alternative use, causing the oppor-tunity cost of capital to be low. This presumably represents a realis-t i c description of many developed fisheries in the short to medium 3) long run . The long run situation, when capital no longer is redun-dant, w i l l not be considered. Externalities are frequently encountered in fisheries. Static externalities 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 positive, e.g. due to sharing of information about locations of f i s h . The latter may potentially be important in the case presently under consideration, because of the importance of the search phase in a schooling fishery. The externalities need to be considered in the estimation of the production function (Brown, 1974). For the North Sea herring fishery, cross-sectional data are available for the years 1968, 1971 and 1975. These data f a c i l i t a t e direct estimation of the production function (Cobb-Douglas). In addi-tion, aggregate time series data are available for the period 1963-77. In this period, the fishery was characterized by open access, so that Clark, Clarke and Munro (1979), in an analytical model, con-sider a case similar to this one. - 5 -myopic profit-maximizing behaviour w i l l be assumed. A Cobb-Douglas har-vest supply functon w i l l be estimated. 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 profit maximization w i l l be performed for the case of a sole owner, and an implicit expression for the optimal stock level i s derived. In this way, the stock externality—the 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 level and the production technology w i l l be analyzed. This model—in 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 fish-ery. By this is meant the optimal stock level 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 practical implementation of such a policy, which involves mat-ters like the adjustment phase, the regulation of effort and transboun-dary issues, although these matters w i l l be discussed. The analysis w i l l be performed in the context of a deterministic model. The thesis i s organized as follows. In Chapter 2, the analytical model is developed, i.e., the model of population dynamics and the pro-duction and profit functions. A dynamic optimization is performed for the case of a sole owner, and the resulting equilibrium condition for the stock level is contrasted to the free entry solution (bionomic - 6 -equilibrium). The relationship between the optimal stock level and the production technology is analyzed. In Chapter 3, the empirical model is specified and estimated. Various functional forms for the model of population dynamics are con-sidered. The particulars of the fishery in question—due to the school-ing behaviour of herring—are discussed and taken into account in the formulation of the production technology. A Cobb-Douglas production function i s estimated based on the cross-sectional data. For the aggre-gate fishery, alternative specifications 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 is com-bined with the empirical results of Chapter 3. Estimates of the bio-nomic equilibrium and the optimal stock level with corresponding harvest quantities are presented. The sensitivity of the optimal stock level to changes in the discount rate, the cost/price ratio and the parameters of the production function is i l l u s t r a t e d . Examples of approach paths to the steady state stock level are given, and matters concerning fishery regulations, transboundary issues and uncertainty are briefly discussed. The major results are summarized in Chapter 5. Some avenues of future research, based on the material in this thesis, are outlined. Appendix 1 contains the technical derivations of the dynamic optimization problem. The data for the empirical estimations are given in Appendix 2. - 7 -2.0 THE BIOECONOMIC MODEL The purpose of this chapter is to develop the analytical model on which this thesis i s based and to perform a dynamic optimization for the sole owner case. Section 2.1 formulates the model of population dyna-mics. The static production and profit functions are treated in Section 2.2. The concept of a sole owner and an intertemporal profit function are introduced in 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 level and the production technology i s analyzed. An attempt has been made to develop a general model, although i t was formulated with a specific application in mind. The applicability 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 in the biomass of a fish 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 illustrated in Figure 2.1 (adapted from Ricker, 1975, p. 25). - 8 -Recruits Natural growth Usable stock 1^^-Natural mortality Harvesting Figure 2.1: Stock dynamics A fish population can generally be divided into several subpopu-lations. 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 will be used to explain changes in the biomass over time: (2.1) St +i = (St - Ht)eG"M + G(St_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, Ht, will be discussed in Section 2.2. - 9 -Clark (1976a) has developed a model which is similar to this one. However, Clark's model of population dynamics is in terms of num-bers and not weight. Another delay-difference model is the one devel-oped by Deriso (1980). The model can be formulated in terms of total biomass, X, by replacing S with X in 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 amplification. The f i r s t part of the right hand side of equation (2.1) denotes changes in the biomass due to natural growth, natural mortality and harvesting. In the model, i t is assumed that harvesting occurs in a short season at the beginning of the period. The escapement, St - Hfc, is l e f t to grow at the net instantaneous growth rate G - M. Therefore, in the absence of recruitment, changes in the spawning biomass over time are given by (2.2) St + 1 = (St - Ht)eG _ M If the fishery took place at the end of the period, we would have (2.2') St + 1 = SteG _ M - Ht Yet another alternative would be the case of a continuous fishery throughout the period, (2.2") St +i = SteG _ M"F, where F is instantaneous fishing mortality. Equation (2.2") is known as the Ricker model (Ricker, 1975). A seasonal fishery (Bradley, 1970; Flaaten, 1983) can be formulated as a further development of this model. The qualitative nature of the model is unaffected by these speci-fications. Thus, the choice between them depends on what is most - 10 -r e a l i s t i c for the fishery in question. Since F in a herring fishery may depend on stock size (Ulltang, 1980), i t is appropriate to focus on har-vest quantity rather than fishing mortality. The second part of the right hand side of equation (2.1) repre-sents addition to the biomass due to recruitment, which is assumed to occur at discrete time intervals. Moreover, recruits w i l l normally join the parent population several years after spawning. We postulate that (2.3) Rt+1 = g(St) where Rt+l i s t n e number of recruits to the juvenile population as a function of the previous period's spawning biomass. A certain fraction, X, w i l l survive the juvenile stage^ and join the spawning biomass, so that (2.4) g(St_y)X is the number of recruits joining the spawning biomass with a delay of Y periods. The delay occurs while the juveniles mature to spawning age. Letting w denote the weight of new recruits, we get (2.5) G(St_T) = g(St_Y)Xw where G(S t-y) denotes recruitment in weight to the spawning biomass. North Sea herring spawn in September every year. The following year a number of recruits, called zero-group herring, join the juvenile population as indicated by equation (2.3). After another two years the survivors (equation 2.4) become sexually mature and join the spawning or adult population (equation 2.5). Thus, in this case, Y = 2, and the The model may be generalized by letting X depend on juvenile stock density. - 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 recruits 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 fishing, equation (2.1) is reduced to (2.1•) St + 1 = SteG _ M + G(St_Y) = Ste6 + G(St_y), 6 = G - M, where 6 is the mean net natural growth rate. If there i s no fishing, changes in the biomass over time are given by equation (2.1'). Under natural conditions, a fish population w i l l grow towards i t s carrying capacity, which i s the upper limit of the stock size as determined by environmental conditions. However, in equa-tion (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 relatively 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 densities. 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 fish like herring (cf. Section 3.1). - 12 -where S is the carrying capacity of the stock; otherwise an upper limit to the size of the biomass would not exist under natural conditions. Equation (2.1) can now be restated: (2.6) St + 1 = (St - Ht) e6 ( St) + G(St_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 . G(S") = 0 * fi(?) = 0 i i . G(S) > 0 + 6(<F) < 0 such that S = G(S)/(1 - e6 ( S )) . The second relationship is probably the more r e a l i s t i c one. When the model is estimated empirically (Section 3.1), we w i l l check which one, i f any, of these two conditions holds. 2.2 A Fishery Profit Function It i s an objective of this analysis to treat the production pro-cess in a manner consistent with economic theory. In fishery economics, the production function i s generally treated in a rather restrictive manner. Much of the literature is based on the Schaefer production function (Schaefer, 1957), H = qES, q constant. Harvest (H) i s linear in both effort (E) and stock size (S) and is 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 is to specify a general production function: (2.7) Ht = H(Et;St,Kt) -> 3) Here, Et i s an n-dimensional vector of variable inputs, while fixed inputs consist of stock size (St) and an m-dimensional vector of other factors (Kt). The latter w i l l typically describe technological aspects of the fleet (gear types, number of boats, size distribution, fish-finding equipment, etc.). S and K are fixed in 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 in the functional form. The production function is assumed to be concave in E and 4) increasing in both variable and fixed inputs (Lau, 1978). Profits can be written as revenues minus variable costs: i (2.8) Pt = ptH(Et;St,Kt) - ^ 4tEi t i = pt[H(Et;St,Kt) - ?ci tE± t] i where pt is (nominal) price per unit output, c|t, i = l,...,n, i s (nominal) price per unit input i and ci t = c;[t/pt is the normalized price per unit input i . Maximizing profits with respect to variable inputs, expressions for the optimal levels of inputs as functions of prices and fixed factors are derived: (2.9) E* = fi( ct; St, Kt) , i = l,...,n 3) An arrow denotes a vector. 4) Weak inequality is implied; similarly for other properties of the production and profit function throughout the thesis. - 14 -Substituting (2.9) into (2.8) and dividing by output price, yields the normalized profit function (2.10) TT* = H(E*;St,Kt) - ^ci tE ^t = G(ct; St ,Kt) i This function gives maximized profits for a given set of values {ct;St ,£(-}. The normalized profit function is (Lau, 1978): 1. decreasing and convex in normalized prices of variable inputs, 2. increasing in the nominal price of output, 3. increasing in fixed factors, and 4. bounded, given St and Kt. In this model formulation, stock size enters into the profit function as a fixed factor, while stock dynamics are disregarded. The reason is that the fishery to be considered was unregulated and charac-terized by open access. Therefore, myopic profit-maximizing behaviour i s assumed. An intertemporal profit maximization, where stock dynamics are explicitly taken into account, w i l l be performed in Section 2.3. The data required for empirical analysis depend on which func-t i o n s ) one desires to estimate. Profits, input demand and harvest sup-ply are a l l functions of normalized prices, stock size and fixed fac-tors, with profits, input quantities and output quantity as the respec-tive dependent variables. The avai l a b i l i t y of data w i l l determine which function i t is feasible to estimate. The profit function is often more amenable to empirical estima-tion than the production function. The problems of simultaneity bias associated with direct estimation of the production function are - 15 -avoided. Estimation of a multi-input and/or multi-output system of equations is also easily f a c i l i t a t e d . In Section 3.2 production func-tions for North Sea herring w i l l be estimated. 2.3 The Intertemporal Profit 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 implicit expression for the optimal stock level derived. This w i l l be contrasted to the open access solution. In addition, the relationship between the optimal stock level and the production technology w i l l be analyzed. 2.3.1 Dynamic Optimization The production function, as defined in equation (2.7), contains inter a l i a a vector of variable inputs. As stated in Section 2.2, this does not in principle represent any problems for empirical estimation of the production function or the dual, static profit 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-fication of the problem (Clark, 1976). Therefore, i t w i l l henceforth be assumed that there is only one control variable, fishing effort"*^ ( Et) . Another reason for considering just one input is that variable inputs commonly are combined in fixed proportions, e.g. capital and labour. Bjjzfrndal (1984) provides empirical evidence for this asser-tion. The concept of fishing effort i s discussed by Rothschild (1977). - 16 -In the empirical work to be undertaken, Et w i l l be defined as the number of boatdays. It i s assumed that the fi s h stock is managed by a sole owner or resource manager whose objective is to maximize the present value of net revenues from the fishery. Normalized net revenues can be restated as (2.11) 7rt = H ( Et; St, £t) - ctEt We assume that the manager of the fishery maximizes the present value of (2.11) subject to changes in the population level given by (2.6) St + 1 = (St - Ht) e5 ( St) + G(St_Y) and the f e a s i b i l i t y constraint (2.12) 0 < Et< Em a x This gives a discrete time, dynamic bioeconomic model with Et and St as control and state variables respectively. The upper limit on effort in a given time period (Em a x) w i l l be determined jointly by the size of the capital stock (the number of boats) and nature (season length). In this formulation we have exp l i c i t l y modelled stock changes over time, but not done so for changes in the K-vector. The reason for not considering the dynamics of the K-vector is that there is great overcapacity in the fleet and a lack of alternative employment opportunities (Bj0rndal, 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 in physical c a p i t a l — i . e . , the K-vector in the pre-sent notation—in addition to the investment problem in the resource i t s e l f . Clark, Clarke and Munro (1979) consider several variations of this problem. In a case of special interest to this thesis, i t is - 17 -assumed that boats have no alternative use and thereby a negligible opportunity cost, but depreciate over time. Therefore, gross investment in the fleet i s constrained to be non-negative. Assuming that the fish-ery previously has been open access and starts out at bionomic equilib-rium, the authors show that optimal management is different in the short and in the long run. In the short run, as capital i s redundant, only operating costs are relevant in the determination of the optimal stock l e v e l . In the long run, however, capital i s no longer redundant and a l l costs are thus relevant. Therefore, the optimal stock level in the long run is higher than that in the short run. In the present model, capital depreciation is not modelled as opposed to the Clark-Clarke-Munro model. If capital through deprecia-tion should become scarce in the long run, and the present model thus only considers the interim period, transversality conditions should have been specified. It i s acknowledged that this may represent a problem with the present model specification. However, the fishery to be con-sidered is a minor one that i s complementary to other fisheries ( c f . Section 3.2). Therefore, investment decisions are l i k e l y to be deter-mined mainly by the prospects in the major fisheries. This fleet 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, capital w i l l always be available and i t s dynamics may be dis-regarded. In other words, one can imagine the following circumstance: The social manager is faced with a cost/price ratio (ct) and a fleet with - 18 -some given attributes (the K-vector). He must then optimize the use of variable effort (Et) over time, taking stock dynamics into considera-tion. In the model, i t i s implicitly assumed that a l l boats are used in 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 in the fish-ery. If there are, for example, start-up or other fixed costs, i t i s conceivable that the manager under certain circumstances would let 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 variable problem w i l l be considered in future research^ \ Returning to the optimization problem, an intertemporal profit function can be derived by maximizing (2.11) with respect to Et subject to the stock dynamics (2.6). This would yield expressions for E* and TTT as functions of {ct, St, Kt> corresponding to those derived for the myopic case in Section 2.2. However, we w i l l here change focus slightly and derive expressions for the optimal stock level and the corresponding harvest. The method of Lagrange multipliers may be used to derive optimal equilibrium conditions. We define the Lagrangean expression L = E {dTTT(Et;St,tt) - qt[St + 1 - (St - Ht)e< S ( St) - G(St_Y)]} t=0 A model of this kind is given by Lewis and Schmalensee (1982). - 19 -where d = l/(l+r) is the discount factor, r the rate of discount and qt the discounted value of the shadow price of the resource. First order necessary conditions for 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 in equilibrium: (2.13) St + 1 - St = (St - Ht)e6*St) + G(St_y) - St = 0 In Appendix 1, the shadow price of the resource is shown to be: qt = dC The following implicit expression for the optimal stock level S* is derived : (2.14) e6^ * ^ — + 1] + 6'(S*)[S* - G(S*)] + dYG'(S*) = 1 + r Since this i s a steady-state stock l e v e l , the corresponding optimal harvest H* is found by rewriting (2.6): (2.15) H* = H(S*) = S* + G ( S*()S*)S* e Moreover, (2.16) H'(S) - 1 + 6'( S ) [ S'G ( S )6;s; G'( S ) " 1 e 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 ratio (ct/pt) is assumed constant and equal to c. - 20 -(2.17) H'(Sm s y) = 0 where Sm Sy i s the stock level corresponding to Maximum Sustainable Yield (MSY): (2.18) MSY = H(Sm s y) The implicit expression for the optimal stock level at f i r s t glance appears somewhat d i f f i c u l t to interpret. This is for two rea-sons. F i r s t , in this model effort i s control variable rather than har-vest, as is common in most models. Second, the model of population dynamics i s f a i r l y complex. These two factors can be used to elucidate the interpretation of the equation. Let harvest (H) be control variable so that the net revenue func-tion i s given by 7Tt = Tr(Ht;St,ft) Performing the dynamic optimization gives the following implicit expres-sion for the optimal stock level: (2.19) e C S h -] + 6'(S*)[S* - G(S*)] + dYG'(S*) = 1 + r Here, the term (TTS + ^R) / T Th is the standard expression for the Marginal Stock Effect in a discrete time nonlinear model (Clark, 1976). The d i f -ference between this term and the corresponding one in equation (2.14) is thus caused by the use of effort rather than harvest as control. From equation (2.6) the following derivative can be obtained: (2.20) 3St + 1/3St = 6'(St)[St - Ht]e, 5 ( St) + e6 ( St) = 6"(St)[St - G(St)] + e6 ( S t ) by equation (2.15) - 21 -This derivative must be understood in terms of stock productivity: the change in stock level in period t+1 caused by a marginal stock change in the previous period. Using this result, equation (2.14) can be restated: (2.14') ; ^j-S- + 3St + 1/3St = 1 + r - dYG'(S*) E This equation can be interpreted in terms of stock adjustment at the margin. The l e f t hand side represents marginal benefits. The term 8 ) C T TS/ TTg is 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 (cf. equation 2.6). In the model, harvesting occurs at 6(S*) the beginning of the time period; hence e represents what is fore-gone in net growth due to harvesting. Therefore, at the optimum, this factor i s applied multiplicatively to the Marginal Stock Effect. Together, the two terms represent the net benefits from a marginal stock adjustment. The terms on the right hand side represent the marginal (oppor-tunity) cost of stock adjustment. The stock adjustment w i l l cause a The Marginal Stock Effect may be explained with reference to the equilibrium condition for the optimal stock level (X*) in the Clark-Munro (1975) model: F'(X*) - c1(X*)F(X*)/[p - c(X*)] = r . F(X) is the natural production function and c(X) the unit cost of harvesting. At the optimum, the own rate of return on the resource—consisting of (i) the marginal physical product (F'(X*)) and ( i i ) c'(X*)F(X*)/[p - c(X*)], the MSE—is equal to the discount rate. The MSE represents the impact of stock density on harvesting costs. Intuitively, i t can be understood by considering that an increase in stock size w i l l increase catch per unit effort and hence reduce unit harvesting costs. The MSE i s anala-gous to the wealth effect in modern capital theory (Kurz, 1968). - 22 -marginal change in recruitment of G'(S*), which can be positive, zero or negative. Since this change takes place with a time lag of T periods, the discounted value amounts to dYG'(S*). If G'(S*) > 0, this term i s a net benefit (stock "appreciation") that is subtracted from the rate of discount, i.e., 1 + r - d^G'(S*) is the "net" cost of stock adjustment, corresponding to a net user cost in capital theory. On the other hand, i f G'(S*) < 0, - dYG'(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 is no discounting (r = 0), equation (2.14) can by use of (2.16) be written as (2.21) C T TS/ T Te + H'(S*) = 0 Assuming ciTg/TT^, > 0, this implies ° ' amsy Furthermore, i f harvesting i s costless (c = 0) or there is no stock effect in the production function (TTs = 0), (2.21) becomes (2.21') H'(S*) = H'(Sm s y) = 0 by equation (2.17). The various cases discussed here w i l l a l l be illustrated in the estimation of the optimal stock level (Chapter 4). Finally, the sole owner solution w i l l be contrasted to bionomic equilibrium, i.e., the free entry solution. In an open access fishery, eff o r t , fleet participation and stock size w i l l a l l be determined endogenously. So far, a net revenue function and stock dynamics have - 23 -been described. A complete analysis of the open access fishery would in addition require an equation describing capital 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 be estimated for given fleet size, which furthermore may be changed parametrically. This approach w i l l be used here. Then, for a given fleet size (K-vector), an open access fishery w i l l be characterized by the dissipation of rents: (2.22) TTT = H ( Et; St, £t) - ctEt = 0 An alternative statement of this condition is given by the equality between the average product and the real cost of effort: (2.22') ct = H(Et;St,Kt)/Et This condition can be used to solve for the amount of effort (Ero) in the open access fishery: (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 implicit expression for SOT, the bionomic equilibrium: (2.24) [S^ - H(c,S0 0,K)]eS ( S o o ) + G(S,J - S«> = 0 It i s commonly asserted that an open access fishery is characterized by an i n f i n i t e discount rate, because fishermen w i l l not be concerned with the effect 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 effort and stock size in the open access fishery. An inspection of equation (2.24) shows that the open access stock le v e l , in addition to the biological factors, depends on the cost/price ratio with increasing in c. The differences between the sole owner and open access solutions are quite striking. While there are discounting and marginal adjustments in the former, this i s not so in the la t t e r . 2.3.2 The Marginal Stock Effect The Marginal Stock Effect (MSE) is defined as (2.25) MSE = cir /TT HI Costs, output price and the production technology influence the optimal stock level through the MSE, as an inspection of the implicit expression for S* (equation 2.14) w i l l reveal. Therefore, the MSE plays an impor-tant role in 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 results about the relationship between the production technology and the optimal stock level may be obtained. Moreover, the effects of changes in vari-ous parameter values on the optimal stock level may be predicted. In this way the present analysis may illuminate the empirical results that w i l l be forthcoming (Chapter 4). - 25 -The analysis w i l l be performed in 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 profit function, *s > 0 However, can be both negative and positive. For harvesting to occur, i t i s a necessary and sufficient condition that T T£ > 0 -* MSE >_ 0 The MSE can also be defined as (2.26) MSE = m(c,a,po,S,K) where K = 71 F i r s t , i t can be ascertained that the optimal stock level is increasing in the MSE. Second, note that from equation (2.25) the MSE is increasing in the cost/price ratio and i t vanishes in the zero cost case. The following results can also be derived: 1. MSE i s decreasing in a provided c > 0 and ir ^ 0 with lim MSE = 0. 1 1 a-'4-" oo 2. MSE i s decreasing in K provided c > 0 and TT ^ 0 with lim MSE = 0. h K> » The intuition behind the f i r s t result is that profits are increasing in the output el a s t i c i t y of effort. For a sufficiently high ot-value, costs become negligible and the MSE vanishes. A consequence of this result i s that productivity improvements over time, reflected by increases in the value of a, would imply a diminishing importance of costs. - 26 -The implication of the second result i s that for a sufficiently large f l e e t , the resource should be managed as i f there are no costs. The profit function i s increasing in K. Also, the optimal level of effort i s decreasing in K. Thus, costs w i l l also be decreasing in K and become negligible for a sufficiently large K. This result may be visualized by considering the search effect. The area that can be searched and/or the intensity of the search w i l l increase with the number of boats in the f i s h e r y * ^ . With modern communication equipment, information about locations of fish can be transferred to the rest of the f l e e t . Thus, the amount of variable effort w i l l be reduced. Another phenomenon can be cited for boat size, the other fixed factor that w i l l be considered in the empirical analysis. It can be noted that the larger the size of the boat, assuming locations of fish to be known, the fewer the number of trips (days) that are needed to take a certain catch. The relationship between MSE and stock size w i l l not be the subject of special analysis. 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. Finally, we turn to the stock output e l a s t i c i t y , which may be the most intriguing case. This i s done by considering the range of values this el a s t i c i t y can take on. It should be recalled that there are no start-up costs. - 27 -1. po = 0 -> TTs = MSE = 0 In this case, the fi s h stock i s not an argument in the production function and harvesting costs are independent of stock s i z e ^ ^ . If the fishery is profitable, i t is established by equation (2.21') that r = o -*• S* = Sm s y, H* = MSY. Moreover, Clark (1982) has shown that r oo + s* = 0, H* = 0. Accordingly, for 0 < r < °°, 0 < s* < sm s y and 0 < H* < MSY. This result may be modified somewhat, because S* may go to zero for f i n i t e discount rates. On the other hand, i f the fishery i s unprofitable, the stock is l e f t unexploited at i t s carrying capacity. For positive po values, two cases may emerge. In the f i r s t one, assuming TT > 0, MSE w i l l f i r s t be increasing and then decreasing in po. This is illustrated in Figure 2.2, with corre-sponding stock levels given in Figure 2.3. Such a relationship could emerge, for example, in the model of Clark and Munro (1975). This case is analyzed by Levhari, Michener and Mirman (1981). - 28 -However, a second case may also occur, which is when MSE and thus S* are decreasing functions of pQ. This result is obtained when the fishery is unprofitable at a zero stock effect, i.e., TTg < 0 given po = 0; see Figures 2.4 and 2.5. The occurrence of this circum-stance depends in a nontrivial manner on the relationship between various parameters of the model. The consequence is that profits are increasing in the stock output elasticity. An increase in po may then Increase profits, which may reduce the optimal stock level and thereby increase the harvest. Figure 2.4: MSE as a function of po. Figure 2.5: S* as a function of pQ. - 29 -In both cases, as po tends to i n f i n i t y , the MSE goes to zero. The intuition behind this result i s the following: As the output el a s t i c i t y of the stock increases, the "contribution" of the stock to i t s own harvest increases. In a sense, the productivity of the stock improves. Therefore, as po increases, less variable effort w i l l be required to obtain a certain harvest quantity. In the l i m i t , as the stock output el a s t i c i t y increases to i n f i n i t y , the amount of effort and hence costs become negligible. Mathematically, the result i s obtained by letting the stock output elast i c i t y go to i n f i n i t y . In practice, po w i l l be in the range zero to one. Thus, the result may be obtained for relatively 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 different fisheries while on the other hand being ideally suited for the fishery to be analyzed in this thesis. We w i l l now briefly address this matter. One of the main characteristics of the model of population dynamics i s the time lag in the stock recruitment function. As such a time lag is 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 is assumed constant, although this may easily be rectified by introducing density dependence. - 30 -This model is 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 is not unrealistic. North Sea herring i s an obvious example in this respect. The model may be less appropriate where the delay between spawn-ing and recruitment i s f a i r l y long. Such species may c a l l for optimiza-tion of two biologically interdependent fisheries, 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 profit function that has been developed is quite general. The presence of fixed factors other than stock size in the production function probably represents reality for many developed fisheries. This is presumably true even in the somewhat long run, due to the lack of alternative employment opportunities for capital. The model formulation is seen to be analogous to a case analyzed by Clark, Clarke and Munro (1979). Where appropriate, the production process may be redefined to be a function of variable inputs and stock size only. Another possibility is to introduce a start-up cost in the profit function. We believe that the bioeconomic model presented in this paper is well suited for an analysis of the management of North Sea herring. In addition, i t is appropriate for the study of a number of other species. The model's general applicability may be further enhanced by suitable refinements of i t s various parts. - 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 is 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 in the English Channel. However, the three stocks mix on the feeding grounds in the central and northern North Sea, rendering i t impossible to distinguish between catches from the three stocks. It is therefore customary to treat the three stocks as one unit, as indeed i s the approach in 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. The herring gen-erally lives until the age of ten. On the biology of North Sea herring, see Saville 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 stock-recruitment 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 fish of a given age become vulner-able at a particular time in a given year, and their vulnerability remains constant throughout their lives; 2. platoon recruitment, where the vulnerability of a yearclass increases gradually, but during any year each individual fish i s either f u l l y catchable or not catchable; and 3. continuous recruitment, where there i s a gradual increase in vulner-a 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 in analyt-i c a l models. The model we have formulated does assume knife-edge recruitment. It may also be noted that available data in many instances presupposes this type of model, as indeed i s the situation for North Sea herring. Ricker (1975, p.281) discusses some properties which are desir-able in a curve of recruits against parent stock: 1. It should pass through the origin, so that when there i s no adult stock there is no reproduction. - 33 -2. It should not f a l l to the abscissa at higher stock levels, so that there is no point at which reproduction is completely eliminated at high densities. 3. The rate of recruitment (recruits/parents) should decrease continuously with increases in parental stock. 4. Recruitment must exceed parental stock over some range of parental stock values (when measured in equivalent units). The following alternative functional forms w i l l be used to estimate the relationship between recruits in period t+1 (Rt +i) and spawning biomass in period t (St): 1. Rt+1 = S tea ( 1-St/ b> 2. Rt+i = aSt/(b + St) 3. Rt+i = as|? 