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Input substitution and rent dissipation in a limited entry fishery : a case study of the British Columbia… Dupont, Diane Pearl 1988

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INPUT SUBSTITUTION A N D RENT DISSIPATION IN A LIMITED ENTRY FISHERY CASE STUDY O F THE BRITISH COLUMBIA COMMERCIAL S A L M O N FISHERY by DIANE P. D U P O N T B.A.(Honours), Carleton University, 1978 M.A., The University of Toronto, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DECREE OF D O C T O R OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Economics We accept this thesis as conforming to the required standard THE UNIVERSITY O F BRITISH COLUMBIA January 1988 © Diane P. Dupont. 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of E c o n o m i c s  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 20 J a n u a r y 1988 DE-6(3/81) ABSTRACT Entry-limiting regulations imposed on common property fisheries have been suspected of encouraging fishermen to substitute unregulated for regulated inputs. This imposes a cost upon society in the form of a reduced amount of resource rent generated by the fishery. Almost no research has been done to provide quantitative estimates of substitution possibilities and the associated degree of rent dissipation. The thesis provides the first estimates of the harvest technology for the British Columbia commercial salmon fishery, one of the first fisheries in North America to experiment with limited entry controls. Estimates of cross-price elasticities of input demand and of elasticities of intensity are given. These elasticities exhibit a greater degree of input substitutability than has heretofore been assumed in the theoretical literature. Two of the four vessel types used in the fishery are observed to be responsible for most of the resource rent dissipation. Potential rent for 1982 is shown to be $73.1 million. This represents 44% of the total value of the landed catch. Actual rent for the 1982 season is estimated to be -$42.8 million. A model of a fishing firm subject to input restrictions is developed in the thesis. The empirical model uses a flexible functional form proposed by Diewert and Ostensoe (1987). The major advantage of the normalized, quadratic, restricted profit function over the translog is its ability to distinguish differing degrees of input substitution between pairs of inputs, while imposing convexity in prices upon the functional estimates. The function is estimated for one output, three variable inputs, and three restricted inputs. Four samples are used which correspond to the vessel types that fish salmon. This allows rent to be calculated for the entire fleet, as ii well as for each of the components. The study of the salmon fishery is completed by addressing the important issue of rent dissipation. The actual amount of rent is established by using the predicted input demands of each vessel to calculate total fleet costs for the number of vessels that fished in 1982. This is compared to the potential rent that would be generated by an efficient fleet. To determine the characteristics of the efficient fleet, the optimal amount of (the restricted) net tonnage for each vessel is determined. Predicted output levels for each vessel are then used to calculate the minimum number of vessels required to take the 1982 harvest. This is done for each of the four vessel types. This exercise is repeated for two alternative scenarios, including the assumption of a greater degree of substitutability per vessel than actually found and a change in the distribution of catch among the vessel types. A comparison of rents generated in each scenario with an estimate of the actual rent from the 1982 fishery suggests that input-substituting activities of the fishermen may cause a substantial amount of rent dissipation. In addition, fleet redundancy and an inefficient catch distribution are found to contribute to the problem. The thesis concludes with a discussion of the implications for effective fisheries management. In particular, the findings of the research endorse the (Pearse) Royal Commission on Pacific Fisheries Policy (1982) recommendation of a fleet reduction scheme to be used in conjunction with a royalty tax on catch. O n the other hand, evidence of input substitutability suggests that a vessel quota restriction might be successful in preventing some rent from being dissipated. iii T A B L E O F C O N T E N T S Abstract " Table of Contents iv List of Tables . vi Acknowledgement x I. Introduction 1 A. Overview 1 B. Hypotheses to be Tested 4 C. Findings of the Research 6 D. Conclusions .'. 10 II. The British Columbia Commercial Salmon Fishery 13 A. Description of the Fishery 13 B. History of License Limitation 15 1. Pre-License Limitation 15 2. The License Limitation Program 16 C. Conclusions 18 III. License Limitation: Theory and Practice 20 A. The Theory Behind License Limitation Programs 20 1. A Survey of the Theoretical Literature 20 2. Requirements for an Effective Program 24 B. Empirical Studies of License Limitation Programs 32 1. Non-Production Function Studies 32 2. Production Studies 35 IV. Modeling the Behaviour of the Regulated Fishing Firm 37 A. The Direct Harvest Production Function 38 1. Specification of the Direct Harvest Production Function 38 2. Problems with the Direct Production Function Approach 41 B. The Dual Approach: The Restricted Profit Function 44 1. A Short-Run, Restricted Profit Maximizing Model 44 2. Relationship Between the Primal and the Dual 51 C. Characterizing Technology Using Duality 53 1. Supply and Demand Functions 53 2. Elasticities of Interest 54 3. Shadow Prices of Restricted Factors 56 V. Econometric Technique and Results 58 A. Econometric Model 58 1. The Normalized, Quadratic, Restricted Profit Function 59 2. Data 65 B. Econometric Technique 71 1. Linear Case 71 2. Nonlinear Case 77 C. Econometric Results 85 1. Own- and Cross-Price Elasticities of Supply and Demand 85 2. Elasticities of Intensity 102 iv 3. Returns to Scale 118 D. Conclusions 123 VI. Measuring Fishery Rent Dissipation 124 A. The Calculation of Fishery Resource Rent 125 1. Theoretical Measures of Fishery Rent 125 2. Empirical Measures of Fishery Rent 129 B. Methodology to Obtain Estimates of Fishery Rent 136 1. Determining the Optimal Levels of Variable Quantities 137 2. Calculating the Optimal Net Tonnage 143 C. Fishery Rent and Rent Dissipation 149 1. Within Sample Rent 150 a. Case I: Actual 1982 Rent 150 b. Case II: Optimal Tonnage Per Vessel 154 c. Case III: An Increase in Substitution Possibilities 160 2. Industry Rent , 162 a. Case I: Actual 1982 Rent 164 b. Case II: Optimal Tonnage Per Vessel 176 c. Case III: An Increase in Substitution Possibilites 179 d. Case IV: Single Vessel Type Harvesting 182 D. Conclusions 183 VII. Conclusions and Directions for Future Research 186 Bibliography 192 Appendix 1 : Data Construction 209 A. Data Sources 209 1. The 1982 Survey of Pacific Coast Vessel Owners 210 2. Sales Slip Data for 1982 211 B. Vessel Selection and Data Transformation 212 1. Vessel Selection 212 2. Data Generation 214 a. The Labour Variable. Price and Quantity 215 b. The Fuel Variable: Price and Quantity 222 c. The Gear Variable: Price and Quantity 223 d. The Output Variable: Price and Quantity 227 e. Restricted or Fixed Inputs 228 C. Is 1982 A Representative Year? 231 Appendix 2 : Parameter Estimates - Nonlinear Case 242 Appendix 3 : Parameter Estimates, Tests, And Results - Linear Case 248 Appendix 4 : Elasticity And Shadow Value Formulae 265 Appendix 5 : Calculations For Chapter 6 269 v LIST OF TABLES Table 5.1-.--Eigenvalues from linear estimation: four vessel types 74 Table 5.2:-Eigenvalues from nonlinear estimation: three vessel types 76 Table 5.3:--Testing for constant returns to scale: four vessel types 79 Table 5.4:--Coodness of fit: four vessel types 81 Table 5.5:-A comparison of the linear and nonlinear log-likelihood functions 83 Table 5.6:--Nonlinear estimates of output-variable own- and cross-price elasticities: seine 86 Table 5.7:--Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet(crs) 88 Table 5.8:-Linear estimates of output-variable own- and cross-price elasticities: troll 90 Table 5.9:~Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet-troll(non-crs) 92 Table 5.10:-Nonlinear estimates of output-constant price elasticities: seine 94 Table 5.11:--Nonlinear estimates of output-constant price elasticities: gillnet(crs) ....96 Table 5.12:-Linear estimates of output-constant price elasticities: troll 98 Table 5.13:—Nonlinear estimates of output-constant price elasticities: gillnet-troll(non-crs) 100 Table 5.14:--NonIinear estimates of elasticities of intensity: seine 103 Table 5.15:—Nonlinear estimates of elasticities of intensity: gillnet(crs) 105 Table 5.16:-Linear estimates of elasticities: troll 107 Table 5.17:—Nonlinear estimates of elasticities of intensity: gillnet-troll(non-crs) 109 Table 5.18:-Nonlinear estimates of output-variable own- and cross-price elasticities: gilinet(non-crs) 111 Table 5.19:-Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet-troll(crs) 113 Table 5.20:-Nonlinear estimates of output-constant price elasticities: gillnet(non-crs) ....115 Table 5.21:-Nonlinear estimates of output-constant price elasticities: gillnet-troll(crs) ....117 Table 5.22:-Nonlinear estimates of elasticities of intensity: gilinet(non-crs) 119 Table 5.23:-Nonlinear estimates of elasticities of intensity: gillnet-troll(crs) 121 vi Table 6.1:--Estimated market rental prices and shadow prices per net ton: four vessel types 148 Table 6.2:--Total within sample rents (using mean vessel): all vessel types, all cases .151 Table 6.3:--Total within sample rents (using all vessels): all vessel types, all cases 153 Table 6.4:-Sample mean net tonnage and predicted optimal mean net tonnage per vessel: all cases 155 Table 6.5:--1982 salmon catch and landed value, by vessel type 157 Table 6.6:--Actual number of vessels and estimated minimum number of vessels (using mean vessel): all vessel types, all cases 161 Table 6.7:-Estimated actual and optimal fleet net tonnage (using mean vessel): all vessel types, all cases 163 Table 6.8:--Estimated total fishery rent (using mean vessel): all vessel types, all cases 165 Table 6.9:--Estimated fishery rent per vessel (using mean vessel): all cases 167 Table 6.10:--Estimated fishery rent per ton (using mean vessel): all cases 169 Table 6.11:--Actual number of vessels and estimated minimum number of vessels (using all vessels): all vessel types, all cases 171 Table 6.12:~Estimated actual and optimal fleet net tonnage (using all vessels): all vessel types, all cases 173 Table 6.13:—Estimated fishery rents (using all vessels): all vessel types, all cases 175 Table 6.14:--Estimated fishery rent per vessel (using all vessels): all cases 177 Table 6.15:--Estimated fishery rent per ton (using all vessls): all cases 180 vii Table A1.1:--Vessel characteristics and expenditures: seine 234 Table A1.2:--Vessel characteristics and expenditures: gillnet 235 Table Al.3:--Vessel characteristics and expenditures: troll • • • • 2 3 6 Table Al.4:~Vessel characteristics and expenditures: gillnet-troll 237 Table A1.5:-Average weekly earnings, British Columbia, current year dollars . - 2 3 8 Table A1.6:--Average weekly earnings and unemployment rate by region 239 Table A1.7:--Fuel prices by region 240 Table A1.8:--Representative bonus rates (%) by species and gear type 241 Table A2.1:--Nonlinear parameter estimates: seine 243 Table A2.2:--Nonlinear parameter estimates: gillnet(crs) 244 Table A2.3:--Nonlinear parameter estimates: gillnet(non-crs) . 245 Table A2.4:-Nonlinear parameter estimates: gillnet-troll(crs) 246 Table A2.5:-Nonlinear parameter estimates: gillnet-troll(non-crs) 247 Table A3.1:--Linear parameter estimates: seine 255 Table A3.2:--Linear parameter estimates: gillnet(crs) 256 Table A3.3:—Linear parameter estimates: troll 257 Table A3.4:--Linear parameter estimates: gillnet-troll(non-crs) 258 Table A3.5:--Linear parameter estimates: gillnet(non-crs) 259 Table A3.6:--Testing for symmetry: all samples (linear estimates) 260 Table A3.7:-Testing for constant returns to scale: all samples (linear estimates) 260 Table A3.8:--Linear estimates of output-variable own- and cross-price elasticities: seine 261 viii Table A3.9:-Linear estimates of output-variable own- and cross-price elasticities: gillnet(crs) 261 Table A3.10:~Linear estimates of output-variable own- and cross-price elasticities: gillnet-troll(non-crs) 262 Table A3.11:-Linear estimates of output-variable own- and cross-price elasticities: gillnet(non-crs) 262 Table A3.12:-Linear estimates of elasticities of intensity: seine 263 Table A3.13:--Linear estimates of elasticities of intensity: gillnet(crs) 263 Table A3.14:--Linear estimates of elasticities of intensity: gillnet-troll(non-crs) . . . 2 6 4 Table A3.15:-Linear estimates of elasticities of intensity: gillnet(non-crs) 264 Table A5.1:--Mean predicted quantities and expenditures per vessel (using mean vessel), all samples: Case I 269 Table A5.2:--Mean predicted quantities and expenditures per vessel (using all vessels), all samples: Case I 270 Table A5.3:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), seine: Case II 271 Table A5.4:—Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet: Case II 272 Table A5.5:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), troll: Case II 273 Table A5.6:—Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet-troll: Case II 274 Table A5.7:-Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), seine: Case III 275 Table A5.8:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet: Case 111 -. .276 Table A5.9:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), troll: Case III 277 Table A5.10:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet-troll: Case III 278 ix A C K N O W L E D G E M E N T For providing the guidance and support over the years taken to complete this thesis, I would like to thank my committee, Phil Neher (supervisor), Gordon Munro, and Bill Schworm. As well, other members of the Department of Economics at U.B.C. have offered assistance, including Don Paterson, Erwin Diewert, and Terry Wales. Many thanks are also due to the personnel of the Department of Fisheries and Oceans, in particular, Heather Fletcher, David Reid, and Paul MacGillivray, and to Frank Flynn, one of UBC's computing centre advisors. This list would not be complete without a mention of friends and family. Among the former are my colleagues in C-wing, especially Shelley Phipps, my office mate of many years, and Mary MacGregor, who never gave up hope. As well, Bunti deserves his share of thanks. Finally, I want to thank my parents, Frank and Ada Dupont, and my husband, Steven Renzetti, for their unfailing support and faith in me. Steven, especially, has shared my joys and sorrows, failures and successes, and his belief in my ability to overcome any problems has given me the courage to finish. It is to him that I dedicate this thesis. x I. I N T R O D U C T I O N A. OVERVIEW My thesis is an empirical study of the relationship between input substitution and rent dissipation by firms in a regulated industry. The chosen case study is the British Columbia commercial salmon fishery. This fishery has been subject to a limited entry licensing program since 1969 and has seen restrictions on the allowable net tonnage of each vessel since 1971. In spite of controls, whose aim is to prevent the excessive use of inputs to catch the salmon, it is often asserted that fishermen are able to subvert the intentions of the program. It is believed that they do so by substituting unregulated inputs for regulated ones. This action may raise the costs of harvesting a given level of catch, thereby reducing the amount of resource rent that may be obtained from the fishery. The issue of interest is the following. Does a restricted access program, that puts an upper bound on the quantity of a key input used in harvesting, effectively forestall rent dissipating behaviour? I examine this issue in two interrelated, but distinct, ways. First, the fish harvesting technology is characterized by means of the output supply and variable input demand functions. They, in turn, yield estimates of the cross-price elasticities of input demand and the elasticities of intensity between variable and restricted factors. These measures are summary statistics of the underlying production function for harvesting salmon and indicate the ease with which inputs are substituted for one another. Second, the estimated supply and demand functions are used to calculate fishery resource rents available under several alternative scenarios. By comparing these rents it is possible to obtain estimates of 1 Introduction / 2 the amount of rent dissipated through several channels, including the fishing firm's ability to substitute inputs against a restriction on a key input. The model in Chapter 4 describes the short-run behaviour of a profit-maximizing, competitive, fishing firm which is subject to restrictions on the use of several inputs.t The firm maximizes seasonal profits by choosing the quantity of output supplied (landed salmon) and the quantities of three variable inputs (labour, fuel, and gear). Constraints on the firm's behaviour arise from a regulation on the amount of net tonnage per vessel, a fixed number of fishing days, and the given abundance of fish during the season. In order to obtain estimates of the parameters of the fishing technology cross-sectional, micro-level data from the 1982 fishing season are constructed for the four vessel types operating in the fishery. These vessel types are: the seine, troll, gillnet-troll, and gillnet. The data appendix (Appendix I) discusses whether 1982 is representative of other fishing seasons. In brief, species prices and stock levels are not unlike those of most previous years, with the exception of the boom years of 1977-1979. However, beginning in 1982 interest rates are much higher than in the decade of the 1970's. Thus, one might expect the costs of some inputs to be higher than in previous years. To the extent that this is true, the rents calculated for 1982 are lower than they might be in previous years. The availability of both output and input data for each vessel in the sample allows the technology to be specified by a short-run, restricted profit function that is dual tFor convenience, the terms vessel-owner, fishing firm, and fisherman are used interchangeably. Introduction / 3 to the underlying harvest production function. The functional form chosen to represent the short-run restricted profit of a given vessel type is the normalized quadratic (Diewert and Ostensoe 1987). It is capable of imposing convexity without loss in flexibility as defined by Diewert (1974).t For each vessel type a separate restricted profit function is specified and estimated using either an iterative linear or nonlinear routine. This is discussed in Chapter 5. The estimated parameters are used to generate cross-price elasticities between pairs of variable inputs and elasticities of intensity between variable and fixed factors. Their values can be used to evaluate the success of the limited entry program. For example, if the regulator can control the output of a fishing firm merely by restricting the use of some key input, and the designated input has neither substitutes nor complements among the other inputs, then a limited entry program will be effective at preventing rent dissipation through input substitution. O n the other hand, evidence of input substitutability argues in favour of the use of a different regulatory scheme. Instead of relying upon direct, or quantitative, controls, the regulators might obtain success with an indirect, or price, control. This takes the form of a royalty that reduces the net price of landed catch to the fisherman, thereby discouraging excessive fishing effort. However, this tool is not without its problems. They include the appropriate choice of the tax rate in the absence of detailed knowledge of behaviour and the inevitable administrative lags associated with a change in the rate (Scott 1979, Crutchfield 1979). Choosing the correct form of regulation in the fishery has been a large part of tFlexibility means that the function is able to distinguish among the elasticities of substitution of the many inputs. Introduction / 4 the theoretical literature on fisheries for the past two decades (Scott 1979, Crutchfield 1979). One way of resolving the debate is to examine the effectiveness of potential schemes in light of empirical estimates of the fishing technology. This thesis generates the first empirical estimates of the four harvest technologies used to take salmon in British Columbia. In addition to examining the harvest technology, this thesis provides the first complete empirical examination of the extent of Type II rent dissipation (Munro and Scott 1985) in the British Columbia commercial salmon fishery. In Chapter 6 the estimated output supply and input demand equations from Chapter 5 are used to obtain projected revenues, costs, and fishery rents. This is done for several alternative cases. A comparison of the potential rents in each case yields estimates of Type II rent dissipation attributable to four sources. They are input substitutability, inefficient tonnage restrictions, fleet redundancy, and an incorrect catch distribution among vessel types. B. HYPOTHESES TO BE TESTED There are a number of specific hypotheses that this thesis tests. They are discussed in turn, along with the role they have in this research. The first three hypotheses involve statistical testing of parameters of the estimated harvest technology. The last four are not statistical tests and relate to the issue of rent dissipation. 1. What are the relationships between the restricted factors and the variable inputs? In order to answer this question I evaluate the elasticities of intensity (Diewert 1974) between the variable and fixed factors and test to see whether they are significantly different from zero. Introduction / 5 Are pairs of variable inputs substitutes or complements in the harvesting of salmon? An examination of the signs and magnitudes of the cross-price elasticities of input demand gives the answer. Used in conjunction with the elasticities of intensity, these elasticities indicate the probable success or failure of input restrictions imposed on the fishing firm. Are fishing firms sensitive to changes in the price of output and how do input quantities vary when the output price changes? Answers to these questions are important for understanding how fishing firms respond to royalties or taxes which lower the net supply price. Is the actual level of the regulated factor (net tonnage) different from the optimal level? First, a solution to the optimal level of net tonnage per vessel is found. Then, the actual and optimal tonnage values are compared. If the fishery is found to have more tonnage than is optimal, then this provides evidence of overcapitalization in the fishery. Given the size of the total 1982 harvest of British Columbia salmon, can the minimum number of vessels be computed? How does the actual number of vessels compare to this minimum? If the actual number exceeds the minimum, then rent dissipation may have occured due to the presence of too many vessels. Several authors have asserted that the fleet could be reduced without impact on its aggregate harvesting ability. For example, Pearse (1982) suggests that the fleet size could be halved. Hilborn (1984) contends that the seine component of the fleet could be reduced by 80-90%. For the gillnet fleet, Loose (1975) estimates that 15-50% of the fleet is redundant. How does the degree of observed rent dissipation depend upon the degree of substitutability in the harvest technology? Introduction / 6 7. What amount of rent dissipation is attributable to each of the four vessel types used in the British Columbia commercial salmon fishery? What does this suggest for further regulation of particular vessel types? C. FINDINGS OF THE RESEARCH One output supply and three input demands are estimated using a linear iterative Zellner technique for simultaneous equations (Judge et al. 1982). This is done for each of the four vessel types used to harvest British Columbia salmon on a commercial basis. The functional form chosen for this study is the normalized quadratic (Diewert and Ostensoe 1987). It has been adopted only in one other applied research project (Lawrence 1987). Linear homogeneity in prices is imposed as a maintained hypothesis. Convexity in prices is tested for each of the four samples and is accepted in only one. It is subsequently imposed upon the three other samples. This procedure entails estimation of a new set of equations which are nonlinear. A nonlinear maximum likelihood technique is used to obtain the new parameter estimates. Using parameter estimates from the appropriate nonlinear and linear cases, the technology may be characterized by means of the own- and cross-price elasticities of input demand and the elasticities of intensity. In general, the values of the own-price elasticities are largest in the troll and gillnet-troll fleets. It is expected that these two fleets respond most to changes in input and output prices, as they are the least regulated components of the entire commercial fleet. In all cases, however, the own price elasticity of gear is very small. This suggests that fishermen are not responsive to changes in the price of gear. Introduction / 7 Whereas the troll fleet shows some degree of variable input substitution, the gillnet-troll fleet has none. O n the other hand, the seine and gillnet fleets give evidence of some amount of substitution among the variable inputs. An analysis of these results is consonant with conditions in the fishery. The net fisheries, seine and gillnet, are more heavily regulated than are the troll fisheries in terms of area openings and times of allowed operation. A natural response by the more heavily regulated segments of the fleet is to exploit a harvesting technology which demonstrates a greater degree of substitution among the variable inputs. Elasticities of intensity are calculated in order to examine the relationships between the variable and fixed factors. Results for the seine and troll samples indicate that labour and gear are direct substitutes for net tonnage. However, fuel is a complement to these inputs. This means that more of all three variable inputs is used in the face of a net tonnage restriction per vessel. This finding suggests that a great deal of rent can be dissipated by these two fleets by virtue of their ability to exploit the technology in order to subvert the intentions of the British Columbia limited entry program. At the opposite end of the scale are the gillnet and gillnet-troll vessels. Given that they have no direct substitutes for net tonnage, there seems much less scope for rent dissipating activities by these fleets. However, they may still dissipate rent by attempting to increase their output levels. This leads to the use of more variable inputs for a given amount of tonnage, thereby raising harvesting costs. In general, the elasticity estimates provide evidence that the commercial salmon harvesting technology possesses a greater degree of substitution than has been Introduction / 8 suggested by observers. Therefore, the next step is to generate estimates of the amount of fishery rent actually dissipated. Using the estimated output supply and input demand equations from Chapter 5 a methodology is developed in Chapter 6 to determine actual and potential fishery rent. I first calculate within sample fishery rent for the 245 vessels used to obtain estimates of the harvest technology. This rent corresponds to the actual rent for 1982 and is generated as the sum of the net profit for each vessel, where net profit is defined as seasonal (restricted) profit minus the fixed cost associated with net tonnage. Seasonal profit is determined for each vessel by using the estimated output supply and input demand equations of Chapter 5 to predict the optfmal quantities of these variables per vessel. The output and input decisions are optimal given a regulatory environment that precludes a first-best solution. Multiplying predicted output by the mean sample-specific output price gives an estimate of revenue; likewise, the expenditure on each variable input is equal to the quantity of each input times its price. Total variable cost is the sum of expenditures on variable inputs. Revenue minus variable cost yields an estimate of seasonal profit per vessel. To obtain the fixed cost associated with each vessel the net tonnage is multiplied by the market rental (flow) price of a ton. Rent to a vessel is seasonal profit minus total fixed costs. Hence, rent per vessel, as defined in this thesis, is static, one period net profit per vessel. Thus, it measures neither congestion nor stock externalities. However, it is capable of distinguishing the inframarginal rents of different vessels. Actual within sample rent for 245 vessels is estimated to be -$572,000 for 1982 in Introduction / 9 current dollars. The within sample results are then extrapolated to determine total industry rent. This is estimated to be -$42,789,000 for the entire salmon fishing fleet of 4528 vessels in 1982. I call this Case I.a. The extent of these negative rents may be compared to the total landed value of the salmon catch. For the 1982 fishing season it is $164,933,000. The troll fleet, comprising about 25% of the total fleet, contributes most to the negative rent. The finding of large negative rents may arise because of the high proportion of fixed to variable costs, although the proportion varies by vessel type as shown by the tables in Appendix 5. The rent for the troll fleet is