4. Rt+i = aSt - bsj 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 specification is a quadratic function. The Ricker function is the only one to satisfy the four properties listed above. For the Ricker function, i . R = S for S = b and 3R a  l x- ^ S = 0 = 6 For the Beverton-Holt function, i . lim R = a - 34 -Moreover, the quadratic function is only well behaved for 0 _< S _< a/b. It should be noted that while recruitment here i s dealt with in deterministic manner, i t is in practice often a highly stochastic pro-cess. Ricker notes that "year-to-year differences in environmental characteristics cause fluctuations in reproduction at least as great as those associated with variation in 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 recruits and parent stock size from observed data. Appendix 2 gives the data on which these regressions are based: annual spawning biomass and number of recruits for the period 1947-81. The data, estimated by virtual population analysis, are given in research reports published by the International Council for the Exploration of the Sea. The spawning biomass consists of agegroups two and older. Spawning takes place in September each year. New recruits enter the juvenile stock one year later as zero-group herring. Figure 3.1 i s a scatter diagram relating the number of recruits-zero-group herring—in year t+1 (Rt +j) to the spawning biomass in the previous year ( St) . We note the high number of recruits in the years 1956, 1960 and 1981, which i s presumably due to exceptionally favourabl environmental conditions. This i s a phenomenon that is known to occur for clupeids at certain intervals. Ulltang (1980) suggests that the stock-recruitment relation exhibits depensation at low stock levels. This means that the rate of - 35 -recruitment (recruits/parents)—or net growth, in a growth model—is increasing over a range of stock d e n s i t i e s ^ . This is 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 effective at higher stock levels, which means that the relative mortality rate can be decreasing in stock size. This phenomenon may give rise to a depensatory stock-recruitment curve (Clark, 1976, ch. 7). If there is depensation, this w i l l influence the speed of recov-ery i f actual stock level i s in 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 herring, however, the present stock level i s probably outside the range where there may be depensation. Thus, we need not be concerned about this phenomenon in our analysis. The estimated regressions for the four stock-recruitment func-tions postulated above are given in Table 3.1. The relations between Rt+1 and St are drawn in Figure 3.1. Table 3.2 gives maximum recruit-ment for the Rt+i = g(St) regressions. Mathematically, depensation corresponds to an inflection point: the second derivative is equal to zero, while the f i r s t derivative chan-ges sign from positive to negative. 2 ) An example is probably given by Norwegian spring-spawning her-ring. - 36 -i Figure 3.1: The Stock Recruitment Relationship - 37 -Table 3.1: Estimated Stock-Recruitment Functions Functional form Regression R2 F2 ) DW Ricker ln(R/S) = 2.87** - 0.81** x 10~3 S (16.68) (-6.90) 0.59 47.62 1.64 Beverton-Holt (1/R) = 0.32 x 10-4 + 0.11** x (1/S) (0.43) (5.17) 0.65 26.77 — Cushing InR = 5.89** + 0.40**lnS (5.99) (2.75) 0.36 7.54 -3) Quadratic R = 12.74**S - 0.004**S2 (6.53) (-4.57) 0.12 - 1.78 1) Time series: 1947-81 (n = 35). R is measured in millions, while S is measured in 1,000 tonnes. The Beverton-Holt and Cushing models have been estimated with f i r s t order autocorrelation, using an iterative Cochrane-Orcutt procedure. In the ordinary least squares regressions of these two models, the Durbin Watson statistics were 1.25 and 1.32 respectively, t statistics in parentheses. ** denotes significant at 95% l e v e l . 2) To estimate the F s t a t i s t i c , the f i r s t observation had to be excluded in the Beverton-Holt and Cushing functions. 3) For the quadratic function, the regression i s forced through the origin. Therefore, the F test i s invalid and the R-square may be incorrect. The function i s well defined for 0 < S < 3.07 million tonnes. Table 3.2: Maximum Recruitment Functional form S 1,000 tonnes R millions Ricker: R = Se2'8 7 ( 1"S / 3'5 5 1> 1,240 8,040 31 450 S Beverton-Holt: R = 0 ?0 <_——— 3,480 + S OO 31,450 Cushing: R = 361.6e^*^ ^  CO CO Quadratic: R = 12.74 S - 0.004 S2 1,530 9,800 - 38 -The Ricker and Beverton-Holt functions explain 59% and 65% respectively of the total variance of the dependent variable. Taking the time span, with possible changes in 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 in 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 predictions. The Cushing function has been estimated for a number of Atlantic and Pacific herrings with point estimates of b in the range 0.2 to 0.7 (Cushing, 1971). The present estimate is also in this range. For this function, recruitment i s continously increasing in stock size. The above regressions relate zero-group herring in year t+1 to the previous year's spawning biomass. However, not a l l recruits survive to agegroup two. The survival rate of equation (2.4) is defined as follows: X = e^V-Mi) where MQ and i s the natural mortality rate of yearclass zero and one 3) respectively. Estimates of M Q and are 0.4 and 0.3 . The mean These estimates include fishing mortality for juvenile herring. M Q and are the sums of average fishing mortality for the period 1976-80 (Anon., 1982) and natural mortality, estimated to be 0.10 (Saville and Bailey, 1980). Fishing mortality i s mainly due to bycatches of juvenile herring. If these could be reduced, the yield from the spawning stock would increase. - 39 -weight of agegroup two herring is 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') G(St_2) = g(St_2)e-( M0+ Ml>w where w is the mean weight of new recruits. If the Ricker function i s used, this becomes: G(St_2) = St_2e2'8 7 ( 1-St-2/3,551)e-0.70 > 1 2 6 = 1 . 1 0 St_2e - ° -8 1 x 1 0"3 st - 2 where G(St_2) is measured in 1,000 tonnes. The stock-recruitment functions for North Sea herring have been estimated on the basis of data for 1947-81. This period is probably not long enough for any major long-run changes in the aquatic environment to 4) have taken place . Moreover, data that can be used to test the effects of possible changes in the aquatic environment on recruitment are not available. 3.1.2 The Net Growth Function The basic postulate about the net natural growth rate, 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 func-tional forms for the relationship between <5 and S are specified: Posthuma (1971) shows that temperature conditions on the spawn-ing grounds during the incubation time seem to affect yearclass strength, possibly through differential egg mortality at different tem-peratures . - 40 -i : S(S) = a + bS i i : 6(S) = a + bS + cS2 The results of the regressions are given in Table 3.3. Table 3.3: Estimation of Net Natural Growth Functions Functional form R2 F DW 6(S) = 0.15** - 0.43** x 10"4 S (7.96) (-3.63) 0.37 20.31 1.44 6(S) = 0.11** + 0.46 x 10"4 S - 0.27** x 10"7 S2 (4.52) (1.25) (-2.51) 0.47 14.87 1.82 1) Time series: 1947-82 (n = 36). Stock size as of January 1 has been used. The Durbin Watson st a t i s t i c for the f i r s t regression is in the indeterminate range, but autocorrelation has not been corrected for. The estimate of the f i r s t order rho i s only significant at the 90% confidence l e v e l . In addition, correcting for autocorrelation causes only marginal changes in parameter estimates, t statistics in paren-theses. ** denotes significant at 95% l e v e l . Table 3.4: Characteristics of Function S^e Functional form Maximum net growth Level of zero growth Million tonnes Stock level Million tonnes Net growth Million tonnes Stea"b St 1.71 0.14 3.49 S t ; ea+bSt-cS2 1.84 0.20 3.05 The linear regression of 6 on S explains 37% of the total vari-ance of the dependent variable. Moreover, 6'(S) < 0 for a l l stock levels. The quadratic regression on S explains 47% of the total - 41 -variance of the dependent variable. The degree of explanation is rela-tively good, considering the time-span and effects of possible changes in the aquatic environment. Furthermore, the net natural growth rate, while positive at low stock levels, is continuously decreasing with increases in the biomass le v e l . In the absence of recruitment and harvesting, changes in the stock level are given by (cf. equation 2.2): st+l - ste This function i s graphed in Figure 3.2 for the two cases under consider-ation, and some of i t s characteristics are given in Table 3.4. There are two main differences between the two functions. F i r s t , although maximum net growth occurs at roughly the same stock levels, net growth i s considerably higher in the quadratic model. The second dif-ference is that net growth remains positive until a considerably higher stock level in the linear than in the quadratic model. These results w i l l have consequences for the model of population dynamics. For the quadratic net growth function, an interesting feature of the combined function is that i t exhibits depensation at stock levels lower than 0.58 million tonnes. This effect i s not present when the linear net growth function is used"^. Thus, there i s a qualitative 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 positive for this function; hence there can be no depensation. - 42 -Ste fi(S) (i) Stea"b St Million tonnes (ii) Ste t t Figure 3.2: The Net Growth Function - 43 -3.1.3 The Complete Model In the absence of fishing, the fish stock w i l l develop according to the delay difference equation St + 1 = S t - e6^ ^ + G(St_2) Various functional forms both for the stock recruitment and the net growth function have been estimated. The different alternatives w i l l now be combined in the model of population dynamics. Two points are of special interest: 1. S, the carrying capacity of the stock; and 2* Sm Sy, the stock level corresponding to Maximum Sustainable Yield (MSY). Sm gy i s found by solving equation (2.17) and MSY is subsequently given by (2.18). Estimates of the carrying capacity and Sm Sy are given in Tables 3.5 and 3.6 respectively. 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 Sm Sy 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. Million tonnes. Stock-recruitment function Net growth 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 Sm<s„ and MSY. Million tonnes. Stock-recruitment function Net growth function Linear Quadratic °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 is due to the one insignificant parameter for this function, which gave a maximum recruitment that was unrealistically high. Similarly, recruitment is continuously increasing in stock size for the Cushing function, which accounts for the high car-rying capacities for this case. - 45 -The carrying capacity i s higher for the linear than for the qua-dratic net growth function, given stock-recruitment function. The rea-son i s that net growth i s positive up to a higher stock level for the linear than for the quadratic net growth function (cf. Figure 3.2). At the carrying capacity, net growth i s in a l l instances nonposi-tive, while recruitment i s nonnegative. The stock of North Sea herring may in recent times have been closest to i t s carrying capacity in the early post World War II years, due to low fishing pressure during the war. Stock estimates for this period range between three and four million tonnes. According to Table 3.5, this indicates that the Ricker and the quadratic stock recruitment functions may yield the more r e a l i s t i c results. The results in Table 3.6 show that Sm s y i s somewhat sensitive to choice of growth function. However, this i s less so for the MSY. In addition, the MSY is relatively stable across stock-recruitment func-tions, except for the Beverton-Holt case. The model of population dynamics that w i l l be used in the estima-tions of the optimal stock level (Chapter 4) w i l l be the combination of the Ricker stock-recruitment function and the quadratic net growth func-tion. There are a number of reasons for that. F i r s t , the Ricker func-tion i s preferred on a priori grounds to i t s alternatives^. 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 phe-nomenon w i l l not be of practical importance in the future. - 46 -or better than their alternatives. Third, the predictions of this model (Tables 3.5 and 3.6) appear to be quite reasonable. The preferred model of population dynamics is illustrated in Figure 3.3. To fa c i l i t a t e graphical representation of a function involving a time lag, a steady state stock level is assumed. The harvest quantity is represented by the difference between St+i and the 45° 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 qualitatively affected by the choice of a particular model of population dynamics. There could be some quantitative differences, however, depending on the combination of stock recruitment and net growth func-tion in question. Moreover, these differences would be greater for stock levels than for harvest quantities. 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 fishing power. The purpose of this exercise i s to derive a conceptual production function. This w i l l be formalized, and the corresponding profit function w i l l be derived. Finally, 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 in the central and northern North Sea. There has traditionally been a prespawning fishery from May to July, and a spawning fishery in August and September7^. Herring is used for reduction into fish meal and fish o i l and for human consumption. The former has traditionally been the more important use in 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 deteriorating. This problem can be aggravated because of warm weather, as the fishery takes place in summer. This circumstance could cause short trips with relatively low catches, so that hold capacity i s rarely f u l l y u t i l i z e d . A very important behavioural characteristic of North Sea herring, as of other clupeids, i s the schooling behaviour. Schooling takes place both to make the search for food more effective and to reduce the effectiveness of predators (Partridge, 1982). It is also important to note that schooling fish 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 effective means of harvesting, especially the purse seine. With modern fishfinding equipment, this means that harvesting can be 'in 1968, 1971 and 1975, respectively 78%, 77% and 62% of Norwegian catches were taken in the months June to August (Bjorndal, 1984). - 49 -profitable even at low stock levels. Murphy notes that the effective harvesting 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 vi r t u a l l y until the end; thus there are no economic brakes on exploitation until collapse" (Murphy, 1977, p. 285). For these reasons, the stock output elasticity is expected to be 8 ) low; in the extreme, i t may even be equal to zero . Moreover, i t has been asserted that a low stock output elas t i c i t y implies that costs have a negligible effect on the optimal stock le v e l . Both these assertions w i l l be tested empirically. If there i s no stock effect in the production function, extinc-tion of the stock i s "optimal" in the sole owner case for a high finite or an in f i n i t e discount rate (cf. Section 2.3). Although this is a spe-c i a l case for the sole owner, i t is asserted that an open access fishery i s characterized by an i n f i n i t e discount rate. 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. If the stock effect is positive but low, open access may s t i l l cause severe stock depletion. These are the reasons why clupeids pose some special management problems. As a real 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 fleet w i l l be used. The purse seiners in question are of For this reason, the Schaefer production function is especially inappropriate. - 50 -120-180 foot length, with a crew size of 9 to 12. The boats are equipped with powerblock, modern fishfinding and communications equip-ment. The latter facilitates the sharing of information about locations of herring, which commonly takes place. The fishery, u t i l i z i n g this technology, started in 1963. In the middle of the 1970s, however, the stock was nearly depleted and the fishery was prohibited at the end of 1977. Severe regulations have been in effect ever since so as to allow the stock to recover (cf. Table A l , Appendix 2). The following factors should be considered for inclusion in the production function: 1. E^: The number of boat-days. 2. St: Stock size. 3. K it: The number of boats in the fishery. This factor is important because we are dealing with a search fishery, where information about locations of fish tends to be shared. This phenomenon may give rise to an external bene-f i t . On the other hand, too many boats may cause overcrowding on the fishing grounds, a negative externality. 4. K2t: Size of boats (measured in gross registered tonnes). This variable may be of importance because catch per trip may depend on boat size. 5. K3t: Fish-finding equipment such as echo-sounder and sonar. This type of equipment contributes to the quality of search effort and may thus be an important variable in the production function. However, already in 1963 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. It has accordingly been omitted from the analysis. 6. K4t: Gear type. This variable would be important in many analyses but is redundant in the present one, because there is only one gear type. In this formulation, one variable input has been specified. One can easily visualize others, e.g. fuel expenditures. 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 like 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 fixed. In the period under consideration, the fishery was characterized by open access, and myopic profit-maximizing behaviour is 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 in 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 in the cross-sectional data sets permits some f l e x i b i l i t y in choice of functional forms, the limited number of observations in the time series calls for a functional form that i s parsimonious in 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 in logarithmic form i s : (3.1) lnHt = InA + -'.ja^nE^ + polnSt + |pjlnKjt Assuming profit-maximizing and price-taking behaviour, the dual profit function can be derived (Lau and Yotopoulos, 1972): (3.2) lnir* = InA* + Za*lncl t + polnSt + .Zp*lnKjt where c^t is the normalized price of input i (cf. Chapter 2). The harvest supply function i s given by (3.3) lnHt = lnAQ + Ea*lnci t + p*,lnSt + zp*lnKjt Since the production and profit functions are self-dual, the parameters of one can be derived exactly from the parameters of the other. The restrictions on the parameters are: A* = f(A,a) AQ = g(A,a) (3.4) u = Za± < 1 i ai = ~ ai / d ~ v ) , i = l,...,n 9) The complex model of population dynamics that was introduced in Section 2.1 may be d i f f i c u l t to combine with a flexible functional form for the profit function. This would represent another independent reason for choosing the Cobb-Douglas function. - 53 -f j = pj/(l - y ) , j = 0,1,...,m It can be shown (Lau and Yotopoulos, 1972) that constant returns in 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 series. Cross-Sectional Results Cross-sectional data for the Norwegian purse seine fleet 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 fishery i s available. The data set includes information per boat on harvest quantity, variable inputs and fixed f a c t o r s ^ \ With this type of data, direct estimation of the production function (equation 3.1) is feasible. Table 3.7 gives a summary of the cross-sectional data. Over the years in question, there has been a decrease in mean harvest per boat and mean number of trips per boat. Average catch per trip increased by more than 50% from 1968 to 1971, but was then more than halved from 1971 to 1975. The 1975 level was therefore only 24% lower than in 1968. There has also been a substantial increase in mean tonnage, especially from 1971 to 1975. ^^Data on harvest quantities and variable inputs were made available by Noregs Sildesalslag (the Norwegian Herring Fishermen's Cooperative). Data on vessel attributes (fixed factors) were supplied by the Directorate of Fisheries, Norway. - 54 -Table 3.7: Summary of Cross-Sectional Data Boats in sample Aggregate harvest Tonnes Harvest per boat Trips per boat Year Mean St .dev. Mean St.dev. 1968 44 44,350 1,008 726 9.86 4.45 1971 23 21,345 930 874 5.87 3.32 1975 102 36,199 355 338 4.70 3.59 Catch per trip 1 ^  Days per trip Gross register tonnes Tonnes Year Mean St.dev. Mean St.dev. Mean St.dev. 1968 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) Based on sample sizes of 41 and 21 respectively. Figures for 1975 are not yet available. An important change in institutional setting also took place in the period under investigation: While the fishery was characterized by open access in a l l three years, tr i p quotas were introduced in 1975. This regulation stated that catch per trip could be a maximum of 100 tonnes plus 40% of the boat's cargo capacity*^ up to 380 tonnes. The effect this may have had on the production function w i l l be investigated. ^The smallest size of the purse seiners in this fishery i s more than 200 tonnes, giving trip quotas of at least 180 tonnes. - 55 -When i t comes to variable inputs per boat, data are available on 1. the number of tri p s , 2. the number of boat-days, and 3. estimated fuel expenditure (1968 and 1971 only). In the regressions to be performed, boat-days wil l be selected as the variable input. The same variable w i l l be used in the time series e s t i -mations. Bj0rndal (1984) estimates production functions with the number of trips and fuel expenditure as variable inputs (1968 and 1971 data). The results show that the output el a s t i c i t y of fuel expenditure i s insignificant 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 fuel 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 steel), 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 pro-duction function: K2 = tonnage (gross register tonnes) K 3 = engine horsepower K 4 = year the boat was built The choice of factors needs some amplification. Tonnage is 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 third factor, year b u i l t , introduces an element of vintage capital as i t serves to distinguish 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 is a product of these three variables. Mate-12) r i a l type w i l l also hardly make any independent contribution to har-vest. In addition, i t is assumed that a l l relevant information about the engine is 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 in the fishery. The reason i s the cross-sectional nature of the estima-tions, which implies that stock size and fleet participation w i l l be 13) constant and equal for a l l boats. Therefore, these two fixed factors can be suppressed in the functional form. The following Cobb-Douglas production function is thus esta-blished*^ : a P2 P3 P4 £ h = AET$ K 3 KVe where h = production (harvest) per boat in a given year and i t i s assumed that e ~ N(0,a2). One would on a priori grounds expect the regulations in 1975—if binding—to affect the parameter estimate of tonnage (p2)> since this i s 12) In 1968, 1971 and 1975 there were fi v e , two and two wooden^ boats respectively. 13) In r e a l i t y , stock size w i l l decline over the year, and fleet participation may vary over the season. Below, results from estimations of whole year and main season fisheries w i l l be presented. The assump-tions of constant St and K jt are presumably more r e a l i s t i c in the latter than in the former case. 14) Subscripts for boat number and year have been suppressed for a l l variables. - 57 -the variable that represents hold capacity. However, Table 3.7 shows that mean harvest per trip in 1975 was considerably lower than the smal-lest trip quotas (cf. footnote 11). This makes the effectiveness of the regulations somewhat questionable. Furthermore, one would a priori expect p2 to be less than one. 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 trips with low catches. The results of the regressions are given in Table 3.8. A hypo-thesis that a l l parameter values in 1968 and 1971 are equal is not rejected according to a Chow test. Therefore, the data for the two years have also been pooled. The result indicates that although stock size and fleet participation changed considerably from 1968 to 1971 (cf. Tables Al and A2), the parameters of the production function remained unchanged. The hypothesis that a l l parameter values in the 1971 and 1975 regressions are equal i s , however, rejected. The output elas t i c i t y for the number of boat-days is s i g n i f i -cantly greater than one in a l l regressions, i.e., there are increasing returns to the variable input. This somewhat surprising result may be explained in the following manner: The herring is not randomly d i s t r i -buted in the ocean. At the beginning of the fishing season, time w i l l be spent searching for herring. Once a catch has been made and deliv-ered, the boat may return to the location of the f i r s t catch in the hope that the herring is s t i l l there. The data appear to confirm that aver-age catch on the f i r s t trip i s lower than average catch per trip for the whole season. In addition, average number of days on the f i r s t t r i p appears to be higher than average days per trip for the season. - 58 -Table 3.8: Estimated Production Function (Cobb-Douglas) for North Sea Herring. Cross-Sectional Data*^  Year ln A a P 2 P3 P4 r2 F n2 ) RSS 1968 -8.98 (-0.44) 1.31** (9.39) 0.72** (1.70) -0.20 (-0.52) 0.01 (0.47) 0.73 24.86 41 18.24 1971 81.22* (1.46) 1.42** (6.74) 0.45 (0.83) 1.08* (1.53) -0.04* (-1.47) 0.78 14.22 21 6.25 1968 & 1971 18.34 (0.05) 1.25** (11.66) 0.57** (1.74) 0.09 (0.30) -0.0002 (-0.02) 0.73 38.00 62 26.76 1975 36.22* (-1.44) 1.16** (12.06) 0.33 (1.24) -0.23 (1.01) 0.02* (1.53) 0.71 41.78 73 17.74 1) t statistics are given in parentheses. * denotes significant at 90% l e v e l . ** denotes significant at 95% lev e l . 2) 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 Only^  Year In A a P 2 P3 P* r2 F n 1968 -18.94 (-0.87) 1.09** (5.68) 0.42 (0.84) -0.19 (-0.45) 0.01 (0.99) 0.60 11.39 36 1971 35.04 (0.66) 1.33** (8.16) 0.95** (1.83) 0.61 (0.96) -0.02 (-0.73) 0.88 19.34 16 1975 -35.20 (-1.29) 0.93** (8.36) 0.12 (0.41) -0.04 (-0.15) 0.02* (1.40) 0.60 24.31 69 1) t statis t i c s are given in parentheses. * denotes significant at 90% l e v e l . ** denotes significant at 95% level. - 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 location. One can therefore imagine a fishery with a main season and pre-and post-season fisheries, as indeed is 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 rise 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 significantly different from one for 1968 and 1975. Although the coefficient is s t i l l significantly larger than one for 1971, the point estimate is lower than in Table 3.7. These results appear to confirm the hypothesis about the interactions between pre-, main and post-season fisheries. For the whole year fisheries (Table 3.8), a l l point estimates of the output el a s t i c i t y of tonnage ( p2) are less than one. For the com-bined 1968 and 1971 regression, i t is signficantly less than one. The low point estimate in the 1975 regression could be due to the regula-tions that were in effect during that year. However, since mean harvest per trip i s considerably less than the smallest trip quotas, this expla-nation does not appear to be very l i k e l y . - 60 -An important change that took place between 1971 and 1975 was in the relative shares of the catch going to reduction and human consump-tion. While about 10% of total catch was delivered for human consump-tion in 1971, the relative share increased to almost 90% in 1975*"*^. Quality considerations when delivering for human consumption typically cause short trips with relatively low catches*^. It appears l i k e l y that this structural change, rather than the system of tr i p quotas, i s the reason for the low point estimate of p2 for 1975. The other two parameters—p3 and p4—are in some instances of the wrong sign and/or insignificant. This phenomenon may be attributed to the high degree of correlation among the fixed factors. However, i t could also be that these two factors in one or more years were rela-tively unimportant. The main results from this section*7^ 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 fishery. 2. It is postulated that the interactions between pre-, main and post-season fisheries can give rise to increasing returns for the year as *"^Source: Annual reports for 1971 and 1975, the Norwegian Herring Fishermen's Cooperative. 16) If the delivery is intended for human consumption, the boat w i l l usually have to leave the fishing grounds within 12 hours of making the f i r s t catch in order to ensure high quality. ^^Bj^rndal (1984) contains further estimations of production functions based on these cross-sectional data. This includes estimation of a translog function. - 61 -a whole, but constant or decreasing returns during the main season. The empirical evidence appears to confirm this. 3. A hypothesis that a l l parameter estimates in the production function are equal was accepted for the 1968 and 1971 regressions, but rejected for the 1971 and 1975 regressions. 4. There appear to be decreasing returns to fixed factors, as was expected. 5. The effectiveness of trip quotas that were introduced in 1975 i s found to be questionable. Time Series Results contain information about fixed factors, including stock size, prices and harvest quantity. With these data, a harvest supply function can be estimated. This is done for the following alternative specifications of the production function: The aggregate time series data for the North Sea herring fishery A l . H t A2. B l . A^sjSf^e B2. where D.. = { 1 for 1963 and 1974-77 0 otherwise and by assumption gt ~ N(0,o^). - 62 -There are two main models. Model A assumes that both the number 18) of boats and boat size matter , while model B only includes the number of boats. In the latter formulation, i t is implicitly assumed that catch per trip i s unaffected by boat size. 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 (St) and fleet participation (K^t) are included as arguments, because they do not remain constant over time. However, boat-days (Et) and boat size (K.2t) are arguments in both the cross-sectional and the time series functions, although the distinction between boat-days at the micro and at the aggregate level should be kept in mind. Both models are specified with and without a dummy variable. The dummy has been introduced because the trip quotas that were in effect during the year 1974-77, might affect our profit-maximizing hypothe-19) sis . In addition, the dummy has been given a value of one for 1963. Since this i s the year the fishery started, the fishermen may for a num-ber 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, specifications with the dummy variable are a priori preferred to those without. In the regressions, i s represented by average boat size per year. 19) In the last few years of the data period, a relatively 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 variable. - 63 -Ideally, the cost of effort should be represented by the opportu-nity cost of this factor of production. It can be argued that the deci-sion to participate in the North Sea herring fishery is a marginal one. Norwegian purse seiners participate in up to five different seasonal fisheries (Bj^rndal, 1981). Of these, the capelin fishery is the major one in terms of quantity and revenues. Only the mackerel fishery, starting in the f i r s t half of July, overlaps to a significant extent with this fishery. Therefore, the North Sea herring fishery may be con-sidered a minor one compared to the capelin fishery or to the other fisheries combined. Furthermore, i t may be viewed as complementary to the other fisheries. Thus investment decisions w i l l mainly depend on the prospects in the major fisheries and be made with an eye to versa-t i l i t y . However, no special equipment w i l l be required for this fish-ery. Once the investment decisions have been made, the situation that we primarily are concerned with, a decision to participate in this fish-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, fuel and materials. While i t i s f a i r to assume that market prices represent opportunity costs for fuel and materials, this is not necessarily so for labour. A particular phe-nomenon of this industry is that the returns to labour vary from year to year due to yearly fluctuations in prices and stock size. Therefore, an If applicable, this should include opportunity cost from the alternative fishery. - 64 -opportunity cost in the form of the wage in manufacturing industry i s used for labour. An index for the cost of effort i s constructed in Appendix 2, which also gives a further analysis of costs. Two types of externalities are associated with open access fi s h -eries. Static externalities 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 static externalities can be positive or negative. On the positive side, the search effect i s l i k e l y to be important in a school-ing fishery. Examples of negative externalities are gear collisions and overcrowding on the fishing grounds. For this fishery in particular, i t should be recalled that i t is an ocean fishery. Overcrowding i s there-fore less l i k e l y than e.g. in a coastal fishery. Moreover, only one type of gear is used, which makes gear collisions less l i k e l y than in some other fisheries. For the time period in question, i t is believed that there might have been some negative interactions between the boats 21) in the latter half of the 1960s, but otherwise not . However, a posi-tive externality through the search effect i s believed to have been pre-sent throughout the data period. The production technology i s estimated on the assumption that the static externality cannot be controlled for. The stock externality, however, can be controlled in the case of a sole resource manager (Chapter 4). Source: Olav Dragesund, professor of fisheries biology, University of Bergen (private communication). - 65 -It i s arguable whether the number of participating boats (K^t) should be considered an endogenous or an exogenous variable. As dis-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 alternative. Hence, this question cannot be resolved purely by reasoning. Therefore, the production function w i l l be es t i -mated with K^t as alternatively exogenous and endogenous variable. The regression results, with K^t exogenous, are given in Table 3.10. The parameters of the production function, derived from the supply function, are given in Table 3.11. The method for estimating the 22) variances of the derived parameters is given by Kmenta (1971) From the supply function, the following results can be derived: 1. The hypothesis of increasing returns in a l l factors—variable and f i x e d — i s accepted in a l l models. The imputed degree of homogeneity is given in Table 3.8. 