estimated at -$33,017,000. Only the gillnet-troll fleet is seen to produce a positive rent equal to $804,000. From the results generated for the industry, it is clear that too many vessels are employed in the fishery. An estimate of the number of surplus vessels is 1078, approximately 25% of the fleet. Taking these vessels out of the fishery would cause total fishery rents to increase to $12,477,000, an increase of $55,266,000. I call this Case Lb. Fleet redundancy is a major contributor to the loss of potential fishery rent. A second exercise is performed for the industry to determine the extent to which potential fishery rent could be increased, if restrictions on the use of net tonnage per vessel were relaxed (Case II). Total industry rent would increase to $31,387,000. This represents an increase of $18,910,000 in rent from the Case l.b, since this latter is achieved by keeping each vessel at its actual level of net tonnage. In both cases, Case l.b and Case II, the minimum number of vessels required to take the 1982 harvest is used in the rent calculation. Introduction / 10 The third case examines the role of input substitution in leading to the loss of potential fishery rent. In particular, a simulated doubling of the substitution possibilities causes total industry rent to fall to -$5,878,000. As suggested by the previous discussion of the elasticities, the greatest amount of rent is dissipated by the seine and troll fleets. A final exercise, Case IV, is performed as a means of answering a question that has been asked by many observers of the fishery. The question is whether seine vessels are more efficient harvesters of the salmon than the three other vessel types. I find that a seine fleet harvesting the entire 1982 catch alone could generate a resource rent of $73,090,000 or 44% of the total landed value of the catch in 1982. The next largest rent, $27,198,000, would be generated by an all-gillnet fleet. The troll fleet is found never to generate a positive resource rent. D. CONCLUSIONS The findings of this thesis are important for two reasons. First, the approach taken is found to be a potentially useful one for empirical fisheries studies of the type called for by Zellner (1970) and, more recently, by Munro and Scott (1985). One of the novel aspects is the use of a restricted profit function to characterize the underlying harvest technology. This allows a complete exploration of the impacts of input controls on each vessel-owner's optimal seasonal behaviour. In addition, shadow prices for the restricted inputs may be calculated, along with the optimal levels of these factors. This type of model also lends itself readily to a new methodology which can generate estimates of fishery rent under a variety of scenarios. This permits a comparison of the partial static equilibrium level of rents Introduction / 11 (obtained by using the actual levels of the restricted factors) to the full static equilibrium level of rents (obtained by using the optimal levels of the restricted factors) (Brown and Christensen 1979). This is a particularly fruitful area of research given the intertemporal nature of a fishery, since it allows a comparison to be made of short-run and long-run solutions (Capalbo 1986). Second, the empirical results themselves have implications for the optimal management of the British Columbia commercial salmon fishery, and, by extension, for other limited entry fisheries. This thesis provides the first empirical estimates of the harvest technologies used in the British Columbia salmon fishery. The calculated elasticities indicate that there is more input substitution than has been previously believed. The ability to substitute inputs may be one conduit through which fisheries rent is lost. There are, however, others. The thesis highlights the role of several other channels through which rent may be lost. They are the presence of too many vessels, the use of inefficient tonnage restrictions, and the regulator's adoption of an inefficient distribution of the catch across vessel types. According to the rent calculations done in this thesis, fleet redundancy appears to be a serious problem in the British Columbia commercial salmon fishery. This finding accords with the accepted wisdom. For example, Pearse (1982) suggests phasing out 50% of the fleet over a 10 year period. The research conducted supports his choice of target. The Fleet Reduction Committee (1982) suggests a similar reduction in the number of vessels. In addition, Pearse calls for the imposition of royalties set at $0.05 or $0.10 per pound of landed catch depending on the species of salmon. The findings of this thesis indicate that this royalty may serve to reduce Introduction / 12 the quantity of inputs used to catch a given level of output, but would have little impact upon the output supply of each vessel type, with the exception of the troll fleet. This policy would seem, then, to achieve the best of two worlds, ie., maintain the level of catch while employing fewer inputs. Given that tonnage restrictions and input substitution activities lead to rent dissipation and given the administrative difficulties inherent in a royalty system, an alternative form of regulation may be suggested. It is quantitative restrictions on catch per vessel, or vessel quotas. The potential usefulness of this tool is currently generating a great amount of interest in the fisheries literature (Moloney and Pearse 1979, Copes 1982). Since each vessel is allowed to take a given amount of the harvest, the onus is then put on the vessel-owner to minimize the cost of taking his allocation. By allowing quotas to be traded, a solution might be found to the inefficiency associated with a suboptimal distribution of the catch among vessel types. This form of regulation has been considered and rejected for use in the salmon fishery, although it has been adopted for use in the halibut fishery. The major difference between the two fisheries stems from the greater degree of uncertainty about stock levels and, therefore, about the total allowable catch in the salmon fishery. Nonetheless, it is arguable that the adoption of a quota system might render ineffective a number of sources of rent dissipating behaviour. II. THE BRITISH COLUMBIA COMMERCIAL SALMON FISHERY A. DESCRIPTION OF THE FISHERY The commercial salmon fishery in British Columbia is the province's most lucrative and important fishery. In 1982 the total quantity of commercial salmon landed accounted for only 40% of total quantity (in metric tonnes), but 69% of the landed value.t Although the wholesale value of the 1982 salmon catch represents only one percent of gross provincial product, the regional importance of this fishery is significant. More importantly, many have suggested that the fishery ranks very highly among the potential resource rent generating fisheries in Canada (Scott and Neher 1981). Salmon is a highly valued resource and has the potential to be caught at low cost. This suggests that the fishery may yield a great deal of resource rent for Canada. Of course, the potential rent must be weighed against the costs necessary to administer the fishery in an efficient manner (Scott and Neher 1981). These costs may be quite substantial. Currently, it is not certain whether the fishery is generating a positive resource rent. Many observers argue that there are too many licensed vessels and that too much harvesting capacity is directed at the fishery. In British Columbia three gear types (seine, gillnet, and troll) are used in the harvesting of the five Pacific Coast salmon species, ie., pink, chum, chinook, coho, and sockeye. Typically, only one type of gear is employed upon a fishing vessel. However, since the 1970's there has been a trend toward the use of multiple gear tProvince of British Columbia. Ministry of Environment. Marine Resources Branch. Fisheries Production Statistics of British Columbia 1982. Victoria, 1983. The British Columbia Commercial Salmon Fishery / 14 types on a vessel. The combination used most frequently is gillnets and troll lines; the vessel is called a gillnet-troller. These vessels are able to target upon a particular species at will, responding especially to changes in regulations. Each vessel type exhibits a variety of characteristics that vary according to the gear used. Seine and gillnet vessels use nets to entrap the salmon, whereas troll vessels make use of lines, hooks, and bait to entice the salmon. The former two vessel types congregate at river mouths; the latter fishes many miles off-shore in the areas where the salmon feed prior to the spawning runs. More often than not troll vessels are equipped with freezers. They maintain the salmon's high quality over the many weeks spent at sea prior to official landing of the catch. Net-equipped vessels deliver their catches to the home port or to packing vessels as frequently as specific harvesting conditions warrant. Trailers have traditionally caught chinook and coho, but have added pinks and sockeye recently to their targeted species. These catches are mostly destined for the fresh/frozen market as final products. The predominantly net-caught species of pink, sockeye, and chum comprise the major inputs into the secondary processing and canning industry. A large part of the fleet, particularly the gillnet and troll vessels, is owned by small-scale operators. Some 10% of total licensed vessels are owned by a few processing companies. The British Columbia Commercial Salmon Fishery / 15 B. HISTORY OF LICENSE LIMITATION This section gives a brief history of regulation in the fishery and responses of the fishermen. Interested readers are advised to consult more thorough discussions found in Sinclair (1978), Morehouse and Rogers (1980), Fraser (1977, 1979), and Pearse and Wilen (1979). 1. Pre-License Limitation Prior to 1969 entry into the British Columbia commercial salmon fishery was unrestricted.t However, for conservation purposes the fishery was managed by means of annual total allowable catch restrictions. The total allowable catch, or TAC, was defined as the 'difference between the estimated run size and the required escapement of fish for spawning purposes. This management tool was derived from the prescriptions of the Schaefer biological growth model which dominated fisheries research for a long period of time. It argued that the harvest rate of the fishery should be regulated to provide the maximum sustainable yield. In practice, this meant setting the annual total catch to achieve a biological optimum. In spite of regulatory effort, annual fish stocks continued to decline in the face of a great deal of excess fishing capacity. In 1960 a commission headed by Sol Sinclair was created to study the salmon and halibut fisheries. Its major objective was to make long-term recommendations to support the continued viability of the fisheries. Sinclair (1960, 1978) proposed "that a limited licensing system be introduced to control the number of boats and tA licensing system was adopted in 1882, but it did not effectively restrain the entry of new vessels. In 1889 a limited number of licenses were created; again the program proved to be ineffective. The program was finally abandoned in 1917 (Sinclair 1978, Scott and Neher 1981, Rettig 1984). The British Columbia Commercial Salmon Fishery / 16 fishermen."t in order to pay for the administration of the program he suggested the introduction of "meaningful" license fees. 2. The License Limitation Program Almost ten years passed before the Canadian government responded to Sinclair's proposals.+ in 1969 a plan was introduced to freeze and then reduce the number of vessels in the salmon fleet. Licenses were issued only to vessels with an established presence in the fishery, a practice called "grandfathering" (Scott 1979, Copes 1980). In the first year of operation more than 6100 licenses were issued, but by 1982 this number had fallen to 4638.* At the program's inception no further plans were made to control any other aspect of fishing effort. Fishermen, although fewer in number, still competed for an uncertain and unspecified share of the annual total allowable catch. During the 1970's salmon prices began to rise rapidly. Together with the restriction on the entry of new vessels and a buy-back program to reduce the size of the fleet, this led to the generation of positive resource rents. Responding to this incentive in the manner predicted a decade earlier by Scott (1962), individual vessel-owners made additions to their old vessels and/or purchased larger and newer ones which incidentally added to the total fishing capability of the fleet. In 1971 tCanada. Fisheries and Oceans Canada. _A_ Licencing and Fee System for the Coastal  Fisheries of British Columbia Volumes I and II by S. Sinclair, Ottawa, 1978. p.26. tUnder the British North America Act ocean fisheries were administered by the federal government, not by individual provincial governments. * Canada. Fisheries and Oceans Canada. Turning the Tide: _A New Policy for Canada's  Pacific Fisheries. Final Report of the Royal Commission on Pacific Fisheries Policy by P. Pearse. Ottawa. 1982 p. 80. Also, op.cit., Fisheries Production Statistics of British Columbia 1982. Table 22. The reduction in the number of licenses can be attributed to several causes including: a buy-back program in the early 1970's and a reduction in the number of temporary licenses as their invalidation dates came due. The British Columbia Commercial Salmon Fishery / 17 regulators attempted to stem investments that took the form of increasing the net tonnage of the fleet. They announced a restriction on the net tonnage of the replacing vessel. The concepts of net and gross tonnage were not well defined. Loosely speaking, net tonnage was a measure of hold capacity; gross tonnage was a measure of the total displacement of the vessel. Unfortunately, the ton-for-ton replacement rule did not stop the net tonnage per vessel from increasing. Fishermen continued to subvert the intentions of the regulations by retiring two vessels (and licenses) and pyramiding their combined tonnage into a single new vessel. This more powerful vessel was generally capable of travelling to a larger number of fishing areas than the two previous vessels, and could do so in a shorter period of time. In addition to tonnage restrictions on each vessel the regulator began to use another management tool. It consisted of closing specific fishing areas to particular segments of the fleet during certain periods of the year. In this way the number of fishing days could be reduced, and access to the fish controlled. From 1977 to 1980 a series of regulations was introduced to make pyramiding more difficult (in 1977) and illegal (in 1980). This effectively put a moratorium on the entry of seine vessels into the fleet. These vessels were the largest and most powerful in the entire fleet. In addition, the year 1979 saw a revision of the ton-for-ton replacement rule to incorporate a length restriction. This was done to counteract the difficulty of relying on the Canada Shipping Act as a source of information and monitoring. This act dealt with the registration of vessels of Canadian origin. However, only vessels over 15 net tons were required to register, a procedure which required official confirmation of net tonnage. This included most The British Columbia Commercial Salmon Fishery / 18 seiners and some very large trailers. Smaller vessels, most gillnetters, trailers, and gillnet-trollers, had the option to register; most chose not to exercise it. Prior to the length-for-length replacement rule many of the smaller boats had an incentive to claim a larger net tonnage for their boat. Without official records these claims were difficult to disprove. During the period of 1981-1982 the salmon fishery was one of the main subjects investigated by the (Pearse) Royal Commission on Pacific Fisheries Policy (1982). Among his many recommendations, Pearse suggested a major revamping of the licensing system to cut the salmon fleet in half. This was to take 10 years and be phased in by means of a competitive auction for licenses. In this way the government would be able to appropriate fishery rent from the fishermen. In conjunction with the fleet reduction scheme, Pearse suggested the imposition of royalties on the landed catch of salmon. Unfortunately, these recommendations did not receive the support of a committee established by the Department of Fisheries and Oceans. C. CONCLUSIONS One important observation can be made. Despite restrictions on the total number of vessels, on the net tonnage per vessel, and latterly, on the number of effective fishing days per vessel, fishermen seem to have found new ways to avoid the intention of the restrictions. This occurs via the substitution of unregulated for regulated inputs. The unregulated inputs can take the form of bigger engines, more electronic equipment, more and different gear types, more manpower, and more fuel to move the vessel faster to a larger number of fishing grounds. The The British Columbia Commercial Salmon Fishery / 19 fishermen have incentives to make these substitutions because of the nature of the limited entry program. It permits fishery or scarcity rents to arise but does not tax them away. In their attempts to appropriate the fish for themselves, the vessel-owners use too many resources, thereby dissipating the potential fishery rent. Some researchers believe that a limited entry program may even accelerate rent dissipating behaviour (Fraser 1977).t The main objectives of this thesis are to provide an empirical basis for testing the input substitution hypothesis and to find other means by which rent may be dissipated. tTownsend (1985) explicitly recognizes this behaviour in a discussion of six incentives faced by fishermen which encourage them to dissipate rent. III. LICENSE LIMITATION: THEORY AND PRACTICE The purpose of this chapter is to review a set of fisheries economics literature which is relevant to the thesis.+ Given the scope of the fisheries literature, this survey confines itself to examining that part of the literature concerned with modeling the rationale for and the efficacy of license limitation programs. General surveys of the fisheries literature are available in Clark (1976), Anderson (1977), Hannesson (1978), and Munro and Scott (1985). This chapter is divided into two major sections. In the first, the theoretical basis of license limitation programs is examined and the requirements for a successful scheme are enumerated. The second discusses the relevant empirical literature. A. THE THEORY BEHIND LICENSE LIMITATION PROGRAMS 1. A Survey of the Theoretical Literature The basis for the development of license limitation and other fisheries management programs is the belief that the equilibrium state of an unregulated open access fishery is socially sub-optimal. This conclusion is derived from seminal articles by Gordon (1954) and Scott (1955). These works demonstrate how the self-reproducing nature of a fish stock combined with easy capture may generate resource rent. Gordon defines this fishery resource rent as the difference between the sustainable revenue and the total cost of harvest effort.^ He then argues that the presence of tThe literature concerning the general design of production studies is reviewed in chapter 4. The set of studies which examines the measurement of rent dissipation is surveyed in chapter 6. +The sustainable revenue is the sustainable (biological) yield curve multiplied by the assumed constant price of fish. This curve often has the familiar parabolic shape because it adopts the Shaefer logistic growth function for the fish stock. 20 License Limitation: Theory and Practice / 21 positive resource rents in an unrestricted or open access fishery induces an increase in the level of effort in the form of new fishing units. Equilibrium in this fishery is characterized by the total dissipation of resource rents and a reduction in the resource stock below its unharvested equilibrium size. Gordon shows that this bionomic equilibrium is sub-optimal in the context of a static model; Scott extends the analysis and findings to a dynamic framework. In either case, the equilibrium is thought to be sub-optimal because the fishery could yield resource rents and maintain a larger biomass. The basis for the divergence of the unregulated equilibrium from the social optimum is the lack of property rights in the fish stock. Since there is no security of tenure, it does not pay a participant in an open access fishery to leave any fish in the sea. Each fisherman continues to increase his efforts to catch a portion of the uncertain stock, so long as positive rents persist. Furthermore, he does not -or is unable toconsider the impact of his activities, either on other fishermen or on the future availability of the stock. An important feature of these and subsequent studies is the characterization of the harvest production function. This is the means by which fishermen combine inputs to produce the output of landed fish. Typically, the output of the fishing unit is modeled as a function of an aggregate index of input use labelled "fishing effort" (Clark 1976, Anderson 1977). Other features of this literature include: the Schaefer growth model for the biological production function, a cost of effort function that is linear in effort, and a perfectly elastic market demand curve for fish. License Limitation: Theory and Practice / 22 Before going on to consider the response of the economic literature to the fundamental conclusions of Gordon and Scott, it is important to clarify what is meant by the dissipation of resource rent.t Munro and Scott (1985) identify two facets of this phenomenon in a fishery with free access or imperfectly controlled access. They identify the Class I type of common property problem as the form of rent dissipation described by Gordon in his seminal work. This occurs when open access competitors drive the fish stock below its optimal sustainable level. This is the level that maximizes sustainable rent. One consequence of this economic overexploitation is that too few fish are left in the sea to contribute to future biomass growth. This implies higher harvesting costs. The Class II type of common property problem describes rent dissipation that occurs when the government restricts the total harvest, so as to maintain the biomass and prevent the Class I problem from arising. If access is not controlled or if it is imperfectly regulated, the presence of above average returns in the fishery encourages entry of new vessels. Further, even if entry of new vessels is prohibited, existing vessel-owners may be able to increase the harvesting capabilities of their boats. In any case, the minimum cost necessary to harvest the available catch is increased. Eventually this may lead to a situation where total costs have risen to the level of total revenue. This represents complete rent dissipation. The distinction between the two forms of rent dissipation is important because the regulatory responses to the two problems differ substantially. It has been suggested that an overall quota on the fish caught in any season solves the biological aspect tA more complete discussion is given in chapter 6. License Limitation: Theory and Practice / 23 of the overfishing problem. That is, the regulator determines the optimal total allowable catch (TAC) and permits operators to take only the prescribed quantity of fish. In theory, this policy is optimal because it allows a sustainable resource rent to emerge. However, unless the T A C is used in conjunction with a restriction on the entry of new fishing units, the presence of resource rents will continue to encourage entry and the dissipation of those rents. The apparent solution to the second type of rent dissipating behaviour is to regulate access to the fishery (Crutchfield 1961). The regulator tries to control the number of boats at some minimum level necessary to take the available TAC. The two most frequently advocated methods of achieving this objective are the imposition of a royalty tax on output and the setting of quantitative restrictions on input use. The royalty tax regulates access indirectly by reducing the net price for the catch, thereby forcing out marginal operators. This scheme has two advantages. First, the resource rent is captured by the government that manages the resource on behalf of the public. Second, the optimal number of boats need not be determined a priori by the regulator. The disadvantages of this program include the difficulty of calculating the correct royalty tax and the lag time between the imposition of the tax and a reduction in the level of effort. Despite these problems, a system of royalties has been proposed for several commercial fisheries, including the British Columbia salmon fishery (Pearse 1982). The scheme's obvious unpopularity with fishermen has prevented it from being put into practice. License Limitation: Theory and Practice / 24 The quantitative restriction directly controls the quantity of inputs employed in the fishery by defining the maximum use of a specified input. In order to adminster this program, a fixed number of licenses, equal to the desired quantity of the input, is created. In this way a license limitation program puts an upper bound on the amount of the licensed input. Theoretically, this scheme is easy to enforce because input quantities are readily observed. In addition, established fishermen can be convinced to support the scheme as a means of excluding "part-time" fishermen. The disadvantage of the program lies in the fact that the resource rent will probably accrue to whichever group of fishermen is permitted to stay in the fishery.t. Furthermore, fishermen still have incentives to subvert the intentions of the regulators by finding substitutes for the controlled inputs. The economic literature on license limitation includes many articles that debate the specific merits of various aspects of a licensing system (Crutchfield 1979, Scott 1979, Copes 1980). The areas of contention include: the choice of licensed input*, the free trading of licenses, and decisions about which groups should be licensed (Anderson 1977). These issues are distributional in nature and are not directly relevant to this thesis. 2 . Requirements for an Effective Program In this section the usual definition of the fishing input, fishing effort, is considered. Next, a more disaggregated view is presented which brings the discussion into line with neo-classical production models. The implications of input substitution between tThe government collects only a small portion of the potential rent in the form of a nominal licence fee ^Ideally, this should be a key input; in practice, it is usually the vessel or the fisherman. License Limitation: Theory and Practice / 25 components in the effort bundle for the success of license limitation programs are then made clear. The section concludes with the requirements for a successful licence limitation scheme. Since Gordon's seminal article in 1954 most economists have adopted the biologist's notion of fishing effort as the single input used by fishermen in the capture of the fish. It is derived from his concern with the impact of harvesting on fish mortality. Thus, as Rothschild (1972) points out, "...fishing effort is defined in terms of the catch: one unit of real or nonnominal effort is simply the numerical fraction of the average population that is caught."+ In practice, this has been defined in (constant) standardized units per vessel, eg., ton-hours of trawling, number of traps per year, etc., (Hannesson 1978, Roy, Schrank and Tsoa 1980). This simple notion of harvest effort as the single input at the fisherman's disposal obscures the true nature of the fishing production function and reduces the channels through which fishermen may respond to limited license restrictions. In fact, the fisherman uses many inputs and therefore, has many ways of subverting the intentions of the regulations. His responses may take three different forms, at least. These are the substitution of unrestricted for restricted inputs, a change in the quality of the restricted input, and technical change. Each may cause rent dissipation. The first and third responses have generated the most discussion in the literature (McConnell and Norton 1980, Townsend 1986). However, given the available cross-sectional data, it is not possible to explore the second and third tB.J. Rothschild, "An Exposition on the Definition of Fishing Effort" in Fishing  Bulletin. 7(3) United States. Department of Commerce. National Marine Fishery Service. 1972. 