2. The supply e l a s t i c i t y of effor t , a*, i s insignificant in model Al and barely significant in model B l . The coefficent i s affected by the dummy variable. Point estimates are of roughly the same magni-tude in specifications with dummy; the same holds for models without dummy. 3. The supply el a s t i c i t y of the stock, p*>, i s significantly less than one in model B, while not significantly different from one in model A. The estimate is affected by the dummy variable. This is really a large sample method. - 66 -Table 3.10: Estimated Harvest Supply Function (Cobb-Douglas) for North Sea Herring. t Exogenous. Time Series Data 1963-77^ Model InA* * a Po p! a* p2 ft r2 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) - - 0.97 B2 -8.48** (-4.86) -0.87** (-2.85) 0.45** (5.12) 1.34** (12.24) - -0.75** (-2.77) 0.98 1) Models A2 and B2 have been corrected for f i r s t order autocorrelation. Model Al has a DW s t a t i s t i c of 1.81 and an F sta 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 statistics are given in parentheses. * denotes significant at 90% lev e l . ** denotes significant at 95% l e v e l . Table 3.11: Derived Production Function Parameters (Cobb-Douglas) for North Sea Herring^  Model A a Po Pi P 2 P5 2 j = o ' Al 0.12 x 10"6 0.25 (0.20) 0.84 (0.43) 1.06 (0.23) 1.16 (1.09) - 3.31 A2 0.26 x 10"4 0.42 (0.10) 0.53 (0.22) 0.75 (0.11) 0.82 (0.57) -0.44 (0.19) 2.22 Bl 0.0037 0.31 (0.17) 0.42 (0.10) 1.01 (0.19) - - 1.74 B2 0.0151 0.46 (0.09) 0.24 (0.12) 0.72 (0.11) - -0.40 (0.13) 1.42 1) Standard errors are given in parentheses. The variance of A has not been estimated. - 67 -4. The estimated supply elas t i c i t y of the number of boats, p^, i s f a i r l y stable across different model specifications and is significantly larger than one in a l l models. 5. The supply elas t i c i t y of boat size, p2, is significant at the 90% level in 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 priori grounds, specifications with the dummy v a r i a b l e — models A2 and B2—were preferred to those without. This expectation was borne out by the empirical results. Furthermore, due to the s i g n i f i -cance of p2, model A2 w i l l be preferred to B2. The other main differ-ence between these two models i s the estimate of the p* parameter. Variables St and K2t are correlated. When K2t i s dropped from the regression, some of i t s effect w i l l accordingly be caught up by p*. When considering the derived output el a s t i c i t i e s of the produc-tion function (Table 3.11), one notes that there are s t r i c t l y decreasing returns to effort (boat-days), with point estimates in the range 0.25 to 0.46. This i s in contrast to the cross-sectional results, which gave constant or increasing returns to effort. The difference may be caused by the different nature of cross-sectional and time series regressions (Stapleton, 1981). The Schaefer production function imposes a stock output e l a s t i -city that is equal to one. An interesting result for herring i s the relatively low stock output e l a s t i c i t y (except for model A l , where the estimate incidentally has the highest standard error). The result must be attributed to the fact that we are dealing with a clupeid fishery. - 68 -Other results give point estimates of p^ larger than one in 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 fishery. The estimates of p2 are somewhat higher than those of the cross-sectional regressions. It has hitherto been assumed that fleet participation is exoge-nous. I f , on the other hand, i s endogenous, there w i l l be a problem of simultaneity bias. However, even i f this i s the case, i t can be argued that the bias i s small. This is because the models are well specified with several truly exogenous variables and the variances of the error terms are small (cf. Maddala, 1977, ch. 11). For these rea-sons, the simultaneity problem may not be serious and this procedure may be preferred to a more arbitr a r i l y defined simultaneous equation system. The consequence of letting 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 in order to ensure consistent estimates. These are the dependent, endoge-nous and exogenous variables lagged once and the cost/price ratio in the 23) mackerel fishery . The inclusion of the latter variable means that the effect of changes in relative prices in the herring and mackerel fisheries are taken into account when determining participation in the herring fishery. The results from the estimations of the supply 23) It i s assumed that the costs in the mackerel fishery are the same as in the herring fishery. This i s reasonable, because the tech-nology and the fishing grounds are to a large extent the same. Statis t i c a l Yearbook (Oslo: Central Bureau of Statistics of Norway) gives a price index for mackerel. - 69 -function are given in Table 3.12 with derived production function parameters in Table 3.13. Treating fleet participation as an endogenous variable affects a number of parameter estimates. In particular both the supply elas t i c i t y of the number of boats and the absolute value of the supply el a s t i c i t y of effort increase. When comparing results, 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 affect estimated parameter values. The interpretation of results i s to a large extent similar to the case when ]sqt was exogenous. Although numerical results vary somewhat, qualitative results remain unchanged. A hypothesis of increasing returns in a l l inputs i s accepted in a l l models. When considering the parameters of the production function (Table 3.13), treating K]^t as endogenous causes an increase in the out-put elas t i c i t y of effort and a decrease in the output e l a s t i c i t i e s of stock size and fleet size. Model Al indicates constant returns to fleet size, while point estimates are lower than one in the three other models. As noted above, boat size (K2) is an argument in both the cross-sectional and the time series (model A) production functions. This faci l i t a t e s the imposition of an estimate of p2 from the cross-sectional analysis on the time series. This is done because the cross-sectional estimations may give a better estimate of p2 than the time series. The reason is that there is much more variation in boat size in the cross-sectional than in 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 jt Endogenous. Time Series Data 1963-77^ Model * InA * a Po * Pi * P 2 * P5 2) Method r2 Al -20.77** (-1.96) -0.62 (-0.99) 1.01** (2.11) 1.62** (4.03) 1.38 (1.00) — 2SLS 0.96 A23> -21.38** (-3.21) -1.26** (-2.46) 0.78** (2.45) 1.57** (5.43) 1.38* (1.65) -0.98** (-3.45) 2SLS + AUTO 0.98 Bl -14.98** (-2.66) -1.37* (-1.65) 0.46** (2.39) 2.14** (3.84) — — 2SLS 0.93 B2 -11.50** (-3.65) -1.43** (-2.52) 0.32** (1.87) 1.64** (5.05) - -0.98** (-3.07) 2SLS + AUTO 0.97 1) t statistics are given in parentheses. 2) 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 inclusion of lagged variables as extra instrumental variables. 3) In this model, the instrumental variable K.2t lagged once was dropped because of nearly perfect collinearity with K2t. Table 3.13: Derived Production Function Parameters (Cobb-Douglas) 1) for North Sea Herring. K^t Endogenous  Model A a Po Pi P 2 P5 2 j = o ' Al 0.4 x 10-5 0.38 (0.24) 0.62 (0.44) 1.00 (0.20) 0.85 (0.99) — 2.85 A2 0.0001 0.56 (0.10) 0.34 (0.19) 0.69 (0.08) 0.61 (0.42) -0.43 (0.20) 2.20 Bl 0.0025 0.58 (0.15) 0.19 (0.13) 0.90 (0.14) — — 1.67 B2 0.012 0.59 (0.10) 0.13 (0.09) 0.68 (0.08) — -0.40 (0.23) 1.40 1) Standard errrors are given in parentheses. - 71 -used. In addition, the cross-sectional data contain relatively many observations over several years. In this case, the following estimating equation for the supply function corresponding to model A2 is used: lnHt = InA* - alnct/(l-a) + p0lnSt/(l-a) + p^nK^/d-a) + p2lnK2 t/( 1-a) + p5Dt/(l-a) + et Here, the relationship 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 direct estimates of the parameters of the production function with corresponding variances. The results of the estimations—for p2 values of 0.57 and 0.75—are given in Table 3.14. Table 3.14: 2SLS Nonlinear Estimation of Production Function Parameters (Cobb-Douglas) for North Sea Herring. B? Value Imposed^ Model a Po Pi P 2 P 5 •+U1 r2 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 statistics are given in parentheses. Due to the nonlinear estimation technique, model A2 could not be corrected for autocorrelation. - 72 -When comparing these results to the production function para-meters 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 po in Table 3.13 have somewhat large standard errors, which presumably explains this dif-ference in results. When p2 is increased from 0.57 to 0.75, a l l para-meter estimates remain stable except for po» This i s caused by the cor-relation between variables St and K2 t. These results give added credibility 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 effort variable was approximated by an index. The index in question i s a weighted average of indices for wages, fuel and materials, the main components of the cost of effort ( c f . Appendix 2). If one alternatively used a wage index, this would cause negligible changes in the results. This i s because wages constitute the largest share in the cost of effort. If a fuel price index is used, the supply elasticity of effort in model Al increases in absolute value and becomes significant. Other parameter estimates would also be affected, but there would be no quali-tative changes in the interpretation of the results. Therefore, the results presented in Tables 3.12 and 3.13 are f a i r l y robust to choice of index for the cost of effo r t . The question of externalities was discussed above, and i t was noted that crowding externalities might have been present during the latter part of the 1960s. An inspection of fleet participation in the - 73 -fishery reveals that i t was at i t s highest during the years 1965-69 (Table A2, Appendix 2). One might imagine the following situation. At low fleet sizes, there w i l l be no crowding externalities. However, i f fleet participation exceeds some threshold value, crowding might result. On the other hand, one would expect the search externality to be present at a l l fleet levels, although to a decreasing extent. The hypothesis about the crowding externality may be tested as follows. Define a new dummy—T>2t—that i s set equal to one for 1965-69 and zero otherwise, and introduce the variable D2tKlt i *1 t n e harvest supply functions. This has been done for the four model specifica-tions. The estimated coefficient was in a l l cases negative and s i g n i f i -cant. This result indicates that there are crowding externalities at high fleet levels. The introduction of this new variable also caused an increase in the output elas t i c i t y of fleet size ( p i ) . This is plausi-ble, because p^ may now represent the "pure" search effect, while the coefficient of the new variable represents the crowding effect at high levels of fleet participation. The empirical literature i s not abundant with studies of fisher-ies profit 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 result 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 effort was constant during the period of investigation (1950-60). Hannesson (1983) and Schrank et a l . (1984) both estimated Cobb-Douglas production functions with effort and stock size as independent variables. Hannesson, in applying this model to Norwegian cod and saithe fisheries, estimated effort output e l a s t i c i t i e s that—with the exception of one gear type—varied between 0.81 and 1.35, none of which was significantly different from one. The stock output e l a s t i c i t i e s were in the range 0.74 to 0.90. Many of these coefficients were not significantly different from one. Schrank et a l . estimated production functions for Newfoundland cod, f l a t f i s h and redfish fisheries. Con-stant returns to effort were obtained in a l l fisheries, with point e s t i -mates of the effort output e l a s t i c i t y varying between 0.88 and 1.09. The stock output elas t i c i t y could only be estimated for the cod fi s h -eries with point estimates of 0.66 and 0.88, which were not s i g n i f i -cantly less than one. The results obtained by Hannesson and Schrank et  a l . are therefore quite analogous: constant returns to effort, but with evidence that the stock output elas t i c i t y may be somewhat less than one. The difference in stock output e l a s t i c i t i e s for the clupeid and the demersal fisheries can be attributed to the different nature of the fisheries. While there may be some empirical evidence supporting the use of the Schaefer function in demersal fisheries, this i s not the case for clupeid fisheries. - 75 -4.0 THE OPTIMAL STOCK LEVEL In this chapter, the empirical results of Chapter 3 w i l l be com-bined with the dynamic optimization model for the sole owner that was developed in Chapter 2. Estimates of bionomic equilibrium and the opti-mal stock level with corresponding harvest quantities w i l l be pre-sented. Section 4.1 w i l l investigate the consequences of various assumptions about costs, while the sensitivity to changes in other para-meters of the Marginal Stock Effect w i l l be considered in Section 4.2. Finally, Section 4.3 w i l l b r i e f l y discuss a number of issues of rele-vance to the management of North Sea herring. 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., fuel and materials; and 3. variable, insurance and maintenance costs. Capital costs have been disregarded because i t has been argued that they may not be relevant for the present analysis. Estimates and a further analysis of costs and output price are given in Appendix 2. The zero cost case is included because i t serves as a useful ref-erence. Moreover, i t has been asserted that a low stock output e l a s t i -city implies that costs have a negligible effect on the optimal stock le v e l . This assertion w i l l be considered in light 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 in his maximization problem. This would make cost category 3 the appro-priate concept. In the following, this w i l l be termed the base case. However, the special case of only variable costs wi l l also be con-sidered. 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 fisheries 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 briefly discussed in Chapter 2. In this fishery, effort (boat-days), fleet size and stock size should a l l be treated as endogenous variables. This complete model formulation represents an avenue for future research. For now bionomic equilibrium w i l l be considered only for given fleet participation. Two figures help to illustrate the dynamics of this fishery. Figure 4.1 i s a phase-space diagram showing combinations of fleet parti-cipation and stock size for the period 1963-77. The early years of this fishery (1963-68) were characterized by increasing fleet participation and decreasing stock size. 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 , this period may represent "mining" of the resource. How-ever, the situation changed in 1968, with generally decreasing fleet participation and stock size in the ensuing years. Presumably the declining stock caused decreased p r o f i t a b i l i t y , which led to exit from the industry. Kl t t - 77 -I 2 St Million tonnes Figure 4.2: Number of Catches and Stock Size 1966-77 - 78 -Wilen (1976) studies the common property exploitation of North Pacific fur seal. This industry went through a dynamic process similar in part to the one illustrated in Figure 4.1. However, exit from the industry caused the stock to recover, and the data illustrated 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 in 1977 when the stock level was as low as 0.12 million tonnes. An open access fishery may have resulted in further stock depletion. In other words, bionomic equilibrium may be very close to zero. In this context i t should be realized that the equilibrium in question was a moving one, since costs and prices were changing over time. In particular, the cost/price ratio was decreasing in the 1970s (cf. Appendix 2). This would attract entry and decrease bionomic equi-librium. Further collection of cross-sectional data would f a c i l i t a t e draw-ing a phase-space diagram for effort (boat-days) and stock size. How-ever, Figure 4.2, which shows combinations of annual number of catches*^ and stock size for the period 1966-77, may throw some light on this relationship. Comparing the two figures, one finds that they essen-t i a l l y t e l l the same story, especially for post-1968 observations. Source: The Norwegian Herring Fishermen's Cooperative - 79 -One of the empirical results of Chapter 3 was that there is a stock effect in the production function, which means that harvesting costs depend on stock size. 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 equilibrium—S^—could s t i l l be very low. For the model developed in this thesis (equation 2.24), S^ , has been estimated for fleet sizes varying between 50 and 150 boats and for 2) the cost/price ratios of the sole owner case . In a l l the estimations Soo was 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. F i r s t , the model is not f u l l y specified. Second, the results involve extrapolation out-side the range of observed values, and estimated coefficients may there-fore not remain constant. 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 results here show that for the case in question this brake is not sufficient to pre-vent severe depletion or possibly even near extinction of the stock in an open access fishery. The consequence is that the u t i l i z a t i o n of a resource like herring poses some serious management problems. The cost/price ratios in the middle of the 1970s were typically lower than the present day ratio. This is 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., until bionomic equilibrium is attained (Scott Gordon, 1954; Scott, 1955). The theory does not predict how fast bionomic equilibrium w i l l be reached. Considering the stock development in the last 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 is specified. We w i l l now proceed to analyze the sole owner case. The estima-tions are based on model A2 for the production function. It should be stressed that results show that the different production functions give the same qualitative results. The dummy variable w i l l be set equal to zero. This i s because profit-maximizing behaviour is assumed. In the estimations of the production technology, only the Norwegian fleet was considered. Ideally, since cooperative management of the resource is analyzed (cf. Section 4.3), the fleets of other European countries that may participate in this fishery should also be included. The reason for not doing so is that data on other countries' fleets 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 l e v e l . This represents 3) a higher fleet particpation than would be forthcoming from Norway . In other words, the fleet may be composed of boats from different European In 1982, the Norwegian purse seine fleet consisted of 193 boats (Brochmann and Josefsen, 1984). - 81 -countries . In addition, sensitivity analyses w i l l be performed to illus t r a t e the effect on the optimal stock level of changes in fleet 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 . , internalize—the stock externality. As equation (2.14) cannot be solved exp l i c i t l y for the optimal stock level (S*), a solution is found by means of simulation. The optimal harvest quantity (H*) is subsequently found from equation (2.15). Results for the zero cost case and the base case are given in Table 4.1. Table 4.1: Optimal Stock Level (S*) and Corresponding Harvest (H*) Discount rate Zero Cost Case Base Case Percentage increase in S* s* Million tonnes H* Million tonnes 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% This assumes that the technologies of other countries' fleets are the same as the Norwegian. This is probably r e a l i s t i c , provided only purse seiners are considered. - 82 -The results in Table 4.1 put the management of this resource in 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 level with a sustained harvest flow ad i n f i n i -tum. From the results in Table 4.1, a number of conclusions can be drawn: 1. The inclusion of costs in the intertemporal profit function causes a considerable increase in the optimal stock l e v e l . This result i s especially noteworthy because the stock output elasticity 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 level i s therefore not necessarily true. 2. S* is not very sensitive to changes in the discount rate and less so in the base case than in the zero cost case. This shows that costs have a stabilizing influence on the optimal stock l e v e l . 3. H* is quite insensitive to changes in the optimal stock level and consequently also to changes in the discount rate. 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 (cf. Figure 3.3). 4. In the zero cost case, S* = Sm s y and H* = MSY for r = 0 (cf. Table 3.6). Moreover, the numerical results show that stock extinc-tion i s "optimal" for a discount rate of about 52%. A consequence of the third result i s that for a change in the discount rate, the harvest quantity—and thus gross revenues—will - 83 -usually be almost unchanged. However, the optimal stock level w i l l be affected, causing an adjustment in harvesting costs. Changes in other parameter values w i l l frequently have the same consequences, with adjustment on the cost side rather than on the revenue side. The optimal stock level i s increasing in the cost/price r a t i o . Therefore, i f a rental price of capital was included in the cost of effort , S* would be higher than the estimates for the base case given in Table 4.1. This conforms to the analytical results of Clark, Clarke and Munro (1979). On the other hand, a secular downward trend in the cost/price ratio would gradually diminish the importance of costs. Bionomic equilibrium for the base case cost/price ratio was es t i -mated to be near zero. However, costs were seen to cause a considerable increase in the sole owner stock l e v e l . These results may seem contra-dictory, but can be attributed to the nonlinear nature of the production function. A necessary and sufficient condition for harvesting to occur in the sole owner case i s : . ,E > 0 This condition can alternatively be stated as VMP^  = aVAP^ > AC„ For the fishery in question, ct is 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 effort is at least twice the value of the average"*^ cost of effort. Bionomic equilibrium, on the other hand, is characterized by VAP„ = ACL, (cf. equation 2.20'). The values of the marginal and the average product of effort are both increasing in stock size. Thus, there may be a threshold level for the stock size in order for the harvesting condition to be satisfied. For the base case cost/price rat i o , numerical results show that TTg < 0 for steady state stock levels less than about 2.0 million tonnes. Accordingly, a sole owner would never drive the stock below this level even at an i n f i n i t e discount rate. These results show that for a sufficiently 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 s t i l l be close to zero. In other words, costs may be an effective brake on stock depletion in the sole owner case but not in the open access case. Furthermore, the results explain why the inclusion of costs caused such a considerable increase in the optimal stock level despite the fact that the stock output elasticity was low. Lastly, i t should be pointed out that these results would not have been forthcoming in the context of a linear model. The optimal stock level for the variable cost case i s given Table 4.2. Even when only variable costs are considered, they cause a marked increase in S*. In this model, average and marginal costs are equal. - 85 -Table 4.2: The Optimal Stock Level for the Variable Cost Case Di scount rate Variable Cost Case Increase in S* compared to zero cost case S* Million tonnes H* Million tonnes 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 level i s somewhat less sensitive to changes in the discount rate at the higher than at the lower cost/price ra t i o . This shows that the higher the optimal stock l e v e l , the less sensitive i t is to changes in the discount rate. The influence of costs i s the cause of this phenomenon, which w i l l also be illustrated in some later cases. In the variable cost case, numerical results show that the opti-mal stock level i s close to but does not reach zero for an infinite dis-count rate. The harvesting condition—-n„ > 0 — i s satisfied for lower stock levels than in the base case. Therefore, costs are less effective as a brake on stock depletion. The base case and the variable cost case represent two discrete alternatives. Table 4.3 illustrates the sensitivity of S* to a marginal change—plus/minus 20%—in the cost/price ratio for the base case. - 86 -Table 4.3: Sensitivity to a 20% Change in Cost/Price Ratio. Base Case Cost Alternative Cost/price ratio: -20% Cost/price ratio: +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 rate, the relative change in S* is only 7-10% as compared to a 20% change in the cost/price ratio. Therefore, the optimal stock level would be only marginally affected by moderate changes in the cost/price ra t i o . The difference between the two cost alternatives may be summa-rized as follows: the choice of cost concept has a substantial effect on the optimal stock l e v e l , as illustrated by Tables 4.1 and 4.2. However, once the choice has been made, S*—and even less so H * — i s not very sen-sitive to moderate changes in the cost/price ra t i o . Gallastegui (1983), in a study of a sardine fishery in the Gulf of Valencia, estimates bionomic equilibrium to be somewhat larger than Sm s v. The sole owner stock level i s for reasonable discount rates more than twice the size of Sm s y. Moreover, S* is not very sensitive to changes in the discount rate. The consequences of discounting in the case of baleen whales are illustrated in a paper by Clark (1976a). An increase in the discount - 87 -rate from zero to five percent reduces the optimal stock level by almost 50%. The reason for this severe effect of discounting as compared to the results obtained in this thesis i s the low growth rate in the whale population. 4.2 The Influence of the Marginal Stock Effect The Marginal Stock Effect was defined in equation (2.26) as inter  a l i a a function of the cost/price rat i o , the output e l a s t i c i t i e s of effort and stock size, and fixed factors. Having analyzed the effect of costs, the relationship between the optimal stock level and the other arguments of the Marginal Stock Effect w i l l now be considered. 1. Changes in the Effort Output E l a s t i c i t y The prediction i s that S* i s decreasing in the effort output e l a s t i c i t y . S* has been estimated for a-values of 0.51 and 0.61 respec-tively. Table 4.4: Sensitivity to Changes in the Effort Output E l a s t i c i t y ( a ) * ^ ct = 0.51 a = 0.61 Discount rate S* Million tonnes 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 1) In the base case, a = 0.56. - 88 -The results in Table 4.4 show that the optimal stock level i s sensitive to changes in the effort output e l a s t i c i t y , as the relative change in the former variable i s larger than the relative change in the lat t e r . This appears to be the case even more for downward than for upward changes in a. The results tend towards the zero cost case with increasing a. Also, a sufficiently low effort output elasticity would render the fishery unprofitable. One observes that at the low effort output e l a s t i c i t y , the optimal stock level i s almost completely insensi-tive to changes in the discount rate. 2. Changes in the Stock Output E l a s t i c i t y Table 4.5 gives the optimal stock level for po-values of 0.29 and 0.39. 1) Table 4.5: Sensitivity to Changes in the Stock Output E l a s t i c i t y (3^) Po = 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 1) In the base case, po = 0.34. For a given discount rate, S* is sensitive to changes in the stock output e l a s t i c i t y and especially so for downward changes in 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 dis-count rate. The result is quite interesting, namely that S* is a decreasing function of po. 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 po decreases, the profitability of the fishery also decreases, lead-ing to an increase in the optimal stock level. For sufficiently low pQ values, the fishery becomes unprofitable. When this is 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 in K of -12% and +11% respectively, where K = K^K^2. - 90 -Table 4.6: Effects of Changes in 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 relative change in S* for a given discount rate is consider-ably less than the relative change in fleet size. The same result would hold for changes in K2 and the combined fixed factors. Moreover, S* is shown to be decreasing in the fixed factors. Sensitivity analyses for changes in p^ and p2 would yield results analogous to those presented here. 4.3 Some Management Issues This section w i l l raise some issues that are important in a prac-t i c a l management plan for North Sea herring. They are: 1. the optimal approach path, 2. the transboundary nature of the fishery, 3. the regulation of effort, and 4. uncertainty. As mentioned in the Introduction, these issues are not considered part of the thesis proper. Rather, they represent issues for future research and w i l l be discussed only b r i e f l y . - 91 -Because the underlying model is nonlinear, the optimal approach to the steady state is 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 rise to an asymptotic approach (Clark, Clarke and Munro, 1979). The conditions characterizing the optimal approach path—the equations of motion—are given in Appendix 1. These conditions come out as part of the dynamic optimization problem. They may also be i l l u s -trated graphically in the form of phase-space diagrams in an analytical model (Neher, 1974). The task of finding the optimal path in an empirical problem i s , however, arduous, and especially so in a model with a time lag. The reason is the high number of possible paths between the present day and the steady state stock levels, to be arrived at at some time in 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. Examples of two such paths are given in Table 4.7. - 92 -Table 4.7: Simulation of Spawning Stock 1983-87 Year Ht = 0 Spawning stock Million tonnes Ht = 0.20 million tonnes Spawning stock Million 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 1) Sg4 =0.50 million tonnes. The rapid increase in stock size from 1982 to 1983 is influenced by the good 1981 yearclass of recruits ( cf. Table A l , Appendix 2). The f i r s t case simulates the development of the stock under natu-ral conditions, which corresponds to the bang-bang approach. This path is both inoptimal and inconceivable, as a complete fishing 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 in order to be able to consider the delay in attaining the steady state, when an asymptotic approach is used rather than the most rapid approach. Accordingly, in the second simulation i t i s assumed there i s an annual harvest quantity of 0.20 million tonnes. Assuming a 6% discount rate, the optimal stock level is 2.21 million tonnes (cf. Table 4.1). With a most rapid approach, this stock level would be attained in 1986, after a four-year fishing mora-torium. An annual harvest quota of 0.20 million tonnes would cause a delay in reaching the equilibrium of two years. - 93 -The two policies can be compared in present value terms. Assume that once the steady state has been reached, the fishery continues on a sustained basis, i.e., with an annual harvest quantity of 0.55 million tonnes ad infinitum. 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 percent^. One would expect that i f the present values of net revenues were calculated, the difference would be of the same magnitude. This result suggests that the loss—measured in terms of present value—of deviating from the optimal approach path may not be great. 8 ) This observation is supported by Clark (1976) and Ludwig (1980) . The consequence is that the resource manager may have some degree of freedom in determining the approach to the steady state stock l e v e l . North Sea herring is a transboundary resource that is harvested by several European nations. Previously, the North West Atlantic Fisheries Commission was responsible for making policy recommendations, but had no power to implement these over the wishes of i t s member coun-t r i e s . This was a major reason for the severe stock depletion that took place in the 1960s and 1970s. The optimal stock level would typically be "overshot" in the year i t was reached (cf. Table 4.7), giving a harvest quantity for that year which is higher than the sustained one. This effect has been neglected in the following calculation. ^^The asymptotic approach has the highest present value. 