671-679. p.671. License Limitation: Theory and Practice / 26 responses. It is certain that they constitute important ways by which fishermen attempt to subvert the intentions of the regulations. Discussions about the efficacy of limited entry programs centre on how successfully the use a key input can be restricted so as to maintain positive resource rents. This depends on the degree to which the underlying fishing technology permits the substitution of unregulated for regulated inputs. For, if there is a great deal of substitutability in the technology, licensed participants may dissipate the entire resource rent through the excessive use of unrestricted and costly inputs.t The literature appears divided on the matter of input substitutability. As early as 1956, before the advent of the modern use of limited entry, fisheries economists recognize the possibility of open-access fishermen substituting one input for another (Crutchfield 1956). In a later article, Crutchfield (1961) asserts that substitution is not as likely in a controlled fishery. Scott (1962), on the other hand, argues that the ability to alter fishing techniques and the incentives to adopt new technical processes are enhanced under a system of limited entry. Pearse (1972), writing after the adoption in 1969 of the limited entry program for British Columbia salmon, claims that there is still a great deal of scope to alter fishing power in a limited entry situation, even with an additional input restriction on the tonnage of the fishing vessel. He also predicts that vessel licensing hastens the substitution of capital for labour in the fishery, but provides no explanation for this assertion. tThis is usually thought of as leading to Class II rent dissipation, although if there is no total allowable catch policy, it may also cause the Class I type of rent dissipation. License Limitation: Theory and Practice / 27 Copes (1980), commenting on the phenomenon, suggests that merely limiting the number of participants in an open-access fishery does not change the fundamental nature of the common property resource. In particular, restricted access does not confer upon the fisherman the right to a pre-specified number of fish. As a result, competition still exists among the fishermen, even though their numbers are reduced from the open-access situation. Each of the remaining fishermen tries to appropriate for himself the fish in the sea. To do so before one's competitors requires increasing the fishing capability of the vessel, in every other possible way. Christy (1974) argues that there are many forms of substitution available to limited entry competitors. Even if the regulators attempt to control the amount of inputs used by the limited number of participants, eg., each vessel's tonnage, length, etc., the fishermen may still be able to substitute away from the restricted components. Recent discussions of actual limited entry management programs argue in favour of the potential for substantial rent dissipation through this type of activity (Scott 1979, Copes 1980, Munro 1980, McConnell and Norton 1980, Neher and Scott 1981, Pearse 1982, Rettig 1984, Munro and Scott 1985). For example, if net tonnage (a measure of hold capacity) is restricted, the fisherman may use more fuel and labour services to make turnaround time faster. Alternatively, he may purchase more gear or electronic equipment, and of a more sophisticated type. In this way a limited entry program with vessel-specific restrictions may alter the fisherman's optimal expansion path. Prior to the regulations the fisherman optimizes by using an increasing amount of some mix of all inputs in order to catch more fish. After the input restriction is imposed, the fisherman uses an increasing amount of a subset of inputs along with License Limitation: Theory and Practice / 28 the fixed amount of the restricted inputs in order to increase his catch. This assumes that the fisherman is able to increase his output level. If this is not possible, for example, because the total stock of fish remains unchanged, then the fisherman has ex posf increased the use of variable inputs without a concomittant increase in output. This constitutes rent dissipation. Although the fisherman's decision is privately optimal (given the incentives and constraints which he faces), it is socially suboptimal. The fisherman uses too many inputs (along with the restricted input) for the realized catch. Crutchfield (1979) terms this behaviour "capital-stuffing". Rettig (1984) describes it as an increase in the ratio of capital expenditure to whatever component of effort is limited. This "seepage" is the ability of effective fishing effort to increase whenever the nominal effort is reduced. Many authors believe that, given sufficient time, complete rent dissipation will occur. Other authors agree in principle with the notion of substitution possibilities and associated rent dissipation, but question the degree to which the phenomenon may manifest itself. Crutchfield (1979) admits the existence of incentives to expand the use of inputs in the face of license limitation and/or additional vessel-specific constraints. However, he expresses doubt about the fisherman's ability to exhaust the entire rent in this way. He argues that fishermen face a sharply increasing marginal cost of expanding against a vessel or tonnage restriction. He views the fishing platform as fixed, or as having little variability. In fact, he appears to subscribe to the view that all inputs used with the vessel may only be employed in fixed proportions, ie., that the harvest production function is Leontief in nature. This implies that there is no License Limitation: Theory and Practice / 29 incentive to increase the use of unrestricted inputs against restrictions on the amount of the fixed input, because expected output cannot increase. In theory, it is as if the fisherman has only one variable input at his disposal.+ Other adherents to this view are Adasiak (1979) and Anderson (1976, 1982, 1985). It is often called the fixed vessel capacity assumption. Anderson (1976) focuses on the behaviour of the individual fishing firm and tries to bring the analysis of the fishery into the realm of micro production theory. He argues that the fishing firm cannot directly control its catch rate, since this is determined by the amount of fish, in conjunction with the number of competitors. Thus, the average catch rate is given to the fishing firm. However, the fishing firm can control the amount of effort it uses. Anderson calls the effort produced by the fishing firm an intermediate product. It is a function of several traditional inputs, eg., labour, capital, materials, gear, fuel, etc. The effort index becomes in Anderson's work the single variable input that the fisherman may control. This analysis is criticized by Huang and Lee (1976). They argue in favour of the need for a more generalized fishery production function. They require this function to permit the elasticities of substitution among the several inputs to depend upon the levels of the variable inputs and other fixed inputs, in particular, the stock of fish. In an exchange with these authors, Anderson (1978) stresses the validity of his assumption of effort as an intermediate output. Huang and Lee (1978) counter by claiming the Anderson view does not address the theoretical possibility and practical tClark (1980, 1982) makes this assumption explicit in a series of models that attempt to describe both competitive and oligopolistic behaviour of fishermen in a limited entry fishery. License Limitation: Theory and Practice / 30 reality of variable input proportions that may be stock-dependent. In 1982 Anderson, writing on the impacts of limited entry in a share system, adopts the fixed vessel capacity assumption, but concludes by asserting that it is a limiting one. However, in a later article (1985), which compares the benefits from license limitation for two different scenarios,+ he adopts the assumption once again. Using static supply and demand analysis he concludes that complete dissipation of rents is stemmed by an increasing marginal cost of effort per boat. This is the assertion made by Crutchfield (1979), ie., that increasing marginal costs do not allow the fisherman to substitute away from the restricted factors. With his assumption of fixed vessel capacity Anderson then asserts that, once the regulators adopt the correct fleet size, they can ignore capital-stuffing. He argues that the initial choice of fleet size incorporates the subsequent actions of the fishing firms. Anderson appears to discount the many continuing incentives for rent dissipating behaviour encouraged by the existence of resource rent. Anderson's static analysis ignores the shifting of individual firm's supply curves due to the increasing use of unregulated inputs. From his work it would appear that Anderson views the substitution possibilities between restricted and unrestricted inputs to be non-existent. The views of those who disagree with the notion that the harvest production function may accomodate substitution between inputs may be summarized. These authors believe that there are substantially increasing marginal costs associated with the substitution of the various inputs used by the fisherman. Thus, few, if any, tThe two scenarios describe situations of constant costs and increasing costs for the fishing industry. License Limitation: Theory and Practice / 31 substitution possibilities exist between pairs of inputs, and therefore, the so-called Leontief fixed proportions production function is a valid description of the fishing technology. It is now possible to discuss the requirements for preventing the dissipation of rent in a limited entry fishery. First, the regulator must identify a key input and put an upper bound on its use. For example, a net tonnage per vessel restriction is one such candidate because it is believed that by controlling this input output per vessel may also be controlled. Next the constraint must be rigidly controlled to prevent the upper bound from increasing. Finally, it is necessary that the key input have neither substitutes nor complements among the other inputs. If it does, then fishermen will re-optimize over the reduced set of variable inputs and use more of the unrestricted inputs rather than increasing the amount of the restricted factor. In this way the fisherman may subvert the intentions of the regulations and dissipate rent. Conflicting views on the effectiveness of restricted access programs arise from individual assessments regarding the substitution possibilities in the underlying harvest technology. If the possibilities for substitution between unregulated and regulated inputs are substantial, then the efficacy of restricted access programs in preventing rent dissipation must be called into question. A resolution of the issue of input substitutability and the concommitant loss of rent is ultimately an empirical matter that must be answered on a fishery-specific basis. This thesis proposes to perform this task for the British Columbia commercial License Limitation: Theory and Practice / 32 salmon fishery. In Chapter 4 a model of individual vessel behaviour is suggested as the appropriate framework. The current chapter continues with a consideration of past efforts to conduct empirical studies of the issues related to license limitation programs. B. EMPIRICAL STUDIES OF LICENSE LIMITATION PROGRAMS In this section the available empirical evidence is surveyed and evaluated. The research may be divided into two broad categories depending upon whether an harvest production function is estimated. Studies of the first kind evaluate the efficacy of license limitation programs in an indirect fashion. These studies typically cite annual statistics on the number of effort units or on the relative fishing power of the units as evidence of the inability of the programs to prevent rent dissipation. They often include information on the trend value of the right to fish (the license) as evidence of incomplete rent dissipation. In contrast, empirically-orientated, harvest production studies, which constitute the second category of research, are in the minority. They use econometric methods to estimate directly the substitution possibilities between pairs of inputs. None of the papers integrates the two methods, ie., estimation of the substitution parameters with calculations to estimate the amount of rent dissipated. This thesis proposes a means of achieving this important objective in Chapter 6. 1. Non-Production Function Studies Before discussing these studies an observation, made by Cicin-Sain, Moore, and Wyner (1978), is offered. In a comparison of the, experiences which many countries have had with limited entry, these authors complain that there have been no License Limitation: Theory and Practice / 33 systematic evaluations of these programs. Furthermore, they criticize the programs for neglecting to generate either the quantity or quality of data necessary to support such analyses. In general, this lack of data still plagues the researcher interested in doing fisheries research (Pearse 1982). A review of the relevant literature bears out this assertion. The articles all cite the lack of suitable data, especially with respect to costs (Fraser 1977, 1979; Pearse and Wilen 1979, Meany 1979, Strand, Kirkley, and McConnell 1981, Huppert 1982, Byrne 1982, Rettig 1984). Campbell (1973) and Sinclair (1978) evaluate the British Columbia commercial salmon licensing program. Recall from the discussion in Chapter 2 that it began in 1969 by licensing first the vessels, then the net tonnage per vessel. The authors comment on the increasing use of newer and bigger vessels, and the use of more sophisticated electronic equipment. They argue that the salmon fleet is being increasingly skewed in the direction of larger boats with a greater than proportionate increase in fishing capacity. Although licenses to fish have acquired positive values, which indicates the emergence of resource rent, Sinclair claims that a substantial amount of rent dissipation has nonetheless taken place. Meany (1979) examines the West Coast rock lobster fishery in Australia. Since 1963 it has been regulated by means of a vessel licensing program with additional restrictions on the number of pots used per vessel. Vessels have exhibited a trend toward the use of a more powerful engine, a greater tonnage, and more equipment on board. Reacting to this evidence of increased fishing capacity, the regulator has shortened the fishing season. Meanyt concludes that there has been tin the same article Meany discusses the Australian prawn fishery as an example of a successful limited entry program. However, Munro (1982) argues that its success is License Limitation: Theory and Practice / 34 excessive reinvestment in gear and that this has caused rent dissipation to take place. This is despite the presence of positive and increasing license values. Rogers (1982) studies this fishery as well, but is less convinced of the ability of the fishermen to dissipate the rent. The abalone fishery in two different countries is the subject of research by Huppert (1982) and Stanistreet (1982). Although the fisherman (or diver) is the licensed factor in both fisheries, reactions to this form of input restriction are very different. It is unclear why this should be the case and the authors offer no enlightenment. Stanistreet concludes that the Australian abalone industryt exhibits little increase in unlicensed factors used by each diver. Huppert's analysis for the California case shows that the crew used per licensed diver has increased substantially since the 1976 inception of the program. Fraser (1982) evaluates the British Columbia roe herring fishery*, which has had a licensing system since 1974. The program is unlike that for salmon because the licensed factor is the individual fisherman. Nonetheless, the fisherman must designate the vessel to be used. Fraser cites increases in the power of participating vessels and in the use of electronic equipment. He also argues that the labour used per vessel has increased.* However, a formal analysis to substantiate these claims is not undertaken. Furthermore, there is no discussion about the abundance of fish over time which might justify an increase in harvesting capacity. t(cont'd) attributable to a government engineered bilateral monopoly. tAfter beginning in 1963-64, this fishery was licensed in 1968. *This is the second largest fishery in British Columbia and many of its fishermen participate in the salmon fishery as well. *This is the likely response to increasingly shortened seasons for the fishery. For example, the Kitkatla roe herring seine fishery for 1987 lasted an historic 6 minutes. License Limitation: Theory and Practice / 35 These studies point to the need for a more formal model of the behaviour of the micro production unit under a license limitation program. In particular, they exhibit the importance of a complete examination of the role played by prices and restricted factors upon the decision-making of the fisherman. Only in this way is it possible to estimate the underlying production technology by means of standard econometric techniques and to provide evidence about the the degree of input substitution and associated rent dissipation. 2. Production Studies Only two studies model production within a limited entry fishery explicitly at the level of the individual vessel (Byrne 1982, Strand, Kirkley, and McConnell 1981). They are important, for it is this level of analysis that can provide estimates of the substitution possibilities that may be used to evaluate the efficacy of limited entry programs with vessel-specific input restrictions. Byrne estimates the relative fishing power of an average prawn-trawling vessel in Australia. The vessel is the restricted input. His model says that the catch rate per vessel in a given area is related to vessel length, horsepower, fishing time and a dummy variable indicating the use of a single or double rig. Using a Cobb-Douglas production function he finds that the average fishing power per vessel has increased by at least 50% in one area and over 100% in another for the period 1970-1978. However, unless the catch rates of these vessels are constant, these observations are irrelevant. Furthermore, Byrne's finding of increased productive efficiency is not related to the effects of relative prices. For example, a technological innovation in License Limitation: Theory and Practice / 36 fishing may increase fishing power and lower costs. The second paper deals outright with substitution of inputs in a limited entry fishery. Strand, Kirkley, and McConnell (1981) estimate a cross-sectional, translog production function for Atlantic surf clams. Output is specified as a function of gross registered tonnage, horsepower, length of dredge blade and hours fished. Isoquants for pairs of these inputs are plotted and found not to support the Leontief fixed proportions assumption. However, the degree of substitution possibilities is not calculated. Once again, input cost data are not available to complete the analysis. IV. MODELING THE BEHAVIOUR OF THE REGULATED FISHING FIRM In this chapter a model is set out describing the behaviour of a small competitive fishing firm that operates in a limited entry fishery and is subject to restrictions on the use of some inputs. The model serves as a basis for the empirical testing of the degree of substitution possibilities in the fishing technology. For, if the technology permits the substitution of unregulated for regulated inputs, this provides quantitative evidence of the ability of fishermen to dissipate fishery resource rents and thereby undermine the intent of limited entry regulations. A second objective is that the model be capable of deriving shadow values for the restricted factors. In order to achieve this goal, techniques from microeconomic production theory are adopted. This is not a common approach for researchers analyzing the fishery. Indeed, very little attention has been paid to economic motivations underlying decisions made by individual competitive fishing firms and most of this work is very recent (Wilen 1979, 1985, Bockstael and Opaluch 1983, Conrad 1984, Kirkley 1984, 1986, Squires 1984, 1985 and 1987b). In the first section of this chapter the typical representation of the harvest function for fish is discussed. Shortcomings of this approach are enumerated. This leads to a search for an alternative method of describing the fishing behaviour of the individual boat owner. In particular, the effects of price changes and input controls are incorporated into the analysis. By using duality techniques it is possible to describe the harvest technology by means of a short-run, restricted profit function. The development of this approach is the subject of the second section. The third section enumerates the advantages of this approach in a discussion of the summary statistics which describe the relationships between output and inputs (both variable 37 Modeling the Behaviour of the Regulated Fishing Firm / 38 and fixed) that may be obtained from the restricted profit function. It is seen that this function is particularly well-suited to studying the behaviour of regulated firms. A. THE DIRECT HARVEST PRODUCTION FUNCTION 1. Specification of the Direct Harvest Production Function Before describing past work relating to the specification of the harvest production function it is necessary to clarify one matter. The commercial exploitation of a fishery is constrained by two production functions. The first is attributable to nature and is inherently time-dependent. It dictates how adult fish produce eggs which in turn become fertile adults (Schaefer 1957, Beverton and Holt 1957). The second production function is in the domain of the commercial fisherman; it describes how fish in the sea are converted into profit-yielding output. It is this latter function alone which is of interest to this research. As a result, the dynamics of the fish population are ignored so that at any period of time the stock of fish available to the fisherman is assumed to be exogenous. This assumption allows the analyst to focus upon the intra-seasonal input decisions of the firm. Both Henderson and Tugwell (1979) and Bj0rndal (1984) maintain this assumption in their empirical studies. Conrad (1984) discusses the appropriateness of this assumption for empirical research and concludes that it is less stringent than the many assumptions used to obtain results in capital-theoretical models of the fishery (Clark 1976, 1985). Much of the fisheries literature of the last three decades reflects the concerns of biologists and mathematicians, (Schaefer 1957, Beverton and Holt 1957, Clark 1976, 1985). Their interests dictate both the types of models and the level of analysis. Modeling the Behaviour of the Regulated Fishing Firm / 39 Because of the biologist's concern with the development of realistic population growth models the role of human exploitation is relegated to the impact on the fishing stock of a single economic variable called fishing effort (Rothschild 1972). Fisheries mathematicians adopt this assumption as a necessary simplification. They use optimal control techniques to analyze the intertemporal nature of the fishery and can not deal with more than two state and two control variables (Clark 1976, 1985; Clark, Clarke, and Munro 1979).t Traditionally the level of analysis focuses on a comparison of the behaviour exhibited by two polar forms of exploitation, ie. monopoly (Clark 1976) and open access (Clark 1980b, Dasgupta and Heal 1978, Wilen 1985). In either case, the ultimate concern is with the impact of the aggregate catch upon the population growth rate. The harvest production function is of the form given in equation (4.1). (4.1) Y(t) = g(E(t),S(t)) In this equation Y(t) is the harvest rate at time t. It is a function of fishing effort (E) at t and the biomass (stock of fish) (S) at time t. This general form is used to describe either an individual vessel's production function or the aggregate industry technology (Anderson 1977), with E(t) specified appropriately. The variable E(t) is usually refered to as an ill-defined composite bundle of capital and labour inputs, although Anderson (1977), Byrne (1982), Strand, Kirkley, and McConnell (1981) and Bj0rndal (1984) indicate that a number of other inputs are often included. In practice, it is made operational by defining a standardized vessel-day tHowever, numerical analysis may be used to infer the properties of such dynamic optimization models, even though they cannot be solved analytically. Modeling the Behaviour of the Regulated Fishing Firm / 40 measure (Roy, Shrank, and Tsoa 1980) through reference to its catchability. That is, one unit of effort is the fraction of the population caught by one vessel-day's effort. Use of this effort variable is usually justified by appealing to Leontief's notion of a fixed-proportions production technology (Munro and Scott 1985). That is, if all inputs are used in strictly fixed proportions then they can be aggregated into a single variable.t In practice, the functional form most commonly adopted for the general fisheries production function given in equation (4.1) is the Cobb-Douglas (Clark and Munro 1975). This is shown in equation (4.2). (4.2) Y(t) = q . E ( t ) a - S ( t ) b In this equation q is defined as the catchability coefficient. This is usually assumed to be constant and the same across all vessels, in the case of more than one vessel. Loose (1975) allows this coefficent to vary across vessels in his theoretical analysis of the B.C. salmon fishery, but then imposes constancy for estimation of the aggregate production function. The parameters a and b are the partial elasticities of output with respect to effort and stock. These are often set equal to one in the theoretical literature (Clark and Munro 1975). t in fact, what is being assumed is that inputs are additively separable. Squires (1987a) discusses the conditions under which fishing effort can be used as a consistent aggregate. Modeling the Behaviour of the Regulated Fishing Firm / 41 2. Problems with the Direct Production Function Approach There are several problems, both theoretical and empirical, with the direct production function as specified by either (4.1) or (4.2). These features make it an undesirable format for the study of input substitution and rent dissipation in a regulated fishery. The major theoretical shortcoming of the direct production function is that it does not permit prices to play a role. It merely describes the technical relationships between output and inputs. Thus, one can derive the shapes of the isoquants between pairs of inputs, but not the equilibrium point(s). This methodology cannot reflect the impact of changes in relative input prices on optimal input allocation. Since the issue of rent dissipation is concerned with the price incentives which encourage overfishing, this is a serious flaw of the direct production function. A related problem stems from the lack of any type of behavioural assumption regarding economic motives. In economic studies of individual behaviour it is usually assumed that agents either minimize costs or maximize profits. There are good reasons to believe that fishermen are economic agents who seek to optimize subject to whatever constraints they face. For example, Eales and Wilen (1986) find that they can reject neither the notion of profit maximizing behaviour nor the notion that fishermen behave in an economically rational way. This leads them to suggest the use of models that describe short-run, profit-maximizing behaviour. As a result, this type of incentive system should be explicitly incorporated into the decision-making model of the fisherman. A second difficulty with the typical production function formulation stems from the Modeling the Behaviour of the Regulated Fishing Firm / 42 assumption that both inputs are variable since a long-run solution is sought. Thus, no distinction is made between modeling of the short-run, where at least one factor is fixed, and of the long-run.t Since no distinction is made between variable and fixed factors, the function cannot be used to describe behaviour when there are regulations that restrict the use of some inputs but leave other inputs under the control of the fisherman. When this feature is combined with the lack of price information it is clear that one can not obtain estimates of the shadow prices of the restricted factors, such as the stock of fish or the licenced vessel characteristics. Finally, the direct production function alone cannot be used to obtain estimates of rent dissipation. To do so requires a model which can incorporate the interaction of production possibilities with the prevailing set of output and input prices. The direct production function also has shortcomings of an empirical nature. First, even if the input set is expanded, it is often difficult to obtain observations on a key input. For example, Comitini and Huang (1967) suggest that skipper knowledge is an important factor in the success rate of the vessel. They are able to test this hypothesis due to the quality of their data set. However, when this information is missing, the estimation may be characterized as having an omitted variable bias (Varian 1978).* Estimates of the parameters and elasticities may be biased away from their true values. The direct producton function is usually specifed as a Cobb-Douglas, or less tThere are a few exceptions (Bradley 1970, McKelvey 1983, Conrad 1984). All of these papers assume that the stock of fish is fixed for the period of interest. tThis means that the right hand side variables are not independent of the error term. Modeling the Behaviour of the Regulated Fishing Firm / 43 frequently as a CES, functional form, (Bj0rndal 1984, Clark 1980b, 1982, 1985; Byrne 1982).t One exception is the paper by Kirkiey, Strand, and McConnell (1981). They estimate a translog production function. There are several properties of the Cobb-Douglas and CES forms which make them unattractive for a study concerned with input substitutability and rent dissipation. Both forms restrict the elasticity of substitution between pairs of variable inputs to be constant. The Cobb-Douglas goes further and restricts all elasticities to be one. This restriction is particularly undesirable as it is common to observe qualitatively different degrees of substitution between input pairs (Fraser 1977, 1979; Scott and Neher 1981, Rettig 1984, Wilen 1985). Furthermore, many authors impose the assumption of increasing returns to scale on the fishing technology. In contrast, Plourde (1971) suggests decreasing returns may be a more appropriate assumption. Munro and Scott (1985) argue that the question of returns to scale can be tested. But this procedure is often not carried out. Squires (1984, 1985, 1987a) is an exception. Estimation of the direct production function is not appropriate for the purposes of this study. Therefore, an alternative means of representing the fishing technology must be used. Recent advances in the area of duality indicate that knowledge of a direct production function may be recovered from the estimation of a dual function (Diewert 1973, 1974, 1982). In contrast to the production function, which uses quantities of inputs as arguments, the dual function depends upon input prices. This approach has the advantage of avoiding a potential simultaneity bias in the estimation since prices are usually assumed to be exogenous to the individual decision maker, whereas quantities are not. The fruitfulness of this direction of tComitini and Huang (1967) cannot reject the Cobb-Douglas form in favour of a CES formulation. Modeling the Behaviour of the Regulated Fishing Firm / 44 research for the fishery has recently been recognized in two doctoral dissertations. Squires (1984) uses a restricted profit model to examine the harvest technology of the New England otter trawl fishery. He specifies a (translog) restricted profit function with three inputs, three outputs, and a technology dummy variable. Kirkley (1986) focuses upon the output-maximizing choices made by fishermen in the New England, Georges Bank fishery. He specifies a single fixed input model and estimates a multi-output (translog) revenue maximization function. The next section describes the dual function chosen for this thesis to investigate the harvesting behaviour of fishing firms. B. THE DUAL APPROACH: THE RESTRICTED PROFIT FUNCTION 1. A Short-Run, Restricted Profit Maximizing Model This model represents the behaviour within a season of a competitive fishing firm in the limited entry British Columbia salmon fishery. It is assumed throughout that the number of participants is less than the number of operators which an unregulated, open-access fishery would attract. In the past, a fixed number of licenses have been issued to qualifying fishermen. Licenses are renewed automatically at the begining of each season upon payment of a nominal fixed fee. I am assuming individual Cournot behaviour on the part of each fisherman. Therefore, it is assumed that fishermen do not seek to form coalitions which might make them better off. As well, the fact that the regulator controls the total allowable catch in a season allows me to neglect intertemporal considerations. It is assumed that a fishing firm owns one vessel and that there is no horizontal Modeling the Behaviour of the Regulated Fishing Firm / 45 integration between vessels. As well, the vessel owner is assumed to be a price taker in both output and input markets. There is only one output and it is defined as the quantity (measured in pounds) of salmon landed. This is an agreggate of the five species of commercially exploited salmon. Application of two types of inputs, variable and restricted (fixed), yields the output according to the production or transformation function. It is possible to define a multi-output problem using this framework but the interest in this thesis is on the input side. Adding complexity on the output side provides no new insights that may be applied to an analysis of input substitution. This approach contrasts with that of Kirkiey (1986). His focus is on output allocation decisions in a multi-species framework. To highlight behaviour in this area he specifies a single fixed input. It is assumed that the vessel-owner is free to vary the use of three inputs within the season. These are labour services, including both crew and skipper, fuel services, and gear services.+ The latter input is the nets, lines, lures, etc., used by the fishermen to either entrap or entice the fish. By increasing the use of any of these inputs, the fisherman is able to catch more fish. For example, by using more fuel the vessel-owner may travel to a greater number of fishing grounds, thereby increasing contact with the biomass. Or, using a greater amount of labour services means that the fish may be brought on board much faster. In addition, this may tThe assumption of the existence of the appropriate rental markets for the latter input appears reasonable for British Columbia. One need only look at the numerous advertisements for the sale of second-hand gear. Squires (1984, 1987a, 1987b) assumes labour, fuel and capital to be variable inputs; he claims that capital (the vessel) is completely malleable because of the existence of rental markets. Modeling the Behaviour of the Regulated Fishing Firm / 46 permit the specialization of labour which will lead to a larger catch. For example, having more men on board permits them to specialize in particular tasks, such as navigation or fish-finding. Finally, by using more nets less time is lost to in-season repairs. This is especially important when the fishing firm is subject to a restriction on the total number of days fished. The restricted (fixed) factors are not controlled within the season by the fisherman, either because the season is too short, or because of regulations, or because of nature. The fixed factors are the stock of fish encountered by the vessel, the net tonnage of the vessel, and the number of fishing days. Nature and the regulator interact to fix the first factor at some level for the season. Nature does so by providing a certain number of fisht; the regulator controls access by permitting fishing only in given areas and for specified time periods. In fact, as the season progresses the stock of fish is reduced from the biomass originally available. McKelvey (1983), Bradley (1970), and Loose (1975) make this explicit in models that deal with intra-seasonal monopoly decision-making in the fishery. This is not an important consideration in this study for two reasons. First, insofar as the individual vessel-owner is concerned, the overall level of stock is so great that reductions in that level have no appreciable impact upon his output. This is because total biomass includes both the Total Allowable Catch for the entire fishery, as permitted by the regulator, and the desired escapement for the perpetuation of the fish. In most cases the escapement far exceeds even the Total Allowable Catch. Second, fishing seasons for salmon depend upon the characteristics tin fact, nature provides a stochastic input called the stock of fish. I ignore the stochastic nature of this variable, since I do not model the intertemporal problem. Modeling the Behaviour of the Regulated Fishing Firm / 47 of the fish themselves. For most species of salmon in British Columbia runs of fish pass through any given area within a two-week period (ESSA 1982). Therefore, the within period stock levels do not fluctuate much. The regulator constrains the net tonnaget of a vessel by decree, as discussed in chapter 2. That is, each license-holder may not use a vessel with a larger net tonnage than the vessel he used at the time that the restrictions were put into place. Over time the vessel-owner may try to increase the net tonnage that he uses by buying a larger vessel. However, he must also purchase a license, associated with a larger net tonnage, from a fellow participant in the fishery. O n the other hand, he may decrease the tonnage by commissioning the building of a smaller vessel. Finally, nature and the regulator act together to control the maximum number of fishing days that a vessel may use. Nature controls the times during which the fish come in to spawn and hence, are liable to be caught. The regulator further restricts the number of fishing days by declaring certain areas closed for given periods of time. It is assumed that the objective of each fishing firm is to maximize seasonal or restricted profit subject to given prices, a production function, and restrictions on the use of some inputs. Restricted (or seasonal) profit is defined as the difference between total seasonal revenue and total cost of the variable inputs. This assumption and that of the exogeneity of the prices appear to be valid in the tRecall from the discussions in chapters 2 and 3 that this is a measure of hold capacity. Modeling the Behaviour of the Regulated Fishing Firm / 48 British Columbia commercial salmon fishery. The participants are mostly small owner-operators and the salmon product is sold on a world market. In addition, the markets for the variable inputs appear to be competitive. The solution to this constrained maximization problem is a partial static equilibrium, since the optimal choices of output and variable input quantities are dependent upon the actual levels of fixed factors (Brown and Christensen 1979, Kulatilaka 1985, 1987). It contrasts with a long-run situation in which all inputs are variable, ie., full static equilibrium. In chapter 6 this partial equilibrium assumption is relaxed with respect to the net tonnage because it is possible that the level of this fixed factor at any given time is not the one that would be chosen in an unconstrained situation. In this way it is possible to answer the question of whether the actual net tonnage is optimal for the current market rental price of net tonnage.+ The assumption of short-run, myopic behaviour seems reasonable given that some factors are fixed for the season. Thus, the fishing firm is concerned with the success of the season at hand. The firm solves a static profit-maximization problem and this provides the optimal allocation decisions for the season. This assumes that the firm cannot alter next year's profit by changing the current catch level. This is partly due to the size of the individual firm relative to the size of the fleet. Furthermore, the presence of the regulator assures the existence of fish next year, since it is the regulator's task to ensure sufficient escapement in the current period to allow recruitment for future periods. Finally, if the current season is not tThis does not propose to offer the final word on the subject, since current-day investment decisions may also depend upon the vessel-owner's expectations about future regulations. They do not enter into the analysis done in this thesis. Modeling the Behaviour of the Regulated Fishing Firm / 49 favourable, the vessel-owner has the option of selling out. The presence of a market for vessels and licenses allows for this possibility.t The production function for the fisherman is defined as the maximum amount of output, y , that the firm can produce given that it has amounts ~x = ( x 1 ( x 2 , x 3 ) of the variable inputs and z = ( z 1 y z 2 , z 3 ) of the fixed inputs, eg., y = F(>c, z).t The vector w=(y, x, z) defines the vector (with dimension seven) of all inputs and the output. As Diewert (1974) shows another way to express this relationship is by the production possibilities set, T, defined by (4.3). Thus, F(y,x,z) and T are dual representations of a production technology. (4.3) T = { (y, x ; z) : y < F(x, z); z < 0 3 } The production possibilities set, T, is assumed to satisfy four regularity conditions. They are given in (4.4). (4.4) (T1) T is a closed, non-empty subset of the 4 +3-dimensional space, (T2) T is a convex set, (T3) if w' e T and w" ^ w', then w" e T, (T4) if (y, x, z) e T, then the components of (y, x) are bounded from above. The first condition is a mathematical regularity condition. It simply says that output tThere appears to be such a market for the British Columbia salmon fishery. $Much of the material for this discussion comes from Diewert (1974). Modeling the Behaviour of the Regulated Fishing Firm / 50 can be produced in some way, but it requires some positive amount of the inputs to produce a positive amount of output. The second requires the technology to have non-increasing marginal rates of transformation, whereas condition three implies the possibility of free disposal and is also called the monotonicity requirement. Finally, the fourth condition suggests that, for a given set of fixed inputs, the set of variable quantities is bounded from above. The fishing firm's problem can be restated in a formal way. It's restricted profit equals the maximum amount of seasonal profit that it can earn subject to the constraints it faces. R —• —' (4.5) 7T (p, w; z) = (max[p-(F(x;z)) - w x ] where z , < z u z2< ~z ^ 2a< ~ In this equation p is the known and certain output price and is strictly greater than zero. The vector of known and certain input prices, w, is also strictly greater than zero. Output, y, and three inputs, x , , x 2 , x 3 , are variable, whereas, the three other inputs, z , , z 2 , z 3 , are fixed. The constrained maximization problem derived from (4.5) may be solved by the Lagrange method of multipliers. From the first order conditions it is possible to obtain implicit output supply and input demand functions that depend upon the prices and fixed factors. However, if the production function is very complicated the exercise is not a simple one. In turn, these implicit equations may be used to generate comparative statics results, eg., how does the demand for an input change Modeling the Behaviour of the Regulated Fishing Firm / 51 when the level of a restricted factor changes. However, without known restrictions on the production function, F(x; z), or knowledge of it's parameters, many comparative statics results are ambiguous in sign. Thus, it is necessary to obtain empirical estimates of the production function to answer many questions of interest. 2. Relationship Between the Primal and the Dual Instead of specifying and solving (4.5) by means of a direct production function for F(x; z), one may appeal to duality theory (Diewert 1973, 1974, 1982) and define a restricted profit function (Samuelson 1953-54, Gorman 1968). This is an explicit function of the output price, three input prices, and three fixed factors, as in (4.6). R —» —* (4.6) 7T (p, w; z ) = max { p « y - w T x ; (v,x,z) eT} x,y for p>0, w > > 0 3 , z<0 3 In this equation, T is the production possibilities set defined in equation (4.3) and w T indicates the transpose of the vector of input prices, w. The function in (4.6) is dual to the underlying production function, F(x ; z), and to the production possibilities set defined in (4.3), if it fulfills a set of properties outlined in McFadden (1970), Lau (1972) and Diewert (1973, 1974). They are the following. (4.7) R —' — (7r1) 7T (p, w; z) is a non-negative function for p > 0, w > > 0 3 , and any z, R —r —r (7T2) it (p, w; z) is non-decreasing in p, R —- —• (7r3) it (p, w; z) is non-increasing in w, (7T4) (w5) (ir6) These conditions imply the following assumptions. The first corresponds to a regularity condition that is not inconsistent with the notion of profit maximization. The second and third conditions imply that a higher output price with fixed input prices and fixed factors causes profit to increase, whereas the opposite is true for higher input prices with a fixed output price and fixed factors. Condition four says that a doubling of all prices will double profits. Finally, condition five says that the profit function is well-behaved for small changes in prices, whereas condition six asserts that there is a maximum profit for each given vector of prices and fixed inputs. Diewert (1973, 1974) proves that given any function 7T satisfying the 6 conditions in (4.7), there exists a unique production possibility set, T, satisfying the 4 conditions in (4.4). Alternatively, if F(x; z) satisfies a set of conditions, given in Diewert (1974), then F and IT are equivalent representations of each other. This means that n and T are equivalent representations of technology, and more importantly, that it may be used to describe the technology and obtain estimates of the elasticities of interest, as well as comparative statics results, in addition, shadow prices of fixed factors may be derived from n . The advantages of the dual approach are several. First, it is easier to specify complex technologies using the restricted profit function, than by using the direct Modeling the Behaviour of the Regulated Fishing Firm / 52 R — — 7r (p, w; z) is homogeneous of degree one in p and w, R _ _ 7T (p, w; z) is convex in p and w, R — — IT (p, w; z) is concave in z for every fixed p and w. Modeling the Behaviour of the Regulated Fishing Firm / 53 production function. Second, the dual restricted profit function is usually linear in prices which reduces econometric computation costs. Finally, comparative statics results are readily generated. C. CHARACTERIZING TECHNOLOGY USING DUALITY 1. Supply and Demand Functions Comparative static results are obtained by differentiating the restricted profit function in (4.6). This begins by defining the output supply and input demand equations. If the restricted profit function satisfies the conditions in (4.7) and is differentiable with respect to the variable quantity prices at p*>0, w * > > 0 3 , then using Hotelling's Lemma (1932) yields the following output supply and input demand functions (Gorman 1968). (4.8) 37TR (p*, w*; z*)/9p = y(p*, w*; z*) Equation (4.8) defines the profit-maximizing level of output (y) given prices p*, w* and fixed inputs z* . (4.9) 97TR (p*, w*; z*)/9w. = x . (p* , w*; z*) for i= 1, 2, 3 In this equation x^, the profit-maximizing amount of variable input i is obtained for given prices, p*, w*, and fixed inputs, z* . The convexity property in (4.7) insures that the supply function slopes upward and the input demand equations slope downward. Furthermore, the matrix of second-order partial derivatives of n with respect to all prices is symmetric and positive semidefinite. Modeling the Behaviour of the Regulated Fishing Firm / 54 In chapter 5 a functional form is chosen for (4.8) and (4.9). The equations are estimated jointly to obtain a empirical description of the optimal output supply and input demand functions, as derived from a profit-maximizing model. These functions depend upon a set of estimated parameters and upon exogeneously given prices and fixed factor levels. If actual mean values for prices and fixed factors are substituted into the estimated equations, it is possible to obtain predictions about the optimal levels of output supply and input demands. These predicted values are used in chapter 6 to obtain estimates of the amount of fishery rent and to determine the degree of rent dissipation attributable to input substitution. 2. Elasticities of Interest Given the profit function defined in (4.6) and the output supply and input demand functions defined in (4.8) and (4.9) it is possible to examine the structure of the harvest technology. For example, how does the demand for a variable input respond to a change in an exogenous factor, either a price or a restricted input. This information is contained in two sets of elasticities. The first is defined with respect to prices and is called the own- or cross-price elasticity of variable quantities. The second is defined with respect to the fixed factors and is called the elasticity of intensity. These elasticities indicate the degree of substitution of inputs in production, and may be used to examine the issue of rent dissipation in the fishery. They are all predicated upon actual conditions in the fishery, that is, with respect to given levels of fixed inputs and prices. The first set of elasticities are defined in equation (4.10), both for own- and cross-price elasticities of demand or supply with respect to changes in variable Modeling the Behaviour of the Regulated Fishing Firm / 55 quantity prices. They are easily interpreted (Varian 1978). For example, a 1% change in the price of fuel may lead to a x% change in the quantity of gear input demanded. The sign of this cross-price elasticity signifies whether the input quantities are substitutes (positive) or complements (negative). The own-price elasticity of supply should be positive, indicating that a 1% increase in the output price causes output to increase by some percentage. O n the other hand, the own-price elasticities of input demand should be negative. (4.10) The elasticity of variable quantity (either supply, y, or demand, x^)with respect to price, p^ e = ( 9x./3 p, )• ( p , /x.) i k l k k l for i,k = 1,..,4 Equation (4.11) defines the non-normalized elasticity of intensity (Diewert 1974).t This explains how a 1% change in the level of a fixed factor effects the supply/demand of a variable quantity. If the elasticity is positive, this indicates a complementary relationship; if negative, a substitution relationship. (4.11) The elasticity of variable quantity (either supply (y) or demand (x.) with respect to a change in the quantity of the fixed factor, z_. £. . = (3x./3 z > (z ,/x.) • i ] i : : i for j = 1,2,3 and i = 1,..,4 tDiewert (1974) defines the normalized elasticity of intensity as the non-normalized elasticity divided by the input share of the fixed factor. Modeling the Behaviour of the Regulated Fishing Firm / 56 3. Shadow Prices of Restricted Factors The dual restricted profit function defined in (4.6) can be used to obtain estimates of the shadow prices for the restricted factors. These shadow prices are defined for given prices of the variable quantities and for given levels of the fixed factors. They describe the marginal increase in profit for a marginal increase in the quantity of the fixed factor (Diewert 1974). These measures are obtained by differentiating the restricted profit function by each of the restricted factors, z . . (4.12) Rj = a7T R(p, w; z)/ 3z_. It is possible to evaluate the expression in (4.12) by substituting in actual prices and quantities of the fixed factors. A comparison of the shadow value obtained in this manner with the actual market price for a unit of the restricted factor permits the researcher to predict whether the optimal quantity of the fixed factor should be greater or less than the actual. If R. > m . , where m . is the current market 3 D 3 price of the "jth" fixed factor, then the fisherman should increase the quantity used of that fixed factor. The converse is true if the sign is reversed. This indicates that the fisherman is not making a sufficient return on the amount of the fixed factor to justify maintaining the current amount. Alternatively, one can examine the question by using the dual to the price space, ie., quantity space (Kulatilaka 1985). In this instance, the researcher solves for the optimal quantity of the restricted factor as that which maximizes the total or long-run profit for the fishing firm for given current rental market prices (Brown and Christensen 1979). It is possible that the levels of the fixed factors at any given Modeling the Behaviour of the Regulated Fishing Firm / 57 time are not those that would be chosen in an unconstrained situation. Presumably, a vessel-owner wishes to have a larger available biomass, a greater net tonnage, and more fishing days, however, he is not able to do so because of the combined actions of nature and the regulator. In chapter 6 this hypothesis is tested for the restricted factor, net tonnage. The framework in that chapter shows how to obtain an expression for the optimal level of one factor while holding the levels of the other fixed factors constant. Having defined the various functions and elasticities that describe the underlying production relationship, the next step is to choose a particular functional form which permits the estimation of the restricted profit function in (4.6). The set of restricted profit functions available is wide; it includes the translog (Christensen, Jorgenson and Lau 1973) the generalized Leontief (Diewert 1973), the quadratic (Cowing 1978), and the normalized quadratic (Diewert and Ostensoe 1987). They are all linear in parameters. This facilitates the estimation procedure and reduces computing costs. This is the subject of the next chapter. V. ECONOMETRIC TECHNIQUE AND RESULTS This chapter begins by discussing the properties of the functional form chosen to represent the restricted profit function defined in Chapter 4 and why they are desirable in the context of this thesis. Next, a brief description of the data is presented and is followed by the equations to be estimated. There are two types. The first is linear in all parameters, whereas the second is nonlinear in some parameters. As such, they require different estimation techniques which are described in detail. In the final section the structure of the harvest technology is presented and analyzed. The section includes a discussion about the elasticities of intensity and the implications of these results for the effectiveness of restrictions on specified inputs per vessel; an analysis of the relationships between pairs of variable inputs, as given by the cross-price elasticities of demand; and measures of the type of returns to scale or size exhibited by each of the four vessel types for which separate restricted profit functions are defined. A. ECONOMETRIC MODEL The previous chapter outlines the conditions necessary for a dual restricted profit function to represent the fish harvesting technology. Since it is desirable to estimate individual pairs of elasticities between inputs, it is necessary to choose a flexible functional form to represent the restricted profit function defined in equation (4.6) (Diewert 1974).t For this thesis the normalized, quadratic, restricted profit function is tA functional form is said to be flexible if it can provide a second-order approximation to an arbitrary twice differentiate function (Diewert 1974). Flexibility implies that the function is capable of defining separate elasticities of substitution between pairs of inputs. Furthermore, it does not restrict these elasticities to adopt prespecified values. 58 Econometric Technique and Results / 59 adopted (Diewert and Ostensoe 1987)t and is assumed to be an exact representation of the true restricted profit function (White 1980).+ 1. The Normalized, Quadratic, Restricted Profit Function Arguments in the restricted profit function are prices, P^, of the (N) variable outputs and inputs, and quantities, of the (M) fixed factors. The normalized, quadratic, restricted profit function is given in equation (5.1). The first price (the output price of salmon) and the first fixed factor (the stock of fish) are chosen as the numeraires. (5.1, * ( F , z > = M - i ° i z i ? / i i - i a i k ( p i p k > / p ' + * hi " i p i ? = i 5 = i b 3 1 < z j z i > / z -N M + s ,s " c. .P.z. The parameters are the a . , a . . . B. , b . , , c . . , however, Diewert and Ostensoe (1987) and Diewert and Wales (1987) note that the a.. , j = 1,...,M and B^, i = 1,...,N parameters may be arbitrarily preset by the researcher without losing flexibility. Diewert (1986) suggests an a priori choice for the is 1/z', where z_j is the fixed factor vector for the first observation. Likewise, the B. is set equal to tThe authors suggest the form should be called the Generalized Fuss restricted profit function because it generalizes a functional form given by Fuss (1977). tThe chosen flexible functional form may be defined either as an exact representation of the true function, or as an approximation. This distinction is important whenever tests of the structure of technology are undertaken, for example, separability or aggregation tests. The problem with assuming that the functional form is an approximation is that one can no longer interpret the estimation error as simply a deviation from the profit-maximizing value of profit. Instead, the error may include a component attributable to approximation error. Econometric Technique and Results / 60 1/p'. This convention is adopted for the empirical work described in this thesis. In order for a restricted profit function to describe the underlying production technology, the chosen functional form is required to have certain characteristics. These are linear homogeneity in prices, symmetry of cross-price terms, monotonicity in the output supply and input demand functions, and convexity in prices for each fixed factor. The normalized, quadratic, restricted profit function in equation (5.1) may satisfy all of these requirements. First, it is noted that the function is linearly homogenous in prices because all prices are normalized by the first price, P x . Thus, throughout the research linear homogeneity is a maintained hypothesis. The choice of a numeraire price is an arbitrary decision, as is the choice of a numeraire for the fixed factors.t Acceptance of the monotonicity condition is verified by checking the signs of the predicted input demands and output supply obtained from the estimation procedure; this is not a statistical test and monotonicity cannot be imposed upon the parameters. The requirement that cross-price terms be symmetric is obtained by defining matrix A = [ a . , ] as symmetric with components equal to the parameters, i = 1,...,N and k = 1,...,N. Because of the linear relationships between the columns in the A matrix the first row and column of A, ee., a„ . through a ^ for all k=1,...,N, are taken to be vectors of zeroes. The symmetry condition is tested and subsequently imposed upon the estimating equations. There is another important characteristic that a well-behaved restricted profit funtion tThe choice of numeraires may affect the results obtained. However, there have been no Monte Carlo studies done yet to evaluate the robustness of the parameter estimates and elasticities when different numeraires are chosen. Preliminary work that I have done reveals that the choice of price numeraire makes little difference either to the parameter estimates or to the elasticities derived from these estimates. Econometric Technique and Results / 61 should exhibit, ie., convexity in prices. Convexity in prices is accepted globally (and locally) with the normalized, quadratic, restricted profit function whenever the A matrix is found to be positive semidefinite. However, if the estimated components of the A matrix do not satisfy this convexity requirement, it may be imposed without loss of flexibility. In this respect, the normalized, quadratic functional form has an advantage over other forms, such as the translog. The latter can only be locally convex; attempts to impose global convexity lead to a reduction in the number of independent parameters that may be estimated. This leads to a less flexible function (Diewert 1976) and destroys the ability to identify individual elasticities of substitution. In order to impose convexity on the normalized quadratic the A matrix is reparameterized as described in Wiley, Schmidt, and Bramble (1973). It is replaced by the product of a matrix E and its transpose, E T , ie., A = E E T . The E matrix is a lower triangular matrix with zeroes in its first column. However, the cost associated with imposing global convexity is that the normalized, quadratic, restricted profit function becomes nonlinear in some of the parameters to be estimated. Since convexity is an important condition, it is worthwhile to discuss what it means to reject it. Convexity in prices is one of the accepted tenets for profit-maximization to occur. If the profit function is not convex, this means that the profit-maximizing output supply and input demand functions are not well-defined and may not be well-behaved, ie., they may have discontinuities or kinks or have the wrong slopes. It may be possible for the producer to alter his output and input decisions, so as to increase his profit. O n the other hand, Wales (1977) suggests that non-satisfaction of convexity in prices may simply reflect the fact that Econometric Technique and Results / 62 the functional form chosen does not provide a good fit for the data over the sample range used. He continues by remarking that good elasticity estimates may still be obtained whenever the profit function is not convex, however, they may not be close to the true values. Given that the estimated output supply and input demand equations generated in this chapter are used subsequently to calculate rent dissipation in the fisher)', it is desirable that they be well-behaved. For, this reason convexity is imposed whenever the linear estimates indicate that it is not accepted.