8) Although Ludwig's results are in 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 fisheries jurisdiction, North Sea herring has been considered a common resource between Norway and the European Economic Community (EEC). Therefore, management decisions are 9) decided upon joi n t l y by Norway and the EEC . These decisions are inter  al i a based on biological advice from the International Council for the Exploration of the Sea. North Sea herring migrates between the Norwegian and the EEC fishery zones. This movement takes place partly due to seasonal migra-tion and partly according to developmental stages. Such migration i s common for many species (Gulland, 1980). This means that the different countries can exploit the resource at different stages in i t s l i f e cycle. The approach taken, however, is that of cooperative management. Indeed, this i s also the approach of this thesis, as the sole resource manager can be compared to the joint Norwegian-EEC body regulating the resource. Munro (1979) used a game theoretic model to analyze the coop-erative management of a fish resource shared by two countries. However, discount rates, costs or consumer preferences may be different between the two countries, giving rise to differences in what the two countries perceive to be the optimal stock l e v e l . The cooperative resolution of this conflict i s studied. In this thesis, 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 in 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 fisheries 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 in a game theoretic framework. Not surprisingly, the steady state stock level turns out to be lower in the competitive than in the cooperative case. Crutchfield (1983) ana-lyzes bionomic equilibrium for a fishery where several nations partici-pate and consider different management alternatives under extended fish-eries jurisdiction. Although the present day management of North Sea herring is meant to be cooperative, there is no guarantee that this w i l l be kept up in the future. Therefore, both cooperative and competitive management schemes should be considered. It may be particularly interesting to include the migratory pattern of the resource in a game theoretic model. This represents an interesting topic for future research*^. The problems with regard to regulating the fishery have been assumed away in this analysis, 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. How-ever, the fishing fleet is in 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 policy, the objective of which is to restore the stock to Here, the interactions between Norway and the EEC have been outlined. This implicitly assumes away the process whereby the EEC-countries arrive at a common fisheries policy, which indeed has proved d i f f i c u l t . ^ O n this topic, see Hannesson (1978), Pearse (1979), Scott (1979), Clark (1982a) and Munro and Scott (1984). - 96 -some equilibrium level over a period of time. This policy can be achieved by setting annual escapement targets with corresponding total 12) allowable catch quotas . As the stock recovers, there w i l l be rents to be made. If the amount of effort is not controlled, excess capacity w i l l result. This can cause rent dissipation and also represents a potential threat for stock depletion. Excess capacity can in theory be prevented by input controls like limited entry programmes and regulations of boat size. 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. Technical change may cause separate complications. Output controls consist of royalties on fish landings and indi-vidual catch quotas. The quotas may either be distributed to fishermen, who then would collect 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 in terms of efficiency. However, this result may be model specific and i s thus not necessarily true in general. A special case w i l l occur i f the stock effect in the production function is zero. If a landings tax is imposed, the fishery w i l l either remain profitable 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 con-siderable problems. - 97 -discontinuously (cf. Section 2.3). Thus, landings taxes may not be effective for some clupeid fisheries. The present analysis i s carried out in the context of a determin-i s t i c model. However, real world management must take uncertainty into account, both with regard to costs and prices, the production technology and the resource i t s e l f . The latter type of uncertainty is vividly illustrated in the estimated stock recruitment and net growth functions (cf. Figure 3.1). Walters and Hilborn (1978) classify major sources of uncertainty in resource management as (i) random disturbances, ( i i ) parameter uncer-tainty and ( i i i ) ignorance about the appropriate forms of the underlying models. So far, most work in 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 in fisheries management. When i t comes to the possibility of extending the present analy-sis to consider the effects of uncertainty, i t should be pointed out that the complexity of the underlying bioeconomic model w i l l make sto-chastic optimization very d i f f i c u l t . Markov processes (Lewis, 1982) and simulation represent alternative methods of analysis. Two other points that may be specific to the resource in question can be made. Since several countries u t i l i z e North Sea herring, moni-toring harvesting quantities becomes especially d i f f i c u l t . This would be an added source of uncertainty. On the other hand, concentrating harvesting on the spawning stock tends to reduce the effect of uncer-tainty in the stock recruitment relationship. This is because more - 98 -information about the size of the new yearclasses can be obtained while they mature to spawning age. In real l i f e , the management issues that have been discussed here are closely interrelated. Excess fleet 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 effort and the monitoring of catches are made more d i f f i c u l t because several nations are involved in the fishery. On the other hand, in an analytical or empirical study, i t may be necessary to treat these issues separately. However, that w i l l be the subject of future research. - 99 -5.0 SUMMARY The objective of this thesis has been to analyze the optimal man-agement of a fish resource—North Sea herring. A dynamic bioeconomic model was constructed. In the development of the model of population dynamics, important biological characteristics were taken into account. The production technology and the net revenue function were specified in manners consistent with economic theory. An analytical solution for the sole owner stock level was derived. This was contrasted to bionomic equilibrium, i.e., the open access solution. Alternative forms of the biological and economic functions were specified and estimated. Esti-mates of the optimal stock level with corresponding harvest quantities were obtained under different sets of assumptions. The presence of fixed factors in the production function—in the form of capital equipment—was acknowledged. This presumably represents reality in many developed fisheries, e.g. the one being analyzed in this thesis, although the situation would be different in a developing fish-ery. It was argued that the fishery in 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 capital dynamics from the dynamic optimization problem. Apart from implicitly assuming a zero depreciation rate, the model resembles a case modelled by Clark, Clarke and Munro (1979), who analyzed simultaneous investment in fish stock and capital stock. The schooling behaviour of the resource in question was expected to have some consequences for management. The empirical results showed - 100 -that the function describing growth in the existing stock exhibited depensation at low stock levels, a phenomenon not uncommon for schooling f i s h . If the fish stock is reduced to a low l e v e l , depensation may cause the recovery to be slow. This has been a case in point for a num-ber of clupeids. For North Sea herring, the stock size now appears to be outside the range of stock values where there is depensation. Modern fishfinding and harvesting equipment make i t possible to locate and catch schools profitably even at low stock levels. Schooling fis h are therefore especially vulnerable to predation by man. It has been asserted that the stock output elasticity w i l l be low for this kind of fishery. This was confirmed by the empirical results, with most estimates of the stock output elas t i c i t y in the range 0.1 to 0.5. It can be shown analytically that a zero stock output el 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 effect is low but positive, which implies that harvesting costs serve as some kind of a brake on stock depletion. However, the empirical results show that this brake is not very effective, as bionomic equilibrium—based on cost/price ratios that were in effect in the middle of the 1970s—is near zero. Although v i r -tual extermination of the species is unlikely, near extinction of the stock and the practical disappearance of North Sea herring as a commer-ci a l species i s a very real possibility. This result puts the management of clupeids in perspective, because the characteristics of this fishery are l i k e l y to carry over to fisheries on other schooling species. The depleted state of a number of clupeids is further evidence of this result. Although open access - 101 -fisheries w i l l cause depletion also in the case of ground fisheries, the stock dependence of harvesting costs w i l l be greater. The possibility of stock extinction is therefore less in these fisheries than for clu-peids. Failure to manage a clupeid fishery can have irreversible conse-quences . When i t comes to the case of a sole owner, i t had been asserted that a low stock output el a s t i c i t y implies that costs have a negligible effect on the optimal stock l e v e l . The results of this thesis show that this assertion is false, at least in the case of North Sea herring. The inclusion of costs in the intertemporal profit function are shown to cause a considerable increase in the optimal stock l e v e l . The results show that bionomic equilibrium is near zero, while the same cost/price ratio causes a marked increase in the sole owner stock le v e l . This apparent contradiction could be attributed to the nonlinear nature of the production function. It was shown that for a sufficiently high cost/price rati o , i t would never be optimal for the sole owner to drive the stock to extinction. Thus, costs may be an effective brake on stock depletion in the sole owner case, but need not be so in the open access case. Two cost alternatives were considered. The f i r s t included only variable costs, while the second included variable, insurance and main-tenance costs. As stock size i s increasing in costs, equilibrium bio-mass i s larger for the second than for the f i r s t cost alternative. Inclusion of capital costs would have caused a further increase in the optimal stock l e v e l . It was also shown that while the optimal stock level i s significantly influenced by the choice between the two discrete - 102 -cost alternatives, once the choice was made i t was not very sensitive to moderate changes in the cost/price r a t i o . The optimal stock level was shown to be not very sensitive to moderate changes in the discount rate. It was illustrated that costs have a stabilizing influence on the stock le v e l . The optimal harvest quantity was shown to be insensitive to changes in the optimal stock le v e l , a result caused by the properties of the estimated model of popu-lation dynamics. For a change in the discount rate, the optimal harvest quantity—and thus gross revenues—will usually be almost unchanged. However, the optimal stock level w i l l be affected, causing an adjustment in harvesting costs. Changes in other parameter values w i l l frequently have the same consequences. The relationship between the optimal stock level and the produc-tion technology was analyzed. Some new results were forthcoming, because the underlying model was nonlinear. It was shown analytically and illustrated empirically that the optimal stock level i s decreasing in the output e l a s t i c i t y of effort and the amount of fixed factors in the industry. Furthermore, i t was illustrated that the optimal stock level—under certain conditions—can be decreasing in the stock output e l a s t i c i t y . This result was also obtained empirically. The practical applicability of this thesis would be enhanced i f the present research were extended in a number of ways. F i r s t , the approach path(s) to the optimal stock level should be investigated more closely. Second, matters concerning the transboundary character of the fishery need to be considered. Third, fishery regulations need to be studied. Fourth, the effects of uncertainty on resource management - 103 -should be analyzed. Although i t may be desirable analytically to ana-lyze these topics separately, i t should be stressed that from the point of view of fisheries management, they are closely interlinked. The concept of a sole owner has played an important part in the analysis. In such an institutional setting, the assumption of price-taking behaviour may seem inconsistent. It should here be borne in 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 in the f i r s t market, i t may be more appropriate to model the sole owner as a monopolist in the consumer mar-ket. This represents another avenue for future research. A final word on the present thesis. The results indicate that the model is quite robust. Different alternatives were specified both for the model of population dynamics and the production function. How-ever, qualitatively speaking, the results are not sensitive to choice of functional form. Although the numerical results may vary somewhat between model specifications, the same qualitative results are forth-coming. 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." 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Oslo: Universitetsforlaget. (1983). "The Bionomic Production Function in Fisheries: a Theoretical and Empirical Analysis." Canadian Journal of  Fisheries and Aquatic Sciences 40:968-982. (1983a). "Optimal Harvesting of Ecologically Interdependent Species." Journal of Environmental Economics and  Management 10:329-345. Hansen, T. (1981). "Analysing the Demand for Fishmeal by a Linear Programming Model" in Applied Operations Research in Fishing. Edited by K. B. Haley. New York: Plenum. Hansen, T. et a l . (1978). En strukturanalyse av sildolje- og sildemelindustrien. Bergen: Centre for Applied Research, The Norwegian School of Economics and Business Administration. Report No. 1: 1978. (in Norwegian). Kmenta, J . (1971). Elements of Econometrics. New York: Macmillan. Kurz, M. (1968). "Optimal Economic Growth and Wealth Effects." International Economic Review 9:348-357. Lau, L. J . (1978). "Applications of Profit Functions" in Production Economics: A Dual Approach to Theory and Applications. Edited by Fuss and McFadden. Amsterdam, New York and Oxford: North-Holland. Lau, L. J . , and Yotopoulos, P. A. (1972). "Profit, Supply and Factor Demand Functions." American Journal of Agricultural Economics 54:11-18. Levhari, D.; Michener, R.; and Mirman, L. J . (1981). "Dynamic Programming Models of Fisheries." American Economic Review 7:649-661. Levhari, D., and Mirman, L. J . (1980). "The Great Fish War: An Example Using a Dynamic Cournot-Nash Solution." Bell Journal of Economics 11:322-334. Lewis, T. R. (1982). Stochastic Modeling of Ocean Fisheries Resource Management. Seattle and London: University of Washington Press. Lewis, T. R., and Schmalensee, R. (1982). "Optimal Use of Renewable Resources with Nonconvexities in Production" in Essays in the  Economics of Renewable Resources. Edited by L. J . Mirman and D. F. Spulber. Amsterdam, New York and Oxford: North-Holland. Ludwig, D. (1980). "Harvesting Strategies for a Randomly Fluctuating Population." Journal Cons. Int. Explor. Mer 39:168-172. - 107 -McKelvey, R. (1983). "The Fishery in a Fluctuating Environment: Coexistence of Specialist and Generalist Fishing Vessels in a Multipurpose Fleet." Journal of Environmental Economics and  Management 10:287-309. Maddala, G.S. (1977). Econometrics. New York: McGraw-Hill. May, R. M.; Beddington, J . R.; Clark, C. W.; Holt, S. J.; and Laws, R. M. (1979). "Management of Multispecies Fisheries." Science 205:267-277. Melhus, T. B. (1978). "Utviklingen i den norske ringnotflaaten" (The Development in the Norwegian Purse Seine Fleet). Fiskets Gang 1978:412-420. (in Norwegian). Mirman, L. J . , and Spulber, D. F. (1982). Essays in the Economics of  Renewable Resources. Amsterdam, New York and Oxford: North-Holland. Morey, E. R. (1983). "A Generalized Harvest Function for Fishing: Allocating Effort 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). "Bilateral Monopoly in Fisheries and Optimal Management Policy" in Essays in 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 British Columbia. Discussion Paper 84-09. Murphy, G. I. (1977). "Clupeids" in Fish 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." Scientific American 246:90-99. Pearse, P. H. (1979). "Property Rights and the Regulation of Commercial Fisheries." Vancouver: Department of Economics, University of British Columbia. Resources Paper No. 42. Posthuma, K. H. (1971). "The Effect of Temperature in the Spawning and Nursery Areas on Recruitment of Autumn-Spawning Herring in the North Sea." Rapports et Proces-verbaux des Reunions 160:175-183. - 108 -Ricker, W. E. Computation and Interpretation of Biological Statistics  of Fish Populations. Ottawa: Environment Canada. Rothschild, B. J . (1977). "Fishing Effort" in Fish Population Dynamics. Edited by J . A. Gulland. New York: Wiley. Sav i l l e , A., and Bailey, R. S. (1980). "The Assessment and Management of the Herring Stocks in 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 in 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. St. John's: Department of Economics, Memorial University of Newfoundland. Schumacher, A. (1980). "Review of North Atlantic Catch Statistics." 