* In addition to its ability to impose convexity, the normalized, quadratic, restricted profit function has two other features that make it a more useful functional form than the commonly estimated translog. First, if it is desirable to obtain parameter estimates for the cross-fixed factor terms in the translog form, for example, because these parameters are needed to calculate elasticities of intensity, then the researcher has two options. The first is to estimate a set of variable and fixed input share equations; the second is to estimate the restricted profit function, along with a set of variable input share equations. Option one is often not possible since it requires the researcher to assume that markets for fixed factors are in equilibrium at their current market rental prices. In other words, the levels of the fixed factors may be costlessly adjusted to their optimal levels. This is most likely not the case for the model discussed in Chapter 4, so this option is not valid for this thesis. The problem with the second option is that, in estimating the restricted profit function, tThere are other reasons why the estimated profit function may not accept convexity in prices. Insufficient price variation and too few observations may lead to poor parameter estimates. These estimates may then lead to the rejection of convexity in prices. Furthermore, aggregating over a number of items, either inputs or outputs, may lead to the rejection of price convexity. Econometric Technique and Results / 63 the researcher may be faced with data-based multicollinearity problems. In addition, there may not be sufficient degrees of freedom because of the large number of parameters to be estimated. Even if the researcher decides to ignore these two problems and use the second option, a further difficulty may be encountered. This occurs when vessel-owners experience negative profits in a season and do not shut down as do competitive firms in other industries. This is because the vessel-owner may have to fish the entire season before determining whether, in fact, the season has been profitable for him. If the translog functional form is adopted and the profit function is estimated along with some of the input demand and output supply equations, most econometric packages replace the log of a negative number with zero. This causes an unknown bias in the parameter estimates. O n the other hand, dropping observations that exhibit negative profits leads to sample selectivity bias.t In using the normalized, quadratic, restricted profit function the researcher may obtain unbiased estimates of all the parameters by using the complete set of input demand and output supply equations. The second advantage of the normalized quadratic is that input and output quantities are expressed as levels, not shares. Since price and quantity information are available for the British Columbia commercial salmon fishery it is desirable to take advantage of this fact. Furthermore, output and input demand predictions expressed in levels facilitate the analysis of rent dissipation in chapter 6. As defined in equation (5.1) the normalized, quadratic, restricted profit function has the minimum number of parameters necessary for it to be flexible when t O n e might expect the finding of negative profits to be fairly common, however, there is no discussion of the associated problems in the literature. Econometric Technique and Results / 64 representing a constant returns to scale technology in M fixed factors (Diewert and Ostensoe 1987). The number of independent parameters in this case is ((N(N-1)/2) + (M(M-1)/2) + NM). Thus, the matrix B = [ b ^ 1 ] , like matrix A, is symmetric and has components given by the parameters b ^ , j = 1,...,M and 1 = 1,...,M. Its first row and column are also vectors of zeroes. If, however, it is desirable to allow for non-constant returns to scale within the context of the restricted profit function, three more sets of terms must be added to those already in equation (5.1). These three terms are given in equation (5.2).t ( 5- 2 ) ? = > i p i ? = i b j z j / Z l + * S . ^ o O i V / z , + J ^ C . P . This adds M + N new parameters to be estimated. They are b j , j = 2,...,M, b 0 , and i = 1,...N. It is noted that b , is set equal to zero. The hypothesis of constant returns to scale in the fixed factors is easily tested. It is a joint test that b 0 = b 2 =... = b =0 and c 1 =... = 0^  = 0. This test generates an F-statistic which may be compared to a critical value for acceptance or rejection of the null hypothesis of constant returns to scale. Alternatively, a log-likelihood ratio test may be used. Since a test of the degree of returns to scale is of interest, the more general version of the function, equations (5.1) and (5.2), is taken as the description of restricted profit. I estimate a system of equations, consisting of a single output supply and three variable input demands. These equations are obtained from the restricted profit function (equations (5.1) and (5.2)) by applying Hotelling's Lemma (Hotelling 1932, tThese terms are obtained by adding an additional fixed factor, eg., Z , to the function and setting it equal to a constant for each observation (Bfewert and Ostensoe 1987). Econometric Technique and Results / 65 Gorman 1968). This states that the first derivative of restricted profit with respect to the output price is the output supply equation. The first derivative of restricted profit with respect to an input price is an input demand equation. The specific equations derived from the normalized, quadratic, restricted profit function are given in the second section of this chapter. Before discussing them I conclude this section with a discussion of the data used in the research. 2. Data The data requirements of a profit-maximizing model of output supply and input demand behaviour are substantial. The necessary components include the unit prices of all variable inputs and output, their associated quantities, and the levels of restricted factors. Furthermore, this information must be available for enough micro units, so as to permit a sufficient number of degrees of freedom. This type of micro level production study has never been attempted for this fishery, due largely to the lack of expenditure and price information in general, let alone for specific vessels. There is no single source of data, although the Department of Fisheries and Oceans is the major one. Economists in the Program, Planning, and Economics Branch have been most generous in providing access to survey data and other background material. The Economics and Statistics Branch has been instrumental in permitting the use of output data. There are two major sets of data. The first is a a cross-sectional survey of Pacific Coast fishermen for 1982. This provides expenditure information by vessel. The second is the 1982 Sales Slips file which supplements the first by providing revenue and output information. An important Econometric Technique and Results / 66 part of this work is the linking of these two data sources in a manner that preserves the confidential nature of each data set. The calculation of each component is described in detail in Appendix 1. A brief description is now given to facilitate the analysis of the empirical results that follow. Four samples are estimated. They correspond to the four types of vessels used to fish salmon in the coastal waters of British Columbia. The first sample contains the seine boats that use nets to entrap the fish and are the largest vessels fishing for salmon. This sample has the least number of observations, ie., 21. The second sample is the gillnet fleet. This vessel type also uses nets but is the smallest in the entire salmon fleet. The number of observations is 80. The troll sample has the largest number of observations, 84, and is next in size to the seine boats. The means of capture is by baited hooks attached to many lines behind and to the side of the vessel as it cruises along the coast. Finally, the gillnet-troll sample is a new kind of vessel that can use both gillnets and troll lines. Its size falls between that of a troll vessel and a gillnet vessel. There are 60 observations in this sample. In each case the output is salmon only. Vessels that catch herring are excluded for reasons outlined in Appendix 1. For each vessel a complete data set includes the quantity and price of one output and three variable inputs, labourt , fuel, and gear, as well as the levels of three fixed inputs, net tonnage, fishing days, and fish stock abundance. Output is tLabour is treated as a fixed input, rather than a variable one, for the gillnet fleet. This is because most gillnet vessels are small and owner-operated. For example, in the sample used in this thesis 19 out of 80 vessels used one crew member in addition to the skipper. Econometric Technique and Results / 67 measured as a positive number and variable input quantities as negative numbers.t Since the framework is that of a single output, i.e., the total quantity of landed salmon, it is necessary to aggregate the five salmon species into a single output or catch for which an aggregate price index must be obtained. From the 1982 sales slips data, which record every sale by a vessel, I can generate a Divisia index of the aggregate output price (Diewert 1976). This is set equal to 1.00 for the first observation in each sample.* The implicit aggregate quantity is obtained by dividing total fishing receipts by the aggregate price index. Quantity is measured in 10,000 pounds of aggregate salmon catch* In order to obtain a unit wage for labour it is necessary to construct an opportunity cost wage, as has been done for many other fisheries (Roy, Schrank, and Tsoa 1982, Hannesson 1983, Squires, 1984, 1987a; Bj0rndal 1984). This is because the actual remuneration to the crew is in the form of a pre-specified share of catch value. The return to labour varies from year to year and more importantly, depends upon the catch rate. It is not an exogenous variable. Thus, by using actual remuneration the researcher runs the risk of introducing a potential simultaneity bias which may make the estimated parameters inconsistent. The opportunity cost or alternative wage is chosen to be the average weekly earnings for an industrial composite category (Anderson 1977). Earnings are obtained for the province of British Columbia on average and for five important regional tThis is a common practice in studies that adopt the restricted profit framework because it makes the results easy to interpret. +AII variable input prices are indexed in this way. *For the seine sample quantity is measured in 100,000 pounds of salmon. Econometric Technique and Results / 68 centres. Since, the survey indicates the homeport, each vessel is assigned to a specific region. The true opportunity cost wage should reflect the difficulty of obtaining employment, so weekly earnings are weighted by the probability of being employed. This probability is region-specific and defined as one minus the average annual unemployment rate in the region. The expected average weekly earnings are then multiplied by each vessel's number of weeks fished in order to generate the opportunity cost for the salmon fishing season. The skipper is assumed to have the same weekly opportunity cost, but the seasonal cost is assumed to be greater in order to account for time spent on repairs and maintenance prior to the beginning of the season. , A Divisia index formula is used to generate the aggregate wage for both types of labour. The number of labour units comes from the survey data and the number of weeks fished from the sales slips data. The implicit aggregate index of labour quantity is determined by dividing the total expenditure on labour by the aggregate wage index. Fuel prices are obtained from Esso Canada for gasoline and diesel products sold in eleven centres. Fuel expenditures are divided by the relevant fuel price to find the quantity of fuel input. The gear input consists of the nets, lines, traps, etc., used by each vessel. Unfortunately, data limitations preclude inclusion of electronic equipment in this measure. Gear is taken to be a malleable capital good whose services are not entirely exhausted in one year. For each gear type a stock (in quantity terms) is Econometric Technique and Results I 69 obtained by adding the existing quantity, as of 1 January 1982, to the newly purchased units (over the 1982 season) of a given type. It is assumed that each unit of gear, no matter its age, provides a constant flow of services. That is, the gear does not deteriorate because the service flow is maintained through annual repairs. A rental cost of gear is calculated for each type of gear using a modified version of the standard Jorgenson capital services price measure (Jorgenson 1963). The modification includes the cost of repairs and maintenance (Schworm 1977). The nominal interest rate is assumed to be the same for all gear types and boats. It is calculated as the average (over the year) of the monthly rates on business loans plus 1%. The business loan rate of 16.81% is taken from the Bank of Canada  Review. The one percent is added because fishermen are usually charged the prime plus one percent on loans for equipment and gear.t A Divisia gear price index is constructed using quantity and unit rental price data. The implicit aggregate index of quantity is obtained by dividing the total value of gear by the aggregate price. With regard to the fixed or restricted factors, the data come from various sources. Net tonnage is taken from the Department of Fisheries and Oceans license records. It has been collected in such a way as to preserve its confidential nature. It is indexed so that the first observation in each sample has a value of 1.00. Although Fisheries and Oceans tries to regulate both the net tonnage and the vessel length tPersonal communication from loan personnel at the Gulf and Fraser Credit Union, Vancouver. Econometric Technique and Results / 70 it seems more appropriate to use net tonnnage in this research. This is because the values of the fishing licences are usually expressed in terms of dollars per net ton. This facilitates the transition between the work done in chapter 5 to that in chapter 6. The survey also has information on the number of fishing days. It is assumed that the number of fishing days per vessel is fixed at a level which is less than that desired by the vessel-owner. This is because of the way the fishery is managed. The regulator may declare an area closed to all fishing or open only to certain vessel types. In addition, nature also has control over the maximum number of fishing days. Between these two forces, the fishing vessel finds its fishing days to be restricted within the season. The stock abundance encountered by each vessel is also assumed fixed within a season. By including this variable it is not the intention to measure the impact of crowdingt or stock* externalities upon the individual fishing firm's technology. Rather, the intention is to observe whether stock abundance is an important determinant of the demand for variable inputs, and further to examine its impact on the output supplied by the firm. The catch of each vessel is so small in proportion to the stock of fish that the assumption of fixity is reasonable. Data used to construct the stock variable come from several publications of the Department of Fisheries and Oceans. For each vessel the stock encountered is calculated as the relative abundance in each fishing area weighted by the number tThis is defined as occuring when too many vessels on a ground may cause damage to equipment and vessels. *This is defined as a reduction in the stock of fish that causes per unit harvesting costs to rise and is typically a long-run concern. Econometric Technique and Results / 71 of weeks the vessel fished in that area. This is a single measure of salmon abundance, so it is an aggregate of the five species. B. ECONOMETRIC TECHNIQUE This section has two parts which correspond to the estimation of the normalized, quadratic, restricted profit function with and without convexity in prices imposed. The linear case, which does not impose convexity, is discussed first. Then the nonlinear case, which imposes convexity, is presented. 1. Linear Case The estimating equations are obtained from the normalized, quadratic, restricted profit function with non-constant returns to scale, formed by adding the terms in equation (5.2) to those in equation (5.1), by differentiating with respect to the variable input and output prices. For the one output, three variable inputs, and three fixed factors case there are four estimating equations that correspond to the single output supply equation and the three variable input demand equations. Output is defined as the salmon catch per vessel, X , . The three variable inputs are labour services, X 2 , fuel, X 3 , and gear services, X t t . The fixed factors are taken to be the stock of fish encountered by a vessel, Z ^, the net tonnage of the vessel, Z 2 , and the number of fishing days, Z 3 . The output price is adopted as the numeraire price and the stock of fish as the numeraire good. The equations for the output supply and input demands are given in (5.3) and (5.4). For convenience they are expressed with symmetry in prices imposed, although a test of this condition is made. In addition, certain parameters that appear in (5.1) and (5.2) are set equal to zero as previously discussed. Econometric Technique and Results / 72 (5.3, i, = -i ? . » a . z . |_. a u ( P j P k )/Pf + « ' 5-*2 !-2 b j l ( Z j Z l ) / Z ' + 5-i C , j Z j + 0, J ! b.Z./Z, + Jbofl./z, + c, J Z J J ( 5 " 4 ) " X i = ?=31 ^ Z j £4 ^ k ^ / ^ + « i 54 U b j i ( z j z i ) / z ^ + 5 = i c i j z j + 0 £ 3 b.Z./Z, + i b o / 3 . /Z, + c. X ] — Z J J 1 1 for i = 2,3,4 There are 28 independent parameters in (5.3) and (5.4) once the necessary restrictions for cross-price symmetry and cross-equation equality are imposed. The three types of parameter restrictions are given in (5.5), (5.6), and (5.7). The first requires parameters in (5.3) to equal corresponding parameters in (5.4). The second describes restrictions on parameters across the equations in (5.4). Type three restricts parameters in all four equations to be equal. (5.5) a.. : X , = a.. : - X . for i = 2,3,4 and k = 2,3,4 i k 1 i k l (5.6) a., : - X . = a.. : - X . for i = 2,3,4 and k = 2,3,4 for all i * k i k l i k k 'o X , = b 0 : ~ X i for i = 2,3,4 Econometric Technique and Results / 73 (5.7) b . J X = b. . : - X . for i = 2,3,4 and 1 = 2,3 J 1 ' b . j l X = b. j l - X . for i = 2,3,4, j = 2,3 and 1 = 2,3 Prior to estimation additive disturbance terms are appended to each equation. The error structure adopted is the following. It is assumed that error terms are normally distributed with zero means and positive variance. Errors may be correlated across equations for an observation, but they are not correlated across observations. Since the normalized, quadratic is taken to be an exact representation of the true restricted profit function, it is possible to interpret the error terms as deviations from the profit-maximizing values (White 1980). Given the assumed error structure and that the parameters enter into the equations in a linear fashion, the estimation procedure is the iterative Zellner technique for seemingly unrelated regressions. The four equations in (5.3) and (5.4) are estimated as a system with the appropriate across-equation restrictions imposed. For the seine, gillnet-troll, and troll samples the equations are estimated as shown above, however, the system of equations is modified to account for the peculiarities of the gillnet sample. For this fleet 19 out of 80 vessels report one crew member other than the skipper. The rest of the vessels are one man operations. It appears that labour is a fixed input for this type of vessel. Thus, the gillnet sample is estimated with one variable output, two variable inputs (fuel and gear) and four fixed factors (stock of fish, net tonnage, fishing days, and crew size, including Econometric Technique and Results / 74 Table 5.1:--Eigenvalues from linear estimation: four vessel types  Sample Eigenvalues Seine Gillnet -non-crs -crs Troll Gillnet-Troll -1.12E-01 6.06E-02 9.37E-02 1.31E-01 7.17E-03 -7.33E-01 -5.99E-04 -7.82E-04 4.36E-01 9.64E-02 9.21E-02 6.83E-04 -5.16E-02 Econometric Technique and Results / 75 skipper). All samples are first estimated for the non-constant returns to scale case. Convergence of the parameter estimates is obtained after from 9 to 17 iterations for all but the seine sample; it requires 60 iterations. A statistical test to verify the appropriateness of this assumption of non-constant returns to scale is performed by setting equal to zero the parameters discussed earlier. The hypothesis of constant returns to scale is accepted only in the gillnet sample, so this sample is re-estimated with constant returns to scale imposed. At this point the estimated parameters of the A matrix are used to check for acceptance of convexity in prices. Parameter estimates of the elements of the A matrix are used to obtain the eigenvalues of the matrix. Because the first row and column of A are composed of zeroes, the matrix is singular. Thus, it is sufficient to check the eigenvalues of the (N-1) by (N-1) matrix to see if they are non-negative. If they are, then the A matrix is positive semidefinite and global convexity in prices is accepted. If one or more of the eigenvalues is negative, then convexity in prices is rejected both locally and globally. The eigenvalues for the A matrix are calculated and are shown in Table 5.1. Only one sample, that corresponding to the troll vessel, accepts global (and local) convexity in prices. No further estimation for the troll sample is required. However, the other three samples are re-estimated with global convexity imposed. This involves using the reparameterization technique discussed earlier that makes the input demand and output supply equations nonlinear in a few parameters. The presence of any form of nonlinearity prevents the Zellner seemingly unrelated regressions Econometric Technique and Results / 76 Table 5.2:--Eigenvalues from nonlinear estimation: three vessel types Sample Seine 1.98E-01 Gillnet -non-crs -crs 6.50E-02 9.50E-02 Gillnet-Troll -non-crs -crs 1.31E-02 2.70E-02 Eigenvalues 4.09E-04 1.97E-04 4.45E-05 5.48E-05 1.05E-01 1.36E-01 6.43E-06 1.87E-05 Econometric Technique and Results / 77 technique from being used. Instead the new equations require nonlinear estimation techniques. A discussion of the linear results for the samples that do not accept convexity is found in Appendix 3, along with the analyses of tests for symmetry and constant returns to scale, and the elasticities of interest. In the next section the nonlinear method of estimation for these samples is given in detail. Following this presentation, all relevant tests are performed on the nonlinear system and their results are shown in this chapter. Finally, the discussion about the harvest technology uses the linear parameter estimates for the troll sample and the nonlinear parameter estimates for the three other samples, seine, gillnet, and gillnet-troll. 2 . Nonlinear Case The new nonlinear equations are given in (5.8) through (5.11). (5.8) X -i( I a Z./PJ) * ( e ? P i + 2 e i e 2 P 2 P 3 + 2 e 1 e « P 2 P « + ( e i + e | ) P i + 2(e 2e„ + e 3 e 5 ) P 3 P « + (eg + e| + e|) Pj) + + (5.9) < £ " " j Z j / P i ) * ( e ^ P 2 + e , e 2 P 3 + e ^ P , ) + + Econometric Technique and Results / 78 M (5.10) -X 3 = ( f = i a j z j / P i ) * ( e , e 2 P 2 + ( e l + e i ) P 3 + ( e 2 e „ + e 3 e 5 ) P « ) + ^ 3 - 5=2 54 b j i ( z j z i ) / Z i + ?= 3i c 3 j z j + 03 £ 3 b.Z./Z, + i b 0/3 3/Z, + c 3 j ^ J J (5.11) - x , = ( S = " a j Z j /P, ) * ( e , e « P 2 + ( e 2 e « + e 3 e 5 ) P 3 + (eg + e§ + e!)P„ ) + 5-2 5 = 2 b j l ( Z j Z l ) / Z ^ + U C 4 j Z j + 0, £ _ 3 b.Z./Z, + i bo04/Z, + c« The 'e' parameters are related to the a parameters in (5.3) and (5.4). These relationships are given in Appendix 3. The set of equations in (5.8)-(5.11) must be estimated as a system using a nonlinear estimation technique. A maximum likelihood procedure is used and the particular algorithm is the Davidson-Fletcher-Powell routine. After a number of iterations, between 80 and 200 depending upon the sample, convergence is achieved. The convergence criterion is fairly stringent, ie., the new coefficient values can be no more than 1.0E-05 different from estimates obtained from the previous iteration. The nonlinear parameter estimates and their standard errors, along with summary statistics, are given in Appendix 2. In order to check that a global maximum has been achieved three different sets of starting values for the parameters are tried. For the first attempt values from the linear estimation are used as starting values for the parameters that are still linear. For the new nonlinear parameters starting values of one are used. Next, starting Econometric Technique and Results / 79 Table 5.3:--Testing for constant returns to scale: four vessel types Sample LLF(R) LLF(U) -2LQG(n) xi V a l u e Decision (0 = 0.010) Seine -124.398 -95.612 57.573 18.475 Reject Gillnet -50.453 -46.319 8.268 18.475 Accept Troll -460.418 -447.365 26.106 18.475 Reject Gillnet-Troll -536.565 -526.621 19.890 18.475 Reject Note: The null hypothesis of constant returns to scale cannot be rejected if the calculated value of -2LOG(y) is less than the critical value. The number of degrees of freedom used to determine the critical value of x2 is given by the number of restrictions. For each sample this number is 7. The log-likelihood values for the troll sample are obtained from linear estimation. Those for the rest of the samples come from nonlinear estimation. For a different level of confidence, ie., where a is 0.005, the critical value of x2 is 20.278. In this instance the hypothesis of constant returns to scale cannot be rejected by the gillnet-troll sample. Econometric Technique and Results / 80 values of one are used for all parameters. Finally, starting values of ten are used for all parameters. In each case the final parameter values are the same to at least 4 digits, thereby ensuring consistency in the results. Once the basic estimates are obtained each sample is checked to make sure that the equations have been specified correctly. This is done by verifying whether convexity in prices is accepted. All samples accept this condition, as shown by the eigenvalues in Table 5.2 for the new reparameterized A matrix.t Incidentally, by imposing convexity in prices in this way, symmetry in prices is also imposed. Many researchers, in testing for symmetry, have been unable to accept the hypothesis. However, it is always imposed subsequently. Along with convexity in prices, the estimated equations should satisfy the monotonicity condition. This is verified by examining the signs of the predicted output supply and input demands calculated for each observation. The former should all be non-negative, and the latter all non-positive. However, this is not a statistical test. The troll and seine samples satisfy this requirement for all observations. O n the other hand, the gillnet-troll and gillnet samples satisfy it for output supply and all input demands, with the exception of gear. The gillnet sample, both with and without constant returns to scale, has one observation with a positive gear demand. The gillnet-troll sample has three observations with positive gear demands. However, their values are close to zero. tit is noted that this table shows two sets of results for the gillnet-troll fleet. This is because the hypothesis of constant returns to scale cannot be rejected using the nonlinear estimates, although it is rejected in the linear case. Econometric Technique and Results / 81 Table 5.4:-Goodness of fit: four vessel types Sample Equation R ; Seine Output 0.3503 Labour 0.3681 Fuel 0.1695 Gear 0.2210 Gillnet -non-crs Output 0.2289 Fuel 0.1645 Gear 0.1568 -crs Output 0.2172 Fuel 0.1438 Gear 0.1134 Troll Output 0.2477 Labour •;. 0.3961 Fuel 0.3814 Gear 0.0084 Gillnet-Troll -non-crs Output 0.1853 Labour 0.3171 Fuel 0.1979 Gear 0.0572 -crs Output 0.2140 Labour 0.2535 Fuel 0.1057 Gear 0.0568 Note: The R 2 values for the troll sample are those derived from, the linear estimation technique. Since the other samples are estimated using a nonlinear technique the R 2 values are calculated as the squared correlation between the predicted and the actual values. Econometric Technique and Results / 82 In all, nonlinear estimation techniques are used to obtain parameter estimates for five samples. The first is the seine sample with non-constant returns to scale. Numbers two and three are for the gillnet sample, both with and without constant returns. Numbers four and five pertain to the gillnet-troll sample, again with and without constant returns to scale. The test for constant returns to scale is performed by using a likelihood ratio test. Each sample is estimated as a system of equations in an unrestricted form, ie., without the hypothesis of constant returns to scale imposed, and a restricted form, with constant returns to scale imposed on the parameter estimates. As mentioned earlier this requires that the following parameters be set equal to zero, b 0 , b 2 , b 3 , c , , c 2 , c 3 , and c u . t A test statistic is formed from the logs of the likelihood functions for the unrestricted and restricted cases. It is calculated as in equation (5.12), (5.12) -2LOG (M)= -2[LOG(R) - LOC(U)], where LOG(R) is the log of the likelihood function in the restricted case. This is distributed as a Chi-square random variable with degrees of freedom equal to the number of restrictions imposed; in all samples the number of restrictions is 7. If the calculated Chi-square is greater than the critical value, which depends upon the number of restrictions, then the null hypothesis of constant returns to scale is rejected. Table 5.3 presents the relevant test results. The seine sample rejects the hypothesis of constant returns to scale decisively, whereas the gillnet sample accepts it. The gillnet-troll sample accepts it at one level of significance (a = 0.005%), while rejecting it at a — 0.010%. For completeness, the test for the troll sample tFor the gillnet sample the relevant parameters are b 0 , b 2 , b 3 , b 4 , c , , c 2 , and c 3 . Econometric Technique and Results / 83 Table 5.5:--A comparison of the linear and nonlinear log-likelihood functions Sample Likelihood function Likelihood function (Linear) (Nonlinear) Seine -81.449 -95.612 Gillnet -non-crs -45.858 -46.319 -crs -49.790 -50.337 Gillnet-Troll -non-crs -516.184 -526.621 -crs -528.890 -536.702 Econometric Technique and Results / 84 using the linear estimation results is included in the table. This sample also rejects the hypothesis of constant returns to scale. Before examining the characteristics of the estimated harvest technology for the four samples it is necessary to discuss how well the estimated equations fit the data. For the troll sample, linear estimation techniques provide summary statistics on each equation. In particular, the calculated R 2 values can be used to evaluate the goodness of fit. For the samples estimated by means of nonlinear techniques R 2 values are not generated. However, a comparable measure may be obtained by calculating the covariance between predicted quantities and actual quantities. It is then divided by the standard errors of the predicted and actual quantities to yield a correlation. Squaring the correlation generates an R 2 - type measure. The goodness of fit statistics are presented in Table 5.4. They show, for example, that the estimated output equation for the troll sample explains 24.77% of the variation and the estimated labour demand equation explains 39.61% of the variation. For cross-sectional data these measures are quite acceptable. The input demand equation that obtains the worst results consistently across all samples is that of gear services. A possible explanation is that this factor has characteristics which make it similar to both variable and fixed factors. It is also instructive to examine the effect of imposing price convexity on the four vessel types used in the fishery. This may be done by comparing the values of the log-likelihood functions obtained from the linear estimation with those from the nonlinear estimation. Strictly speaking the models are not comparable, one being linear and the other nonlinear. However, it is possible to see whether imposing Econometric Technique and Results / 85 convexity is an unreasonable restriction. The relevant values are found in Table 5.5. The larger (negative) log-likelihood values, obtained from the nonlinear estimation, reflect the fact that a restriction has been imposed, although the same number of parameters is estimated in each case. If the likelihood values are not too different, the convexity restriction is not an unreasonable one. In most cases the values do not differ much. C. ECONOMETRIC RESULTS The discussion that follows compares results for the seine, gillnet-troll, and troll fleets in the non-constant returns to scale case, and for the gillnet fleet with constant returns to scale. The estimates for gillnet-troll with constant returns to scale and for gillnet with non-constant returns do not differ much from the ones to be presented. A brief discussion of their differences follows the analysis. All samples, with the exception of the troll sample, are estimated using nonlinear techniques. The troll sample is estimated by standard linear techniques. 1. Own- and Cross-Price Elasticities of Supply and Demand The estimated parameters from equations (5.3) and (5.4) or (5.8) and (5.11) can be used to calculate output-variablet own and cross price elasticities of supply or demand. As discussed in Chapter 4 they are defined as the first derivative of restricted profit with respect to an output or input price times the ratio of price to quantity. Appendix 4 derives these elasticities for the normalized, quadratic, tThese elasticities have both a substitution effect and an output effect, since elasticities obtained directly from a restricted profit function do not hold output constant. They are in contrast to output-constant elasticities obtained directly from a cost function. Econometric Technique and Results / Table 5.6:--Nonlinear estimates of output-variable own- and cross-price elasticity seine Quantity/ Output Labour Fuel Gear Price Output 0.290 -0.144T -0.288 0.142T (0.246) (0.089) (0.274) (0.092) Labour 0.050T -0.025 -0.050T 0.024t (0.031) (0.021) (0.029) (0.014) Fuel 0.112 -0.055T -0.111 0.055 (0.106) (0.032) (0.123) (0.044) Gear -0.025T 0.013T 0.025 -0.012T (0.016) (0.007) (0.020) (0.008) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 87 restricted profit function. Tables 5.6 to 5.9 display these elasticities. They are calculated at the mean values of each of the samples.+ In general, the majority of the elasticities for the seine, gillnet, and troll fleets are found to be significantly different from zero at a 90% significance level. Not as many are significant for the gillnet-troll fleet. The complete set of elasticity estimates is given in Tables 5.6 through 5.9. Own-price elasticities for the troll sample, Table 5.8, show that this vessel has the largest and most significant output supply response at 0.480, ie., a 1% increase in the price of landed salmon elicits an increase in the quantity supplied by a troll vessel of 0.480%. In descending order after the troll fleet response are those of the seine, Table 5.6, gillnet-troll, Table 5.9, and gillnet fleets, Table 5.7. The latter two are significantly different from zero, but very small, ie., they are 0.153 and 0.098. These results are not surprising since the troll fleet is the least regulated of the salmon-fishing fleet. Hence, it is able to choose fishing areas and times with relatively little intervention from the regulator. It is also able to target the more preferred, ie., higher priced species, by using special bait or hooks. At the other end of the spectrum, the gillnet fleet is more controlled by regulations about where it may fish. Furthermore, it is the least mobile component of the entire salmon-fishing fleet. Thus, it is not able to target productive areas or move effectively in response to price variation. It had been anticipated that the response from the seine vessels would be greater tAsymptotic standard errors are generated from the formula for the variance of a random variable comprised by either adding or multiplying several random variables that are not independent (Kmenta 1977, Judge et al. 1982). Econometric Technique and Results / Table 5.7:-Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet(crs) Quantity/ Output Fuel Gear Price Output Fuel Gear 0.098T (0.064) 0.524T (0.298) -0.054T (0.022) -0.126t (0.072) -0.672T (0.333) 0.069t (0.029) 0.028t (0.011) 0.148T (0.062) -0.015t (0.010) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 89 since they have the largest scale of operation and the most powerful engines. However, fishing times and areas are rigidly controlled for this segment of the salmon-fishing fleet. Although these vessels have the ability to respond to price variation across species and areas, they are prevented from taking advantage of their size. A comparison of the own-price elasticity of demand for labour reveals that it is significant only for the troll sample and of moderate size, ie., -0.371. This fleet, along with the seine fleet, exhibits the characteristics of an industrialized production unit. Thus, it is surprising that the own-price elasticity for labour is not significantly different from zero in the seine fleet. This may be caused by an insufficient number of observations for this sample, or from insufficient variation in the calculated labour prices. O n the other hand, imprecise estimates are often caused by multicollinearity in the data. Attempts to verify this possibility reveal no obvious signs of multicollinearity among the regressors. Turning to the own-price elasticities of demand for fuel, the notion is reinforced that the troll fleet is best able to respond to price changes. It has the largest and most significant elasticity, eg., -1.368. This is to be expected given the relatively large own-price elasticity for output. It indicates that the troll fleet is very responsive to area fuel prices since it travels to and from many areas. As well, fuel is an important component of the troll technology, since the operation requires much cruising up and down the coast. Both gillnet-troll and gillnet vessels have significant elasticities, ie., -0.677 and -0.672. These results argue in favour of the hypothesis that fishermen are very sensitive to fuel prices. In the three fleets Econometric Technique and Results / 90 Table 5.8:--Linear estimates of output-variable own- and cross-price elasticities Quantity/ Output Labour Fuel Gear Price Output 0.480T -0.112T -0.357T -0.011 (0.088) (0.029) (0.090) (0.018) Labour 0.427T -0.371t -0.065 0.009 (0.111) (0.072) (0.103) (0.031) Fuel 1.464T -0.070 -1.368t -0.025 (0.369) (0.112) (0,430) (0.039) Gear 0.015 0.003 -0.009 -0.010 (0.025) (0.011) (0.014) (0.019) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 91 mentioned fuel expenditures are a large component of total variable costs. Once again, the seine response is very small; eg., an elasticity of -0.111. It is not significantly different from zero. This may reflect a lack of variation in fuel prices or the impact of controls on the fleet's ability to respond. In general, these results show that it is important to include fuel as a decision variable for the fisherman. Estimates of own fuel price elasticities may be used to evaluate the effects of higher or lower (eg.subsidized) prices upon fishing behaviour. In turn, this has implications for the ability of the fisherman to dissipate fishery rents in the form of increased fuel costs. The last of the own-price elasticities describes the response of gear usage to a change in the price of gear. For each sample it is the smallest and least significant of all the elasticities. This suggests that gear is a complex factor that exhibits both variable and fixed characteristics. Only the gillnet and seine elasticities are not significantly different from zero, but the responses are very small, ie., -0.015 and -0.012. Cross-price elasticities obtained directly from the restricted profit function have two components, a pure substitution effect and an output effect. The output-variable input demand elasticities (e) in Tables 5.6 through 5.9 are related to the output-constant input demand elasticities (v) in Tables 5.10 through 5.13 in the following way (Sakai 1974): <5"» "il= 'Ik"- . « ' t t - u > ' « u «» i.k -2 .3 ,4 . Econometric Technique and Results / 92 Table 5.9:--Nonlinear estimates of output-variable owiv - and cross-price elasticities: gillnet-troll(non-crs) Quantity/ Output Labour Fuel Gear Price Output 0.153T 0.005 -0.160T 0.002 (0.108) (0.045) (0.082) (0.008) Labour -0.015 -0.044 0.060 -0.001 (0.128) (0.076) (0.107) (0.007) Fuel 0.590T 0.078 -0.677T 0.009 (0.301) (0.140) (0.277) (0.025) Gear -0.0002 -0.00006 0.0003 -0.000004 (0.0008) (0.0003) (0.0007) (0.00002) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a=0.10. Econometric Technique and Results / 93 In (5.13) v is the output-constant elasticity between variable inputs i and k and 6 i k ' S ^ 6 o u t P u t " v a n a D ' e elasticity derived directly from the estimated restricted profit function, e is is the cross-price elasticity between output and input k and D S £ k l ' S ^ e c r o s s " P n c e elasticity between input k and output. Finally, e is the own-price supply elasticity. From the signs of the cross-price elasticities of demand it is possible to find out the relationships between pairs of variable inputs. For example, if the sign is positive, this implies that the inputs are substitutes; if the converse is true, then the inputs are complements. The degree of substitution or complementarity is indicated by the magnitude of the elasticity. Used in conjunction with the elasticities of intensity, these elasticities indicate the ability of the fisherman to subvert the intentions of input restrictions and dissipate resource rent. Beginning with the output-variable elasticities, in the seine sample labour and fuel are found to be complements with a cross-price elasticity of labour with respect to fuel price changes of -0.05. Although very small, this value is significantly different from zero. O n the other hand, labour and gear and fuel and gear are found to be substitutes, with elasticities for the former pair significantly different from zero. The gillnet sample has only two variable inputs and a substitute relationship is found between fuel and gear. For example, a 1% increase in the price of gear leads to a 0.148% increase in the fuel demanded. In both cases, the vessel-owner may choose to travel to more areas or remain in one and use more nets to catch different species of salmon. Many observers believe the gillnetter to have few options for altering the input mix. However, the results from this sample exhibit Econometric Technique and Results / 94 Table 5.10:--Nonlinear estimates of output-constant price elasticities: seine Price/Quantity Labour Fuel Gear Labour -0.127E-12 -0.018 0.035T (0.791E-07) (0.019) (0.024) Fuel 0.039 -0.84E-13 0.117 (0.073) (0.575E-07) (0.117) Gear 0.018T 0.027T -0.428E-14 (0.010) (0.019) (0.696E-08) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a=0.10. Econometric Technique and Results / 95 the largest values for the cross-price elasticities across the four samples. Looking at the troll sample's results it is observed that none of the output-variable cross-price elasticities are significantly different from zero. An examination of individual pairs of variable input elasticities reveals that labour and gear alone appear to be substitutes. But the elasticity is very small, eg., a 1% increase in the price of gear leads to a 0.009% increase in the quantity of labour demanded. The choice is whether to make the vessel more labour-intensive or "capital"-intensive in the form of more troll lines, winches, etc. The signs of the elasticities for the labour/fuel and the fuel/gear pairs suggest complementary relationships. Unlike the seine vessel the troll vessel obtains a complementary relationship between fuel and gear. As mentioned earlier the troll operation consists of cruising up and down the coast with lines playing out and to the side of the vessel. For this fish harvesting technology fuel and gear are obvious complements. A similar set of results obtains for the gillnet-troll sample, ie., none of the cross-price elasticities is significantly different from zero. As in the gillnet fleet, fuel and gear are found to be substitutes, although, at 0.009, the gillnet-troll elasticity is much smaller. Insofar as the other input pairs are concerned, labour/fuel and labour/gear are found to have the opposite relationships to those suggested by the troll fleet. In particular, labour and gear are found to be complements. The reason may be that each gear type, gillnet and troll lines, requires the use of specialized knowledge and labour skills. O n the other hand, it seems that labour and fuel are substitutes. This might reflect the proportion of time spent using troll equipment - which is fuel-intensive versus the time spent switching back and forth - which is labour-intensive. Econometric Technique and Results / 96 Table 5.11:--Nonlinear estimates of output-constant price elasticities: gillnet(crs) Fuel Price/Quantity Gear Fuel -0.591E-11 (0.866E-06) 0.820t (0.374 Gear 0.084t (0.037) •-0.276E-11 (0.405E-06) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a=0.10. Econometric Technique and Results / 97 In conclusion, it appears that the troll and gillnet-troll fleets have much less substitutability between variable inputs than do the seine and gillnet fleets. There is a very simple and interesting explanation for this dichotomy. In the past few years the net fleets, ie., the seiners and gillnetters, have become increasingly regulated in terms of fishing areas, times, and species. These two vessels compete on the same grounds with one another and for the same species of salmon. When a run comes in the gillnetters are usually given permission to fish the first few days. Then, the seiners, which are often capable of taking the entire allotment in a short period of time, are given their chances at an area. It seems that these two fleets have developed ways of substituting variable inputs as a means of getting around the strict controls on fishing days and areas. O n the other hand, the troll and gillnet-troll fleets have been the least regulated of the fleets. In fact, the gillnet-troll fleet arose mainly because of restrictions on gillnet vessels. When an area is closed to the gillnetter, the combination gillnet-troll vessel merely moves further out to sea and uses troll gear. In this way it can avoid area and species regulations. These fleets have been less controlled and have not needed to develop substitution amongst variable inputs still under control of the captain. Earlier in the chapter a distinction is made between output-variable and output-constant elasticities. The latter are presented in Tables 5.10 through 5.13. In general, these elasticities are somewhat larger than their output-variable counterparts. The number of significant elasticities does not change nor do the the signs of the elasticities, with the exception of the troll sample. The output-constant elasticities for this sample suggest that substitution possibilities between labour and fuel, and between gear and fuel are quite large and significant when output is held constant. Econometric Technique and Results / Table 5.12:-Linear estimates of output-constant price elasticities: troll Price/Quantity Labour Fuel Gear Labour -0.217T 0.035 0.101T (0.077) (0.068) (0.061) Fuel 1.017T -0.281T 1.062T (0.441) (0.092) (0.378) Gear 0.004 -0.008 -0.010 (0.011) (0.013) (0.018) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 99 This suggests that the troll fleet exhibits a fish harvest technology that permits a great deal of variable input substitutability when output is held constant. For the other fleets, the conclusions derived from the output-variable elasticities are unchanged. It is instructive to compare the results obtained in this study with those from previous ones. Squires uses a similar framework to study two different New England fisheries, the otter trawl fishery in (1984, 1987a, 1987b) and the sea scallop fishery (1985). Since the former is a multi-species fishery, he employs a translog restricted profit function defined over three variable outputs, three variable inputs (labour, fuel, and capital) and a dummy variable for the stock level in one of the two years for which he has data. He finds positive price elasticities of supply and complementary relationships between outputs. The own price elasticities of demand are larger than those calculated for this research, eg., they range from -0.8663 for capital to -1.1485 for labour. As well, this fishery shows evidence of complementary relationships between the variable inputs. For example, a 1% increase in the price of fuel would lead to a 0.272% reduction in the quantity of labour demanded and a 1% increase in the price of capital would lead to a 0.2743% reduction. These values are somewhat larger than those obtained by this research, but the fisheries are very different. Furthermore, Squires uses two years of data, thereby picking up more price variation. In addition, the fishery studied by Squires does not have any regulations on the entry of new vessels or tonnage. Hence, there are no incentives to exploit substitution possibilities between variable inputs. For his second study, that of a single-output fishery, Squires adopts a normalized Econometric Technique and Results / 100 Table 5.13:--Nonlinear estimates of output-constant price elasticities: gillnet-troll(non-crs) Price/Quantity Labour Fuel Gear Labour -0.035T 0.048 -0.0008 (0.065) (0.091) (0.009) Fuel 0.555T -0.048 0.499T (0.264) (0.092) (0.232) Gear -0.40E-4 0.0002 -0.123E-05 (0.0002) (0.0006) (0.139E-04) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 101 quadratic functional form.t Results obtained using one year of data on the sea scallop fishery are reported in Squires (1985). They are much smaller than those obtained by this research. For example, the own price elasticity for capital is -0.242 and for fuel, -0.001. The signs for the output supply elasticity and the labour demand elasticity are incorrect, but the former is significantly different from zero and large, ie., -0.956. Squires does not accept the condition of price convexity, nor can he impose it with the functional form he uses. For this fishery, he provides evidence of substitutability between capital and labour and capital and energy. The cross-price elasticities cited in the paper are of the same order of magnitude as those generated in this thesis for the British Columbia salmon fishery. It is possible to further characterize the relationships between output and each of the inputs through two types of elasticities. The first elasticity indicates how output supplied changes when the price of an input is altered. This suggests how intensively the input is used to produce the output. Inputs can be ranked according to relative intensity of use. The second gives information about the degree by which input demand changes when the output price increases. This provides knowledge about the quality of inputs to the firm, eg., if the value is greater than unity, the input is said to be superior. If the value is greater than zero, but less than one, the input is normal. An inferior input has a negative value. The elasticities necessary to describe these relationships are given in Tables 5.6 through 5.9. For all vessel types fuel is the input used most intensively, followed +Although this has the same name as the form chosen for this thesis, the two functional forms are very different. Econometric Technique and Results / 102 by labour and then gear. This says that, as the price oi fuel decreases, output supplied increases more than if the price of another input were to decrease. Thus, the intensity of use can only be ranked if the sign of the elasticity is negative. The results described above are mirrored in the classification of inputs. Fuel is either superior, for the troll fleet, or normal, in all other samples. Gear, on the other hand, is inferior in all samples, with the exception of the troll sample. Labour is normal for the seine and troll fleets, but inferior for the gillnet-troll fleet. 2. Elasticities of Intensity In this section I investigate the importance of the relationships between the variable inputs and the restricted factors by analyzing the non-normalized elasticities of intensity (Diewert 1974). The formulae derived from the normalized, quadratic, restricted profit function are given in Appendix 4. These elasticities are to be interpreted in the following way. A 1% increase in the level of the restricted factor leads to a x% change in the quantity demanded of the variable input. A negative elasticity indicates a substitute relationship and a positive elasticity, a complementary one. The degrees of these relationships, given by the magnitudes of the elasticities, have implications for the success of a limited entry program that uses input controls per vessel in order to prevent rent dissipation. In order to understand the effects of the input restriction program upon the behaviour of the fishermen, one must first ask what the regulator's objective might be. Recognizing that the lack of property rights in the fish creates incentives for each participant to increase his own output, the regulator hopes to effectively constrain output per participant by prohibiting the use of a key input beyond a Econometric Technique and Results / 103 Table 5.14:—Nonlinear estimates of elasticities of intensity: seine Quantity/ Fixed Factor Output Labour Fuel Gear Stock of Fish -0.767t (0.520) 0.114t (0.070) 0.114 (0.582) -1.526t (0.890) Net Tonnage 0.067 (0.162) -0.010 (0.029) 0.336t (0.180) -0.425t (0.251) Fishing Days 0.483t (0.210) 0.004 (0.051) 0.167 (0.237) 0.406 (0.336) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 104 specified level. Experience shows that imposing an input restriction per vessel is not an effective means of imposing a catch restriction per vessel. In the face of an input restriction, the fisherman merely re-optimizes over the set of inputs under his control; in this way, he can increase his output. There are many means that the fisherman may use to circumvent the restriction. The typical view is that the fisherman increases the use of inputs that are substitutes for the restricted input. This is a direct effect. However, what is not so well understood is that a secondary or indirect effect can also take place and lead to higher harvesting costs. This secondary effect occurs as the use of substitutes for the restricted factor is increased and this, in turn, induces an increase in the use of complementary variable inputs. In this way, the use of variable inputs that are direct complements to the restricted input may increase, as well. This, too, leads to higher harvesting costs in the sense that the costs of taking a given level of output would be less if the fisherman could operate in an unconstrained manner. Thus, the expansion path for the fisherman is skewed by the use of input controls. There is one case where rent dissipation through input substitution is prevented. This occurs when all variable inputs exhibit a zero elasticity of intensity with respect to the fixed input, and there is no output effect. In other words, the technology is truly of the Leontief fixed proportions type. I begin the discussion of these elasticities by analyzing those obtained between the variable inputs and the net tonnage input, since the Department of Fisheries and Oceans has put an upper bound on the use of this input per vessel in order to control the actions of fishermen. Tables 5.14 through 5.17 present all the elasticities Econometric Technique and Results / 105 Table 5.15:—Nonlinear estimates of elasticities of intensity: gillnet(crs) Quantity/ Stock Net Fishing Labour Fixed of Factor Fish Tonnage Days Output 0.052 0.105 0.535T 0.190 (0.141) (0.176) (0.098) (0.231) Fuel -0.257T 0.144 0.578T 0.734T (0.157) (0.215) (0.126) (0.279) Gear 0.225 0.018 0.226T 0.965T (0.230) (0.234) (0.144) (0.301) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 106 of intensity for the four samples. I observe that fuel and gear inputs used by the seine fleet have elasticities of intensity that are significantly different from zero. This is not the case for the labour input, Table 5.14. Fuel is a complement with the net tonnage, whereas, gear and labour are substitutes. The elasticity values are fairly large, ie., that for fuel is 0.336 and that for gear, -0.425. The direct effect suggests that rent dissipation by this fleet occurs when vessel-owners substitute toward the use of an increased amount of gear and labour. O n the other hand, the indirect effect of the increased use of labour is in the direction of an increase in the use of fuel, since these inputs are complements. However, since gear and fuel are substitutes, the net result depends upon the relative strength of the elasticities. Nonetheless, this vessel type