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 Possibilities from Cross-Section and Time-Series Data" in Modelling and  Measuring Natural Resource Substitution. Edited by E. R. Berndt and B. C. Field. Cambridge and London: The MIT Press. Ulltang, 0. (1976). "Catch per Unit Effort in 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 Pacific Fur Seal." Vancouver: Department of Economics, University of British Columbia. Resources Paper No. 3. Young, R. D. (1979). "Some Aspects of Production from an Ocean Fishery. Ph.D. diss., 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'Cd^Et^t.tt) - qt[St + 1 " (St - Ht) e6 ( St) - G(St_Y)]} t=0 where Ht = Ht(Et;St,Kt), qt = shadow price of the resource, d = 1/(1 + r) = the discount factor, and r = the discount rate. First order necessary conditions for an optimum are (Clark, 1976): i . -||- = 0, t = 0,1,2,... i i . ||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> f ^ = d S - « t HEe6 ( S t )= ° (3) | | - = d^;s + qt6-'(S)(St - Ht) e6 ( S ) + qte6 ( S )( l - Hg) - q,--! + qY + tG*(St) = 0 (31) d^s + qt6'(S)(St - Ht) e6 ( S ) + qte6 ( S )( l - Hs) + qY + tG'(St) = qt_! The equations of motion are given by (2) and (3). The shadow price of the resource is found from equation (2): - I l l -( 4 ) "t • * ' - ^ f e r E The shadow price i s inserted in (3'), which i s simplified to become: (5) e6 ( S )[HE ^ - H8 + 1] + 6'(St)(St - Ht) e6 ( St) + dYG'(St) = ± E The normalized net revenue function is defined as irt = H(Et;St,Kt) - cEt The derivatives are given by TT = H - c E E S nS These results can be used to rewrite (5): (5') e6 ( S )[ ^ + 1] + 6'(S)(St - Ht) e6 ( St) + dV(St) = 1 + r E For the stock to be in steady state, one requires: (6) St + 1 - St = (St - Ht) e6 ( St) + G(St_y) - St = 0 Equation (6) can be solved for the steady state harvest quantity: S - G(S) (7) H = H(S) = S fi(S) e Utilizing equation (7) in (5'), the following implicit expression for the optimal stock level S* is derived: (8) e6^*^-^- + 1] + 6'(S*)[S* - G(S*)] + dYG'(S*) = 1 + r E Define (9) MSE = CTTS/TTE where MSE i s the Marginal Stock Effect. - 112 -2. The Cobb-Douglas Case The Cobb-Douglas production function is given by (10) H = AEasP°K, K = TTK^ , j where time subscripts have been suppressed. The net revenue function becomes (11) TT = AEaSp°K - cE and the derivatives are given by TT = ctH/E - c Ji TTS = pQH/S The MSE is accordingly (9.) M S E =_ |fH / S _ The level of effort needed to obtain the harvest corresponding to a given steady state stock level is found by use of equation (7): H = AEaSp°K = S - S ~(^S ) e (12) E = ( A K r1 / aS - p o / « [S - S ]1 / a e It can be ascertained that i . | | < 0 and i i . lim E = 0. K > c o - 113 -3. The Marginal Stock Effect The MSE i s defined as (9) MSE = CTrs/TrE = m(c,ct,p0,S,K). The MSE vanishes for c = 0. By the properties of the profit function, The interactions between the MSE and the different parameters w i l l now be investigated. The Cobb-Douglas production function is assumed with A = 1. The Effort Output E l a s t i c i t y Assume for simplicity that po = 1 and insert the appropriate derivatives in (9): M S E = - ^ « Ea *SK - c Then 3MSE _ ... c2EaKlnE + cE2 a~1SK2  U ; 3° " [dEa-1SK - c ]2 This derivative is negative provided i . c > 0, i i . Hg ^ c, i.e., TTE ^ 0. If E < 1, a third condition would need to be satisfied in 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 a -»• 0 0, provided E > 1. The limiting value is found by applying L'Hopital's rule: - 114 -i • ucD i • cEaKlnE lim MSE = lim a-»-oo a^-oo Ea _ 1SK + aEa - 1SKlnE , . cKlnE n = lim ; = 0 a->co E-1SK(1 + alnE) Fixed Factors Assume for simplicity that a = pQ = 1 and insert the appropriate derivatives in (9): MSE = C E K SK - c Then (14) 3MSE = c2. [SK - c ]2 This derivative i s unambiguously negative provided i . c > 0 i i . TTE i= 0. Moreover, lim MSE = lim ^  = ~ lim E = 0. K->°° K->°° K->oo The Stock Output El a s t i c i t y Assume for simplicity that a = 1. Then MSE cSJESpOK 3 P°K - c n . sr j30 = 0 ••» MSE = 0 This case can be resolved by L'Hopital's rule: - 115 -lim MSE = lim ^ a E S ^ P o - " " Po c3nSr° KlnS -+<*> sP°KlnS cgnE = c P o — S P o — ' po = lim - — lim pol i m E By use of equation (12), i t can be shown that this limit is zero provided S > 1. i i i . 0 < p„ < co 3MSE = [cEsP°K + Cp n EsP°KlnS][SP°K - c] - Sp°KlnScpnEsP°K  3 P o " [ s P o K - C]2 = c E s P°K [ s P°K - cp^lnS - c ] [SP°K - c ] 2 The sign of this derivative is ambiguous, depending on the term in brackets in the numerator. Therefore, different relationships between the MSE and p o may occur, as outlined in Section 2.3. In particular, i t is claimed in Section 2.3 that the MSE can be a decreasing function of p o . Consider the derivative when evaluated at p o = 0: 9 MSE Po _ cEK po=0 ~ K - c A necessary and sufficient 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 effect, i.e., i T g < 0 at p o = 0, then the MSE—at least l o c a l l y — w i l l be decreasing in p Q . - 116 -APPENDIX 2: DATA Table Al: Biological Data Recruits the Spawning stock following Harvestj v Instantaneous per September 1 year quantity net naturals Year 1,000 tonnes Millions 1,000 tonnes growth rate 1947 2,945 4,720 587 -0.1159 1948 2,581 4,100 502 -0.0380 1949 2,618 5,680 509 -0.0531 1950 2,428 6,900 492 0.0053 1951 2,169 7,690 600 -0.0307 1952 1,908 9,100 664 -0.0174 1953 1,707 8,070 699 0.0807 1954 1,745 7,700 763 0.0216 1955 1,821 4,768 806 0.1193 1956 1,741 21,429 675 0.0770 1957 1,593 5,641 683 0.0051 1958 1,236 7,555 671 0.1763 1959 2,063 1,954 785 0.1454 1960 1,871 16,686 696 0.0230 1961 1,601 7,085 697 0.0926 1962 1,132 8,740 628 -0.0252 1963 1,800 10,907 716 0.1397 1964 1,829 5,709 871 0.0903 1965 1,340 5,289 1,169 0.1000 1966 1,116 7,581 896 0.0893 1967 817 7,623 696 0.0669 1968 390 3,820 718 0.1754 1969 359 9,081 547 0.1470 1970 318 7,146 564 0.1306 1971 219 4,975 520 0.1514 1972 269 2,398 498 0.1817 1973 228 5,583 484 0.1593 1974 166 773 275 0.1404 1975 117 720 313 0.1382 1976 141 1,064 175 0.2265 1977 123 899 46 0.1607 1978 154 2,582 11 0.0237 1979 208 3,423 25 0.0385 1980 238 12,414 61 0.0891 1981 368 14,958 95 0.0589 1982 498 * * 0.1137 - 117 -1) Harvest of juvenile herring i s included. 2) The net growth rate i s estimated according to the formula 6 — F St + 1 = Ste t t + 6 f c = F t + l n ( St + 1/ St) ; Ft = fishing mortality. For these estimates the stock size as of January 1 has been used. * Not available. Sources: St: Anon. (1977) for 1947-54 Anon. (1982) for 1955-74 Anon. (1983) for 1975-82 Rt and Ft: 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 Participation in the Herring Fishery Year Norwegian harvest Tonnes Number of participating purse seiners Index of average size of 2) purse seiners Without power-block With power-block 1963 3,454 16 0 1.000 1964 147,933 148 47 1.060 1965 586,318 150 134 1.124 1966 448,511 72 262 1.192 1967 334,449 14 312 1.264 1968 286,198 0 352 1.340 1969 134,886 0 253 1.420 1970 220,854 0 201 1.505 1971 210,733 0 230 1.677 1972 136,969 0 203 1.900 1973 135,338 0 153 2.015 1974 66,236 0 165 2.115 1975 34,221 0 102 2.173 1976 33,057 0 92 2.331 1977 3,911 0 24 2.406 1) Column 3 illustrates the rapid introduction of powerblock in the fle 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. 2) In the regressions, actual boat size has been used rather than the index listed here. It is immaterial which one of these series i s chosen. Sources: Number of participating purse seiners for 1971, 1972 and 1977: Noregs Sildeselslag (Norwegian Herring Fishermen's Cooperative) Index of average boat size: Based on Melhus (1978) Other data: The Directorate of Fisheries, Norway - 119 -COSTS AND PRICES The Cost/Price Ratio for the Harvest Supply Function In the harvest supply function, the cost/price ratio enters as a variable. While data on the output price are available, this i s not the case for data on the cost of effort. Therefore, a time series needs to be constructed for this variable. The cost per boatday, c^, includes labour, fuel and material costs. In other words, only variable costs are considered. A l l that is needed is an index of the change in cj- over time and not the absolute cost figures. Price indices for wages, fuel and materials are readily available. The index for cj. w i l l then be a weighted average of these indices, 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 fishing 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 fuel Share of materials 1969-73 79.20% 13.80% 7.00% 1974-77 72.15% 21.75% 6.10% A simple inspection of the annual data shows that the effects of the f i r s t shock in o i l prices became noticeable in 1974, which is the reason for the division in 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 different indices are given in Table A3, while price data are presented below in Table A6. - 120 -Table A3: The Cost of Effort Index Raw materials Cost of Wage index^ Fuel 2v price index price^s, index effort.N • j 4) index Year 1963 100.0 100.0 100 100.0 1964 106.2 104.0 104 105.7 1965 115.8 102.0 109 113.4 1966 124.2 102.0 108 120.0 1967 133.7 108.9 108 128.5 1968 144.4 111.9 106 137.2 1969 158.0 110.9 111 148.2 1970 177.1 113.9 124 164.7 1971 198.8 128.7 130 184.3 1972 216.5 132.7 128 198.7 1973 239.5 141.6 136 218.7 1974 281.0 200.0 168 256.5 1975 336.6 221.8 193 302.9 1976 391.8 250.5 206 349.7 1977 434.6 274.3 212 386.2 1) Index of hourly earnings in manufacturing companies a f f i l i a t e d with the Norwegian Employers' Confederation. 2) Wholesale price index for fuel and e l e c t r i c i t y . 3) Wholesale price index for raw materials (except f u e l ) . Materials include, in order of magnitude based on 1977 figures: miscellaneous costs, telephone expenses, har-bour fees, some items of hired labour, including social costs, ice, etc. 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 costs, i t would not matter i f another index were used. 4) The weights for the period 1969-73 were used for the years 1963-68 as well. Source: Stat i s t i c a l Yearbook, Central Bureau of Statistics of Norway The particular wage index that i s chosen is average earnings in the manufacturing industry. We have used this rather than e.g. ex post seasonal returns in the fishing industry. This is because the - 121 -manufacturing industry represents important alternative employment opportunities for fishermen. For this reason changes in wages in the fishing industry w i l l over time have to correspond to those in industry. The Cost/Price Ratio for the Sole Owner While an index is sufficient for the estimation of the supply function, this i s not the case for the cost/price ratio that is used in the estimation of the optimal stock level (S*). In this case, the abso-lute value of the cost/price ratio matters, although S* i s , of course, linearly homogeneous to proportional changes in the cost of effort and the output price. The supply function was estimated using data for 1963-77. The optimal stock l e v e l , however, is the target one would aim at achieving at some time in the future. Therefore, present day, or ideally, expec-ted future, price and cost data should be used, rather than historic data. The optimal stock level 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 in the estimation of the supply function, this may be different 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 capital. - 122 -Category 1 essentially corresponds to the cost of effort, c^, that was discussed above. In Section 2.3 i t is argued that the opportunity cost of capital i s low. In this analysis, i t w i l l be set equal to zero. Two cost alternatives w i l l be considered: 1. Variable costs only. This essentially corresponds to the case that has been considered. 2. Variable costs plus insurance and maintenance costs. The remuneration of fishermen in the Norwegian purse seine fleet 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 share—to be determined s h o r t l y — w i l l accrue to the capital owner, i.e., the sole owner of this analysis. This means that variable costs w i l l include only fuel and materials. Table A4 gives average annual operating costs for purse seiners for the period 1975-81, while daily costs—expressed in real prices—are given in Table A5. - 123 -Table A4: Average Annual Operating Costs for Norwegian Purse Seiners 1975-81. Figures in kroner  Year Fuel Materials Insurance Maintenance Number of operating days Wholesale price index 1975 327,405 86,892 156,780 646,427 244 60 1976 468,228 110,914 201,189 896,663 237 65 1977 478,629 148,922 191,702 844,734 221 69 1978 501,840 141,754 221,974 799,659 243 72 1979 642,034 157,552 249,371 779,323 225 78 1980 752,668 175,181 202,023 734,086 198 90 1981 1,010,641 224,894 207,871 835,897 206 100 Sources: Cost data: Annual Cost Investigations 1975-81, The Budget Committee for the Fishing Industry (Budsjettkomiteen for fiskerinaeringen) Price index: Statistical Yearbook 1983, Central Bureau of Statistics of Norway Table A5: Average Daily Operating Costs for Norwegian Purse Seiners. Figures in 1975 kroner  Year Variable costs Insurance and maintenance Total costs 1975 1,698 3,292 4,990 1976 2,256 4,276 6,532 1977 2,470 4,078 6,548 1978 2,208 3,503 5,711 1979 2,734 3,517 6,251 1980 3,124 3,152 6,276 1981 3,599 3,040 6,639 1) Conversion into real prices by means of wholesale price index given in Table A4. - 124 -A number of comments are required. F i r s t , as noted in Section 2.3, the boats in question participate in a number of fisheries. It is here assumed that daily operating costs are equal in the different fisheries. Second, insurance and to some extent maintenance costs are really fixed costs. The rationale for treating them as variable i s that they, in a given year, need to be covered by incomes earned during the operating days. Third, expenditures on maintenance are to some extent influenced by an incentive to reduce taxes. For this reason, and in 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 in a given year pertain to that year only. The figures in Table A5 show that daily variable costs have doubled in real terms over the period in question. This is essentially due to the development in o i l prices. On the other hand, there is no particular trend for insurance and maintenance costs. 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 daily cost of effort figures, expressed in 1975 kroner: Variable: cl = kr. 3,362.00 Variable, insurance and maintenance: c2 - kr. 6,913.00 It i s noticeable that the inclusion of insurance and maintenance costs more than double daily operating costs. - 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 in Table A2. The high prices in the last years of the period are due to small quantities combined with a high proportion of deliveries for human consumption. Table A6: Norwegian Price Data Average price of herring Year Kroner/tonne 1963 387 1964 339 1965 342 1966 357 1967 235 1968 213 1969 309 1970 437 1971 408 1972 358 1973 640 1974 830 1975 1,226 1976 1,422 1977 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% for 1979-811\ 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 shift towards more capital intensive boats. Some crew members (skipper, engine chief and cook) receive fixed remuneration in addition to their shares. The error that i s caused by treating these fixed payments as variable is negligible. - 126 -2) fish . This means that the sole owner's share amounts to 61%. As discussed above, herring is used for reduction into fish meal and fish o i l and for human consumption. The fish meal market and the demand for fi s h meal is analyzed by Hansen (1979). With the harvest quantities corresponding to the optimal stock levels estimated in Chapter 4, i t is quite l i k e l y that the major part of the harvest wil l be used for reduction purposes. Hansen et a l . (1978) establish price forecasts for fish meal and fi s h o i l and use these to derive f i r s t hand prices for different species. Ideally, a more recent forecast would be preferred, but apparently this work has not been updated. It is judged to be beyond the context of this thesis to undertake such a task. The price forecast is derived as follows: Price forecast, based on Hansen et a l . (1978) for the period 1981-85, per tonne North Sea herring (1975 kroner): kr. 380.00 20% extra for consumption deliveries: kr. 76.00 Average f i r s t hand price: kr. 456.00 Crew share and product fee (39%): kr. 178.00 Net price: kr. 278.00 This gives rise to the following cost/price ratios, where the price per 1,000 tonnes has been used: 1. ci/p = 0.0121 2. c2/p = 0.0249 This completes the analysis of costs and prices. 2) Source: Cost investigation for 1979, Budget Committee for the Fishing Industry. The fee is supposed to cover employers' pension contributions on behalf of their employees. 

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