appears to have a number of channels that it may use to dissipate rent. Most observers are convinced that seiners are the vessels best able to dissipate the rent of the British Columbia salmon fishery. In contrast, the elasticities of intensity for the gillnet fleet are not significantly different from zero. Since both are positive this indicates a complementary relationship between fuel and tonnage and gear and tonnage, Table 5.15. Thus, there is no direct effect for this fleet. O n the other hand, in order to increase output this type of vessel uses fuel quite intensively. It appears that there is little scope for rent dissipation by this segment of the salmon fleet, but it should take the form of too much fuel. Most observers of the fishery would agree with this finding, since the gillnetter has the smallest scale of operation, and can really expand only toward the use of more fuel as it travels to a greater number of areas, in an attempt to increase its output. Econometric Technique and Results / 107 Table 5.16:--Linear estimates of elasticities: troll Quantity/ Fixed Factor Output Labour Fuel Gear Stock of Fish 0.278T (0.085) 0.247t (0.072) 0.101 (0.091) 0.136 (0.668) Net Tonnage 0.525T (0.199) -0.354T (0.194) 0.744t (0.242) -1.222 (1.569) Fishing Days 0.623T (0.162) 0.022 (0.139) 0.246 (0.175) 0.323 (1.290) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 108 Turning next to the troll fleet, fuel and labour are found to have significant elasticities of intensity with respect to net tonnage, although they are of the opposite sign, Table 5.16. These elasticities are fairly large, ie., an elasticity of -0.354 for labour indicates that it is a substitute for net tonnage, whereas fuel is a complement with an elasticity value of 0.744. In addition, the elasticity for gear is very large, ie., -1.222, but not significantly different from zero. The direct effect encourages substitution toward the use of more labour and gear on board the vessel. In addition, there is an indirect effect that induces an increased amount of fuel usage, since fuel is a complement to both labour and gear. The troll fleet appears to be capable of dissipating resource rent by increasing the use of all three variable inputs against a restriction on the use of net tonnage. In fact, it is commonly accepted that troll vessels are among the worst offenders of rent dissipating behaviour. Results for the gillnet-troll fleet suggest that all three variable inputs are complements to the net tonnage. Furthermore, the elasticity values are all very large and significantly different from zero. They range from 0.452 for labour to 1.608 for gear, Table 5.17. Thus, there is no direct effect. Furthermore, although the cross-price elasticities show evidence that labour and gear are complementary inputs, they are extremely small and not significantly different from zero. It appears that this vessel type does not have many ways of dissipating rent in the face of a restriction on net tonnage. I want to look next at the restricted factor, fishing days. For the seine fleet, the elasticities of intensity indicate complementary relationships between fishing days and Econometric Technique and Results / 109 Table 5.17:--Nonlinear estimates of elasticities of intensity: gillnet-troll(non-crs) Quantity/ Stock Net Fishing Fixed Factor Fish Tonnage Days Output 0.098 1.167T 0.247T (0.183) (0.384) (0.123) Labour -0.016 0.452T 0.277T (0.077) (0.170) (0.065) Fuel 0.003 0.571t 0.507t (0.157) (0.356) (0.129) Gear -0.872T 1.608 -0.115T (0.476) (1.253) (0.381) Note: Asymptotic standard errors are in parentheses and the symbol "t" : that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 110 the three variable inputs, so there is no direct effect, Table 5.14. Furthermore, by restricting the number of fishing days, the regulator is able to restrict the output per vessel, since the elasticity of output with respect to days is very large and positive. In light of these results it appears that D F O has been able to prevent potential rent dissipation by controlling the seiners' total access to the fish. This is also true for the gillnet fleet, Table 5.15. Both gear and fuel are complements with fishing days and have fairly large and significant elasticity values, 0.578 (fuel) and 0.226 (gear). Like the results for the seine and gillnet samples, variable inputs used by the troll vessels also have complementary relationships with the number of fishing days, Table 5.16. These elasticities are large, but not significantly different from zero. Once again, the regulator appears to have prevented a great deal of rent dissipation by controlling the number of fishing days. The gillnet-troll fleet has somewhat different results than those obtained for the other three samples. Although it exhibits a complementary relationship between labour and fishing days and fuel and fishing days, with both elasticities significantly different from zero and of medium magnitude, gear appears to be a substitute for fishing days, Table 5.17. The direct subsitution effect is shown by an increase in the use of gear, instead of fishing days. The indirect effect is an increase in labour, along with the increased use of gear, since these inputs are complements. However, for this fleet there is some doubt as to whether the constraint on fishing days is truly binding. The need for a combination boat arose in response to restrictions on fishing days imposed upon the single-operation gillnetters. In their ability to switch gear types, the gillnet-trollers also extend the number of days that they may fish. Thus, it appears as if there is some potential for rent dissipation by Econometric Technique and Results / 111 Table 5.18:-Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet(non-crs) Quantity/ Output Fuel Gear Price Output 0.075 -0.099T 0.025T (0.061) (0.069) (0.011) Fuel 0.413T -0.550T 0.137T (0.289) (0.326) (0.059) Gear -0.048T 0.064T -0.016T (0.021) (0.028) (0.010) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 112 this segment of the salmon fleet. For much of the last two decades the Department of Fisheries and Oceans has concentrated upon controlling the behaviour of the largest operators, the seiners. It appears that it has succeeded in preventing much rent dissipation by this type of vessel. However, the regulator has all but ignored the actions of the other segments of the fleet, in particular, the gillnet-trollers and the trailers. My results suggest that there is still scope for rent dissipating behaviour by these two components, especially the latter. One means of controlling their behaviour might be to introduce area licensing which would restrict access to certain areas. In fact, a two area scheme has recently been implemented for the troll fishery. I conclude by discussing the elasticities of intensity between the variable inputs and the stock of fish. The information contained in this set of elasticities is important for assessing the Salmonid Enhancement Program which is so popular in British Columbia. The program attempts to artificially increase the stock of fish. However, its success depends critically upon the ability of the regulator to prevent the use of more inputs than are necessary to take the catch. The seine results indicate that, as the stock size grows, more labour and less gear is used, Table 5.14. Both elasticities are significantly different from zero, whereas that for fuel is not. O n the other hand, the elasticity between stock and output is negative and very large, as well as being significant. This suggests that increasing the stock of fish leads to a smaller level of output and to the use of more inputs. This causes the cost of harvesting a given catch to rise. One possible Econometric Technique and Results / 113 Table 5.19:--Nonlinear estimates of output-variable own- and cross-price elasticities: gillnet-troll(crs) Quantity/ Output Labour Fuel Gear Price Output 0.191T -0.011 -0.184t 0.004 (0.117) (0.049) (0.087) (0.008) Labour 0.030 -0.057 0.027 0.001 (0.137) (0.073) (0.118) (0.009) Fuel 0.678T 0.035 -0.725T 0.012 (0.319) (0.155) (0.296) (0.025) Gear -0.0004 0.00004 0.0004 -0.000008 (0.0008) (0.0003) (0.0007) (0.00003) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 114 explanation of this result is that seiners fish under highly regulated conditions. Access to certain areas is controlled by the number of days in which seine fishing is permitted. Hence, at any one time, there may be a large number of vessels on a given fishing ground. The larger the abundance of fish, the more vessels, and the smaller the catch per vessel. The negative elasticity may reflect the existence of a stock externality (Huang and Lee 1976). However, I have not included this possibility in the model developed in Chapter 4. This result is not repeated for the other three vessel types, but they fish under different conditions. This finding suggests that an investigation of the stock externality problem might be a fruitful area for further study of rent dissipation in the fishery. For the gillnet fleet the elasticity of intensity between gear and the fish stock is positive, but not significantly different from zero, Table 5.15. O n the other hand, the elasticity between fuel and stock is significant, it has a negative sign, indicating a substitute relationship. It is also observed that output increases as the stock of fish increases, but not to a great extent. Thus, in order to assess the impact upon harvesting costs per unit one would need to know the relative costs of the two inputs, and the magnitude of the change in output. For the troll fleet an increase in stock size leads to a greater use of labour services, eg., a 1% increase in stock results in an increase in labour demanded of 0.247%. Neither the demand for fuel nor the demand for gear changes appreciably as the stock size changes, Table 5.16. Output is very responsive to increases in the stock size, thereby indicating that the Salmonid Enhancement Program might be effective with regard to this fleet. This would also be the case insofar as the Econometric Technique and Results / 115 Table 5.20:--Nonlinear estimates of output-constant price elasticities: gillnet(non-crs) Price/Quantity Fuel Gear Fuel -0.457E-12 0.686t (0.272E-06) (0.368) Gear 0.080t -0.214E-12 (0.036) (0.127E-06) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 116 gillnet-troll fleet is concerned. For this sample, a 1% increase in stock size leads to a decrease in the quantity of gear demanded of 0.872%, Table 5.17. Labour and fuel demands are unchanged when the stock size changes, but output is somewhat responsive. It seems that the cost of any given size of catch would be reduced if the stock of fish were to be increased. These pairwise comparisons are suggestive, but a complete analysis of the Salmonid Enhancement Program requires a full simulation, so that the total costs of harvesting could be compared with the increase in output or the value of the catch made possible by a greater stock. O n the other hand, this model is not capable of doing a full cost-benefit analysis on the program, since this would require having knowledge about the costs of administering the program, as well as the opportunity costs of the resources used by it. One final set of elasticities of intensity remain to be discussed. They pertain to the gillnet fleet and describe the relationships between the variable inputs and the fixed labour input, Table 5.15. Given the small scale of the gillnet operation, it is no surprise to find that both fuel and gear have a very strong complementary relationship with the amount of labour used on board the vessel. As well, these elasticities are significantly different from zero. Before ending this section, a very brief mention is given to results for the gillnet fleet with non-constant returns to scale and the gillnet-troll fleet with constant returns to scale. Results for the gillnet vessel do not differ when a comparison is made of variable factor cross-price elasticities, with the exception that the constant returns to scale case has more significant elasticities, Tables 5.18 (output-variable) Econometric Technique and Results / 117 Table 5.21:--Nonlinear estimates of output-constant price elasticities: gillnet-troll(crs) Price/Quantity Labour Fuel Gear Labour -0.056 0.028 0.003 (0.067) (0.106) (0.018) Fuel 0.689T -0.071 0.666T (0.328) (0.088) (0.286) Gear 0.50E-4 0.0004 -0.133E-06 (0.0003) (0.0008) (0.486E-05) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 118 and 5.20 (output-constant). In regard to the elasticities of intensity the fuel/stock elasticity becomes insignificant, Table 5.22. The elasticities of output and inputs with respect to the stock change signs, but are not significant. For the gillnet-troll fleet the elasticities between variable factors are slightly higher in the constant returns to scale case, but the number of significant results does not change, Tables 5.19 (output-variable) and 5.21 (output-constant). Labour and gear become substitutes, but not significant ones. The elasticity between gear and net tonnage becomes larger and significant, whereas the elasticity of labour with respect to stock changes sign to indicate a complementary relationship, although insignificantly different from zero, Table 5.23. 3. Returns to Scale Since some samples reject the hypothesis of returns to scale it is interesting to calculate the degree of scale returns for them (Laitinen 1980). This may be found by evaluating the following expression. (5.14) R T S = ( Z ^ O T T ( P , Z ) / 9 Z . ) * Z . ) / T T ( P , Z ) ) The sample mean of predicted restricted profit is used in the denominator and sample means of the actual fixed factors are used in the numerator. The first derivative of restricted profit with respect to a fixed factor is the shadow price of that factor. It is computed by using the sample means of the prices and fixed factors, along with the estimated parameters. If the number calculated by this expression is found to be greater than one, the sample is said to have increasing returns to scale. If less than one, this indicates decreasing returns to scale. Econometric Technique and Results / 119 Table 5.22:-Nonlinear estimates of elasticities of intensity: gillnet(non-crs) Quantity/ Stock Net Fishing Labour Fixed of Factor Fish Tonnage Days Output -0.059 -0.038 0.438T -0.100 (0.166) (0.199) (0.111) (0.295) Fuel -0.197 0.240 0.699T 0.949T (0.193) (0.247) (0.151) (0.355) Gear 0.703 0.049 0.338T 1.128T (0.748) (0.264) (0.154) (0.445) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Econometric Technique and Results / 120 It should noted that the measure calculated in the manner described above and called returns to scale is not the typical notion of returns to scale used in the literature. Because the fixed factors are not necessarily at their optimal levels, the measure of returns is more appropriately called local returns to size (Laitinen 1980). This measures the percentage change in restricted profit for a proportionate change in the levels of all fixed factors. The expression in (5.14) is used to calculate the degree of local returns to size for the seine, gillnet-troll, and troll samples using the appropriate parameter estimates. Since the gillnet sample accepts the hypothesis of constant returns to scale, no further measure need be calculated. Nonlinear parameter estimates are used Tor the seine and gillnet-troll samples, whereas linear parameter estimates are used for the troll sample. See Appendix 2 for the nonlinear estimates and Appendix 3 for the linear ones. For the seine sample the value of the expression is 0.2860; for the gillnet-troll, 2.0390, and for the troll, 1.3497. Only the seine sample exhibits decreasing returns to scale. That is, a 1% increase in all of the fixed factors leads only to an increase of profits of 0.2860%. At first glance this finding might appear to run counter to accepted wisdom in the fishery which suggests that the seine fleet alone has untapped economies of scale and could take the entire catch with only a few boats in a few days. However, the explanation is very simple. This segment of the salmon fleet is highly regulated, both in terms of vessel capacity and choice of fishing areas and times. These results suggest that what is observed, in fact, is the effect of regulated diseconomies of scale. Econometric Technique and Results / 121 Table 5.23:--Nonlinear estimates of elasticities of intensity: gillnet-troll(crs) Quantity/ Fixed Factor Output Labour Fuel Gear Stock of Fish 0.031 (0.179) 0.028 (0.077) 0.014 (0.166) -0.830t (0.523) Net Tonnage 0.753t (0.218) 0.734t (0.098) 0.620t (0.207) 1.910t (0.623) Fishing Days 0.238t (0.119) 0.240t (0.066) 0.403t (0.133) -0.091 (0.364) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a=0.10. Econometric Technique and Results / 122 O n the other hand, both the troll and gillnet-troll fleets have increasing returns to size. This is not a surprising result for the troll fleet, since it is the least regulated of the entire salmon fleet. As well, the nature of the troll operation means that it benefits from a larger vessel and a greater stock abundance. Greater vessel capacity means the trailer can stay on the fishing grounds for longer periods, thereby minimizing the trips back to port. Greater stock abundance also means easier and faster catchability. The same arguments are valid for the gillnet-troll vessels The finding of such a large degree of returns to size for the gillnet-troll vessel is somewhat disturbing given that the hypothesis of constant returns to scale is accepted at the 0.005 level of significance, although rejected at the 0.010 level. The a priori expectation is that the evaluated expression in (5.14) should be close to one. The explanation lies in the imprecise parameter estimates obtained by this sample. Not many parameters are significantly different from zero. Since the test for constant returns to scale relies upon certain parameters being insignificant, the gillnet-troll sample is biased towards acceptance of the hypothesis. One cause of lack of precision is data-based multicollinearity. As indicated earlier, an analysis of the data reveals no apparent evidence of this phenomenon. Only two sets of results by other researchers are comparable to those obtained for this thesis. Squires (1984, 1987a), calculates a returns to scale measure for the multi-species New England Otter Trawl fishery, and finds evidence of decreasing returns to scale for the two years of his study. For 1980 the degree of scale returns is estimated as 0.4513, whereas for 1981, it is 0.3889. In his model stock abundance is the fixed factor and modeled by means of a dummy variable for Econometric Technique and Results / 123 1981. In a later study of the New England sea scallop fishery Squires (1985) obtains a calculated value of returns to scale equal to 5.92. Obviously, these measures are truly fishery-specific. D. CONCLUSIONS The results indicate that the salmon fishing technology has more substitution possibilities than previously believed. In particular, the seine fleet appears capable of substituting toward the use of all three variable inputs in the face of a per vessel net tonnage restriction that attempts to control the catch of the vessel. In this way, seine vessel-owners may dissipate a great deal of potential resource rent. The troll fleet may do likewise. O n the other hand, the gillnet and gillnet-troll fleets appear to have little scope for rent dissipation through the use of more variable inputs, against a net tonnage restriction. The limited entry program with net tonnage restrictions per vessel appears to have been moderately successful for two of the four vessel types used to catch salmon. Of the two types of restrictions employed to control behaviour in the fishery, it appears that a restriction on the number of fishing days is most effective at preventing rent dissipating behaviour. For example, an increase in the number of fishing days means a greater use of variable inputs. This raises the costs of taking the given harvest. The fishing days restriction is also popular with the regulators because of the absolute control it gives them to stop fishing activity. VI. MEASURING FISHERY RENT DISSIPATION This chapter presents a method for the empirical measurement of fishery rent. Although in the spirit of the seminal work by Crutchfield and Pontecorvo (1969), the procedure has several novel aspects. First, it uses the estimated output supply/input demand equations from Chapter 5, along with given prices and levels of the restricted factors, to obtain predictions for the optimal levels of the variable quantities controlled by each vessel-owner. This is done for each of the four vessel types. This information describes the profit-maximizing or economically efficient choices for each vessel. In Case I actual static rent for the fishery in 1982 is calculated as the sum of profit per vessel, where profit is defined as total revenue minus the sum of total variable and fixed costs. Second, as shown in Chapter 4, an estimate of the optimal amount of net tonnage per vessel may be obtained from knowledge of the parameters in the restricted profit function. Substituting the optimal level of tonnage for the actual permits the researcher to solve for the associated profit-maximizing levels of the variable quantities. A comparison of the actual rent (Case I) to the rent derived from this exercise (Case II) reveals the extent of rent dissipation attributable to tonnage restrictions imposed by the regulator. Third, the role of input substitutability is investigated by performing another exercise (Case III). This entails the doubling of the parameter values for the price terms in the estimated equations. This is an arbitrary means of increasing the substitution possibilities. Then, new optimal levels of net tonnage are found and used to solve for the associated profit-maximizing variable quantity choices. Rent calculated using the data obtained from this scenario 124 Measuring Fishery Rent Dissipation / 125 can be compared to that generated in the second. The difference indicates the amount of rent dissipation attributable to input-substituting activities of the fishermen. Fourth, in each case it is possible to solve for the minimum number of vessels required to take the actual 1982 salmon catch. The difference between the minimum and the actual number of vessels that fished can be used to find an estimate of seasonal fleet redundancy deadweight loss (Munro and Scott 1985). No attempt is made to estimate the intertemporal loss of rent that occurs through the excessive depletion of the fish resource. Furthermore, two further forms of (within period) rent dissipation are ignored. They are both attributable to overcrowding on the fishing ground. The first is the stock externality (Huang and Lee 1976). It usually manifests itself as a reduction in the catch per vessel caused by the presence of other vessels in a particular fishing area. The second is called the congestion externality, also called gear-fouling, since it refers to the possible damage done to the gear and equipment of one vessel by other vessels. Data restrictions do not allow for the measurement of these effects. As well, the model of the harvest technology, described in Chapter 4 and estimated in Chapter 5, ignores these aspects of the rent dissipation problem. A. THE CALCULATION OF FISHERY RESOURCE RENT 1. Theoretical Measures of Fishery Rent The basis for the notion of fishery rent comes from two features specific to the fishery. The first is the self-reproducing nature of the fish population; the second is Measuring Fishery Rent Dissipation / 126 that the minimum cost of catching the fish is less than the market value of the catch. For the static case the maximum fishery rent is simply defined as the maximum difference between sustainable revenue and cost (Cordon 1954). For the dynamic case a present value calculation is used instead (Scott 1955). As a means of putting the work described in this chapter into context it is instructive to present the Cordon-Schaefer model of the fishery.- This begins with a simple differential equation that describes the net natural growth rate of the fish population, F(X(t)). Typically, a logistic growth function is used. To this is appended a harvest rate production function. (6.1) h(t) = g(E(t),X(t)) Generally, the Cobb-Douglas functional form is adopted for (6.1), as discussed in Chapter 4. Then, the net growth of the fish stock is described, in equation (6.2), as the difference between the net natural growth rate and the harvest rate, (6.1). (6.2) x(t) = F(X(t)) - h(t) It is now possible to define the sustainable yield from the fishery as the catch associated with a zero net growth rate of the stock. That is, harvesting does not reduce the standing stock of fish. This describes the relationship between the sustainable yield and the effort used. To understand the notion of sustainable resource rent, prices and costs must be Measuring Fishery Rent Dissipation / 127 appended to the biological model. Cordon assumes both a constant price for landed fish and a constant unit cost of effort. Furthermore, the market price of fish represents the marginal social benefit of a unit of fish, and the cost of a unit of effort measures its marginal social cost. Then, sustainable revenue is defined as the (constant) price times sustainable yield. The total cost of fishing effort is given by the product of the unit cost times the amount of effort. The maximum sustainable resource rent is found simply as the maximum distance between the two functions describing sustainable revenue and the total cost of effort. Gordon uses this model to describe the optimal state of the fishery where the sustainable resource rent is maximized. He compares the amount of effort this requires with the open access level of effort. Open access is when there is competition for the common property resource and there are no controls put upon the amount of effort used. Sustainable resource rent is driven to zero as new entrants are attracted to the fishery by the above normal returns comprised of fishery rent. This is the so-called bionomic equilibrium where the fish resource rent is entirely dissipated. One may re-express Gordon's model in a manner that makes clear that there are two types of rent dissipation (Munro and Scott 1985). These authors compare sustainable revenue to the associated biomass levels. Total costs are now a function of the biomass, ie., they increase as the level of the stock falls. Maximum sustainable resource rent is defined as before; namely, the maximum difference between the sustainable revenue curve and the total cost function. This defines the optimal level of the fish stock. In contrast, the open access case of zero rent Measuring Fishery Rent Dissipation / 123 obtains at a lower stock level. Economic overexploitation occurs because open access competitors drive the fish stock below its optimal level. Munro and Clark term this the Class I Type of Common Property problem. I refer to this as the Class I form of rent dissipation. The Class II Type of Common Property Problem occurs when the government restricts the total harvest in order to maintain the biomass at its optimal levelt, but either does not or cannot control the amount of effort directed at the fishery. Rent dissipation occurs when an excessive number of vessels and fishermen compete for the allowable harvest. They are encouraged to behave in this fashion by the presence of fishery rents. In this manner total harvest costs may continue to rise until equal to total revenues.t I call this the Class II form of rent dissipation. Cordon's static model does not represent the complete picture if intertemporal considerations are important. Following the suggestions of Scott (1955), optimal control techniques are developed for analysis of the fishery (Plourde 1970, Quirk and Smith 1970, Clark 1976). Instead of maximizing static fishery rent the objective is to maximize the present value of the flow of rents over a period of time. The optimal solution describes the time path of the stock variable (biomass) that maximizes the objective. The monopoly exploiter achieves this goal through changes in the control variable (harvest rate). In order to simulate Class I rent dissipation in this situation the discount rate is set to infinity. tThis is called a total allowable catch policy; it is used to regulate the total harvest of the British Columbia commercial salmon fishery. tMunro and Scott mention that rent dissipation may also result from crowding externalities or through the processing sector. In the former instance competing vessels disrupt each other's harvesting efforts. In the second the fishing season may be shortened as the harvest is taken ever more quickly by a greater number of vessels than optimal. This means that processing plants operate at full capacity only for the duration of the fishing season, then lie idle for the rest of the year. Measuring Fishery Rent Dissipation / 129 In order to address the Class II form of rent dissipation in a dynamic setting it is necessary to adopt a 2-state, 2-control variable model (Clark, Clarke, and Munro 1979). The two state variables are taken to be the biomass level and the number of vessels in the fleet. Under the assumption of non-malleable capital* the optimal solution becomes analytically complex. Class II rent dissipation may still occur at the optimal biomass level if the fleet size is too large. 2. Empirical Measures of Fishery Rent The empirical measurement of rent dissipation attributable to the Class I type of common property problem is not a new area of research. For example, both Loose (1975) and Gardner (1980) provide present value estimates of the intertemporal flow of rent that could be earned in the British Columbia salmon fishery under a system of optimal management.* Their results are presented for the purpose of a comparison with the ones obtained in this thesis. In each case a biological reproduction function is estimated using historical data. Next, a harvest production function is estimated. This is specified at the level of the individual gillnet vessel by Loose and at the aggregate fleet level by Gardner. However, the latter repeats the calculation for two vessel types and adjusts his measure of effort by an index of technological change. The modeling is completed by adopting the criterion of present value profit-maximization. For Loose the solution set includes an estimate of optimal escapement and the tNon-malleable capital implies the existence of fixed costs that cannot be salvaged by a shifting of the resources used to another activity. *However, Gordon considers only the Fraser River sockeye fishery and allows two types of vessels to fish, whereas Loose examines only the Skeena River gillnet fishery which harvests two species of salmon. In neither case do the authors attempt to generate estimates for the industry as a whole. Measuring Fishery Rent Dissipation / 130 optimal amount of fishing effort (modelled as vessel days). Loose compares two scenarios. The first allows the monopoly owner of the fishery to hire the necessary fleet on a weekly basis (Case 1), while the second constrains the monopolist to follow an annual hiring policy (Case 2). Using a series of different interest rates Loose simulates the net present value of the fishery. For Case I and a 6% rate of interest the net present value varies from 23.6 to 44.8 million in constant 1975 dollars. For Case II the comparable numbers are $31.9 and $54.4 million. In addition, he calculates the optimal number of boats for each optimal harvest rate. For the first case, the optimal number is 52% less than the actual number of boats in 1975; for the second, it is 15% less. However, he does not calculate the cost of this fleet redundancy. Gardner solves for the optimal escapement and the least cost spatial combination of vessel types to take the optimal catch. He uses a discount rate of 3.5% and finds the increase in profits (or fishery rents) in 1951 dollars ranges from 1.23 to 2.55 million. These number represent increases of 19%-44% in profits over the four year life-cycle of the salmon. His methodology also permits him to ascertain the least cost combination of vessel types needed to take the actual catch, rather than the optimal one. He finds that fishermen already earn rents with the sub-optimal combination of vessel types, but that the optimal combination would permit resource rent to increase. This is a measure of type II rent dissipation. It is estimated to be between 0.51 and 2.2 million in 1951 current dollars. This represents an increase of from 38%-208% in fishery rent over the salmon cycle. There are several studies of relevance to the measurement of rent dissipation Measuring Fishery Rent Dissipation / 131 associated with the Type II common property problem (Crutchfield and Pontecorvo 1969, Huppert 1982, Fraser 1977, 1979; Pearse and Wilen 1979). The first two are most closely related to the methodology developed in this thesis; the latter three are included because they are concerned with rent dissipation in the British Columbia commercial salmon fishery. The seminal work (Crutchfield and Pontecorvo 1969) estimates the amount of resource rent that could have been recovered from the Alaska Bristol Bay and Washington Puget Sound salmon fisheries. In both cases the authors assume that the total annual catch is determined optimally by the regulators and that the landed price of the catch is independent of fishing costs.t Starting from the assumption that free entry in both fisheries has resulted in complete rent dissipation they argue that, "any reduction in costs to the individual unit and in the aggregate will simply accrue as explicit or implicit rent".* In order to calculate rent for the Bristol Bay fishery the authors follow a five step procedure. In essence they calculate total revenues less the minimum total costs required to harvest a given catch. First, they calculate the gross revenues of the fishery in each year as the landed price times the actual catch. Second, a base year is chosen such that the inputs, distribution of fishing effort, and the harvest rates approximate those that would represent the minimum required inputs. They choose the period 1942-1943 as their base, arguing that the war prevented excessive harvest effort from being directed at the fishery. For each of the remaining years the inputs used are indexed to the base year, thereby creating an tThis assumption means that the authors need not calculate the value of consumer surplus (Copes and Cook 1984). * J . A. Crutchfield and G . Pontecorvo. The Pacific Salmon Fisheries: _A_ Study in_ Irrational Conservation. Baltimore: The Johns Hopkins Press for Resources for the Future, 1969. p . m . Measuring Fishery Rent Dissipation / 132 index of fleet efficiency. The base year also provides an estimate of the maximum yield per unit of effort. Third, assuming a constant yield-effort relationship over the years 1934-1959, the minimum quantity of inputs needed for each of the recorded annual harvests is calculated. Fourth, it is assumed that all factors are paid their true opportunity costs. Total costs are computed for two scenarios, the actual amount of inputs used in each year and the minimum number as indicated by the base year requirements. Finally, a measure of the potential rent is obtained by taking the difference between the actual annual gross earnings of the fleet and the projected gross earnings that would have arisen had the fleet been limited and operated at the highest level of efficiency. Since the revenues are taken as given, this step is equivalent to calculating the amount of dissipated rent by comparing the net profits from the fishery in the two scenarios: the actual and the efficient. As an alternative to the fourth step the authors suggest that an estimate of the yield per boat per day may be used. Dividing this yield into the total catch provides an estimate of the minimum number of boats required to land the entire catch. Estimates of the amount of dissipated rent are substantial and range from 1.850 million (1934-1939) to 3.608 million (in 1959) in constant 1951 American dollars. The relative amounts of redundant gear range from 33% to 83% over the 1934 to 1959 period. Data for the Puget Sound are not as well developed as those for the Bristol Bay fishery. In particular, it is not possible for the authors to designate a base year Measuring Fishery Rent Dissipation / 133 period. This necessitates a less ambitious attempt to measure the loss in potential fishery rent. Crutchfield and Pontecorvo use a computer simulation of salmon runs, along with physical efficency and area utilization studies for the three major net gears.t This permits the authors to estimate the effect upon landings and fishing days of arbitrary reductions in the amount of gear. Using actual price and cost data the rents available with alternate reductions in the quantity of gear are calculated. With a 50% reduction in each of the three gear types the potential cost savings in current dollars amount to 2.639 million in 1955 or 3.771 million in 1958. The authors claim that these figures may in fact underestimate the true amount of potential rent because a relaxation of the regulations governing the use of gear and fishing times would likely reduce harvesting costs to a greater extent. They claim that the annual flow of net benefits from the fishery could reach between 5 and 7 million dollars. The authors then claim that the British Columbia salmon fishery is similar to the one in Washington State and use data from the latter fishery to estimate the potential net economic yield for the British Columbia fishery. It is set at $15.0 million on the basis of prices and catches for the period 1960-1965.* The second paper dealing with the Class II form of rent dissipation is an examination of the California herring roe fishery (Huppert 1982). This fishery is managed by a limited entry vessel program and seasonal quotas per boat. Huppert follows the methodology used by Crutchfield and Pontecorvo. Total revenue is taken as given for 1978 and 1979. The potential catch per vessel is inferred from the tThe gears are the drift, gill, and seine and the data are generated in a study by W. Royce, D. Bevan, J. Crutchfield, C . Paulick, and R. Fletcher "Salmon Gear Limitation in Northern Washington Waters", University of Washington Publications in Fisheries, new series, vol 2, no 1, 1963. ^Recall that only data from the net fisheries are used in their calculations. Measuring Fishery Rent Dissipation / 134 1976 and 1977 catch statistics, since the catch per effort figures are the largest for those years. An estimate of minimum cost per vessel is obtained as the average earnings per vessel in 1979, for in this year the limited entry program is not enforced. Therefore, the author suggests that all rent for that year is dissipated and that total revenues equal total opportunity costs. The increase in potential profit (or economic rent) for the fishery is estimated at 3.15 million dollars for 1978, an increase of 100% and 0.661 million dollars for 1979, an increase of 12%. These figures are estimated under the assumption that the existing gear and season regulations, and catch distribution among the gear types does not change. The positive resource rent is achieved by reducing the size of the fleet. Huppert argues that the projected rent is most likely underestimated because a reorganization of the gear types would probably result in lower harvesting costs. Furthermore, he suggests that each vessel has no incentive to use more than the minimum necessary inputs because of the existence of seasonal quotas per vessel. There are several shortcomings in the approach taken by Crutchfield and Pontecorvo and Huppert. These include the arbitrary choice of a so-called technically efficient vessel as the basis of comparison, the lack of an explicit behavioural model of the individual fisherman, the assumption of a constant catch per unit effort, the failure to incorporate the effects of gear restrictions and prices on input choicet, a disregard for the role of input substition, and the maintained assumption of a single, homogeneous vessel type.* In addition, rent dissipated through crowding tHowever, the authors note that gear and season restrictions may contribute to higher harvesting costs, thereby lowering potential rent (Smith 1969, Crutchfield 1979). *Not only is inframarginal rent taken to be zero (Copes and Cook 1984), but differences among vessels and/or gear types are ignored. Measuring Fishery Rent Dissipation / 135 externalities on fishing grounds or by processing companies (Munro and Scott 1985) is not measured. I address many of these deficiencies in the methodology proposed in this thesis. This provides a more complete analysis of rent and rent dissipation than has heretofore been conducted for a fishery. Before turning to a discussion of the methodology used in this thesis to calculate fishery rent, it is worthwhile to discuss the other three papers mentioned earlier, ie., two studies by Fraser (1977, 1979) and one by Pearse and Wilen (1979). They consider the issue of rent dissipation in the British Columbia commercial salmon fishery. In order to examine this issue they calculate the real value of the fleet's capital and its growth over time. They begin their analysis with the 1969 season, which is characterized by open access and, therefore, by complete rent dissipation. Fraser compares a 49% increase in capital costs from 1969 to 1977 to a 35% increase in the real value of the resource. He concludes by suggesting that rent dissipation has taken place. Pearse and Wilen agree that capital costs have increased since the imposition of the license limitation program. However, they argue that the average annual growth rate of the value of capital has decreased since the inception of the program. Over the period of license limitation they claim the growth rate of captal costs to be 3.3%, as compared to the average rate of growth of revenue at 4.4%. Thus, they sugest that rent dissipation is incomplete. These studies suffer from an overly simplistic implicit model of the harvest technology. This means that the roles played by other inputs are ignored. More importantly, the authors do not distinguish between the effects of increasing capital prices and increasing capital costs. In addition, they do not address the issue of Measuring Fishery Rent Dissipation / 136 rent dissipation attributable to fleet redundancy. B. METHODOLOGY TO OBTAIN ESTIMATES OF FISHERY RENT Total fishery rent is measured as the sum of profit per vessel, where profit is calculated as revenue minus total cost, including the fixed cost associated with the net tonnage of the vessel.t Thus, the resource rent that I calculate is a measure of the net profit to the fishermen and includes a return to two fixed factors, the number of fishing days and the fish stock.* In order to obtain estimates of both the costs of and the revenue from fishing I must first solve for the profit-maximizing levels of the variable quantities, ie., output supply and input demands. They depend upon the actual levels of the fixed factors, including the net tonnage. I call the rent associated with this set of parameters the actual rent to the fishery. However, the rent for this case is based upon the actual or restricted levels of net tonnage used by each vessel. I also want to obtain an estimate of the potential rent that could be obtained when the fisherman is free to use his desired amount of net tonnage, since tonnage restrictions may cause rent dissipation. To do this I must first solve for the optimal level of net tonnage per vessel. Given the current market price of one net ton, I find the currently optimal quantity of this restricted input as the solution to a long-run profit maximization problem Substituting this value for the actual one, the corresponding tFollowing this section is a discussion of how the appropriate market rental price of a ton is obtained. *lt is difficult to separate the returns to these two factors, since returns to fishing days occur because of the existence of the fish stock. So, in some sense, the total return is attributable to the fish resource. Measuring Fishery Rent Dissipation / 137 profit-maximizing levels of output and inputs are found, along with total cost and revenue per vessel. Rent is calculated as before, ie., estimated seasonal profit minus the total fixed cost. The details of the steps followed to obtain for each vessel the optimal amounts of the variable quantities and the optimal level of the restricted input, net tonnage, are given next. 1. Determining the Optimal Levels of Variable Quantities In order to obtain estimates of the rent per vessel, it is necessary to know the optimal quantity of output that an efficient vessel-owner catches and the optimal input quantities used to produce the output. Predictions for these quantities can be obtained from the estimated output supply and input demand equations of Chapter 5, ie., equations (5.3)-(5.4) for the linear estimation and equations (5.8)-(5.11) for the nonlinear. If sample mean prices and mean levels of fixed factors are substituted into the estimated equations*, the researcher obtains a solution to the optimal input demand and output supply allocation decisions for the mean vessel of each type. They are the result of . a static, private, profit-maximization exercise. If the prices reflect the true social opportunity costs, then the private solution coincides with the socially optimal one. This is in contrast to the approach, used by Crutchfield and Pontecorvo and Huppert, which determines the technically efficient vessel as the one that has the greatest historical catch. Profit maximization requires technical efficiency, but the converse is not true. When the predicted quantities are multiplied by sample mean prices, estimates of tNonlinear parameter estimates are found in Appendix 2 and are used for the seine, gillnet, and gillnet-troll samples. Linear estimates used for the troll sample are found in Appendix 3. Measuring Fishery Rent Dissipation / 138 variable (or seasonal) profit are obtained for the mean vessel as the difference between seasonal revenue and the expenditure on variable inputs. An estimate of total fixed cost is given by the product of the mean net tonnage and the market rental (flow) price per ton. Subtracting this fixed cost from variable profit gives an estimate of the resource rent earned by the mean vessel. Total rent for each of the four different vessel types is equal to the product of the number of vessels in the sample and the rent per mean vessel. If all vessels are identical, then it is a sufficient statistic to know the rent of the mean vessel. However, as chapter 5 shows, the four vessel types do not all exhibit constant returns to scale. Thus, in order to generate an estimate of total rent per sample, it is necessary to obtain an estimate of the rent of each vessel in the sample and then to sum these rents. In order to take into account differences in vessels, the procedure described above for the mean vessel is modified slightly. Within a sample it is assumed that each vessel faces the sample mean prices for output and the inputs. Predicted output supply and input demand quantities for each vessel are obtained by substituting the sample mean prices and vessel-specific levels of fixed factors into the estimated equations. Revenues and variable costs are calculated as described earlier. Total fixed cost per vessel is given by the product of the market rental (flow) price per ton and a vessel's own net tonnage. Rent per vessel is defined as before, ie., seasonal profit minus total fixed cost. Thus, vessels do not necessarily earn the same rent. Furthermore, it is possible that some vessels may earn negative rent, while others earn positive rent. In order to obtain the total rent for the sample, I sum over the vessel-specific rents. Measuring Fishery Rent Dissipation / 139 Since it is also of interest to know the amount of total rent for the entire fleet of each vessel type, as opposed to rent for a sample of vessels of each type, the within sample results are used to extrapolate to the industry. This is done both for the rent obtained per mean vessel in a sample and for the rent obtained using the entire sample of vessels. The initial steps are identical to those used to obtain rent per sample but one extra step is required. Recalling that the optimal output per vessel is determined from the output supply equation, one can use published information on the actual total catch of 1982 to solve for the minimum necessary number of vessels. If data supplied by the mean vessel is used, then total catch divided by the optimal catch of a single representative vessel gives an estimate of the number of vessels needed in the fishery. O n the other hand, if the output levels for each vessel are used, the sum of output for the entire sample can be divided into the total catch. The result is then multiplied by the sample size to obtain an estimate of the number of vessels required. This assumes that the sample of vessels used to estimate the harvest technology is representative of the entire population. Since the survey data from which I obtain information on each vessel is supposed to be a random sample, this is a reasonable assumption. Once the number of vessels is known, either using the mean of the sample or its entire distribution, it is a simple matter to construct total fleet costs per vessel type, both variable and fixed. Estimated fleet rent per vessel type is given by the difference between total fleet revenue and total fleet cost. Total industry rent is merely the sum over the four vessel types of the total rent per fleet. The rent calculated in the manner described above is the actual, static (one period) Measuring Fishery Rent Dissipation / 140 rent associated with conditions prevailing in the fishery in 1982. I call this Case I. This rent is obtained by using the given levels of the fixed factors, ie., the net tonnage per vessel, the number of fishing days, and the stock of fish. The profit-maximizing solution set obtained by holding at least one factor fixed is called a partial static equilibrium (Brown and Christensen 1979). It is possible that the actual levels of the fixed factors differ from the optimal levels that would be obtained at the current market prices, if the factors were variable. If the two levels are equal, then the optimizing choices for all inputs constitute a full static equilibrium solution (Brown and Christensen 1979). Using a methodology developed by Brown and Christensen (1979) and most recently employed by Kulatilaka (1985, 1987), one can solve for the optimal levels of the fixed factors at their current market rental prices. I perform this exercise for one restricted factor, net tonnage. Recall from the discussion in Chapter 3 that the regulator of the British Columbia limited entry program tries to control the net tonnage used per vessel by imposing an upper bound on its use. The intention is presumably to prevent tonnage from increasing and leading to an increased ability to catch more fish. In fact, this input restriction may be a source of inefficiency for the fisherman. That is, vessel-owners may not be able to respond as they would like to a change in market prices. Instead of using the correct net tonnage, they use more variable inputs. Thus, costs are higher and profits smaller than would be generated by the use of the correct amount of the restricted factor. In this way, fishery rents are lower than they might be if the optimal amount of net tonnage were used per vessel. Measuring Fishery Rent Dissipation / 141 Once the optimal net tonnage has been calculated*, both for the mean vessel in each sample and for each vessel in the sample, the estimated equations for output supply and input demand are used to obtain predictions for the new optimal levels of these variable quantities. Within sample rents and extrapolated industry rents are calculated as described earlier and called potential rent, since they refer to the rents that could be obtained by the use of the optimal tonnage per vessel. This is called Case II. In doing this simulation, I keep the other two fixed factors, stock of fish and number of fishing days, at their actual levels. I do not solve for their optimal levels.* The optimal or potential rent of Case II is compared to the estimate of actual rent in Case I. The difference measures the amount of dissipated rent attributable to input inefficiencies induced by tonnage restrictions. In an effort to isolate the amount of rent dissipation that may be attributed to the ability of the vessel-owners to exploit input substitution possibilities I perform another simulation called Case III. This is done in an admittedly ad hoc manner. +The next section contains a detailed discussion of how this is done. *The optimal level of the fish stock cannot be determined within the static framework used in this thesis. Rather, the operative assumption is that the regulator chooses the stock according to his own criteria. However, allowing the stock to change would be an interesting exercise that could be used to evaluate the impact of the Salmonid Enhancement Program upon catches and input usage. In reality, the actual number of fishing days may not be optimal, as this input is also regulated (for the entire fleet) to prevent overfishing of the total allowable catch. It is not clear what would happen to this input should the size of the fleet be reduced. In part, this uncertainty is due to the role played by nature in determining the maximum number of possible fishing days. Measuring Fishery Rent Dissipation / 142 The coefficients a . , (for i = k and i,k = 2,3,4) are doubled in absolute value. This i k means that the cross-price elasticities double in absolute size (see the equations in Appendix 4). This exercise is meant only to simulate the effects upon rent dissipation of a technology characterized with a greater degree of substitution possibilities than actually observed. It is necessary to solve for the optimal net tonnage per vessel associated with the new parameters for the restricted profit function. The optimal tonnage is used in turn to predict the new output supply and input demand levels per vessel. Rent is constructed as before using both the mean vessel of each type and the entire distribution of vessels in each sample. Finally, the rent for this Case is compared to the rent in Cases I and II. Cases I, II, and III are undertaken to obtain both the rent for the vessels within a sample and the rent obtained by extrapolating to the entire fleet of each vessel type and hence, to the industry. One final simulation is undertaken for the industry level calculations alone. This is called Case IV. It relaxes one assumption maintained in all of the previous cases. These cases take the distribution of total catch among vessel types as given. As an alternative, it is suggested that the following exercise be conducted. It is assumed that the entire salmon catch is landed by a single vessel type. For each vessel type the parameter estimates from Chapter 5 are used; that is, the actual degree of input substitution is a maintained hypothesis, but it is assumed that each vessel uses the optimal net tonnage. Then, using the predicted levels of optimal output, the minimum number of vessels required to take the entire annual catch is determined. This is done in turn for each of the vessel types. The total costs of each of the four vessel types are computed. The total rent associated with each vessel type is taken to be an estimate of the potential Measuring Fishery Rent Dissipation / 143 rent which the fishey could earn if only one type of vessel were permitted to fish salmon. The objective of this exercise is to observe whether the fishery would earn more rent by following a scheme to allow only one vessel type to take the entire catch, than it does by permitting multiple vessel types to fish. It is assumed throughout that the basic harvesting technology does not change. In particular, the possibility of the development of any new vessel types is ignored. As well, it is assumed that the vessel owners continue to act competitively. Thus, no strategic or collusive behaviour is allowed. In the next section, I discuss the method used to obtain the optimal tonnage per vessel and the data work required to generate a market rental price of net tonnage. 2. Calculating the Optimal Net Tonnage The solution to the profit maximization problem discussed in chapter 4 determines the maximal amount of restricted or seasonal profit. If, however, the firm can vary the amount of a restricted input, total or long-run profit may be defined as restricted profit minus expenditures on the fixed input: T —' — R — —• (6.3) it (p,w ;z) = it (p,w;z) - m « z 2 In this equation m is the unit rental price of the fixed factor, z 2 , where z 2 is T R defined here as net tonnage, it is total profit and it is restricted profit. It is assumed that the other two restricted factors remain at their actual levels. To solve for the maximum total profit in (6.3) it is necessary to first obtain the optimal Measuring Fishery Rent Dissipation / 144 amount of the tonnage input. The first order condition with respect to net tonnage is given by (6.4) T —• — R — — (6.4) 97r (p,w;z)/9z 2 = 97r (p,w ; z ) / 9 z 2 - m =0 Equation (6.4) is merely an application of the envelope theorem (Samuelson 1954). It states that the shadow price of a fixed factor is equal to the market price if the level of the fixed factor is optimal (Diewert 1974). Equation (6.4) defines an implicit function for the profit-maximizing demand for the net tonnage input. Thus (6.4) , may be solved for the optimal amount of net tonnage, z^: (6.5) z* = h(p,m,w;z) For the normalized, quadratic, restricted profit function, the specific form of (6.5) is given in (6.6). Unlike for the translog case, this expression is linear and may be solved in closed form. (6.6) z* = - ( z 1 /(*> 2 2L i =* PLVL) -[m2 + l / 2 . ( I i = 2 L K = 2 a i k ( P A / P * » 0 .P. . ( ( b 2 3 z , ) / z 1 + b a / z l ) +2. . 4 C _ P . ] 1=1 i 2 l Long run profit is found by substituting the optimal level of z f into (6.3). (6.7) 7r (p,m,w; z) = ir (p,m,w; zf (p,m, w;z); z)-m*z* 2 (p,m, w;z) Measuring Fishery Rent Dissipation / 145 I turn next to the problem of obtaining a suitable market rental price for tonnage. Ideally, the market price should reflect the opportunity cost of a newly constructed stripped-down vessel. Unfortunately, this information is not available for the vessel types used in the British Columbia commercial salmon fishery. As an alternative, two measures of the unit rental price are derived that provide a reasonable range for this price. The survey from which expenditure information for vessels is obtained asks each vessel-owner for the current market , value of his vessel, including any onboard equipment. The responses refer to the value of the stock of vessel in 1982 current dollars. A high estimate of the market price per unit of net tonnage is obtained by averaging over the current market values per net ton given for newly constructed vessels. I call this cmval 2 because it refers to the second fixed input in my model, net tonnage. Vessels are included in this estimate if they were built during the period 1979-1981. The number of vessels in each sample used to calculate the high price is 2 (out of 21) for the seine sample, 10 (out of 80) for the gillnet sample, 10 (out of 84) for the troll sample, and 11 (out of 60) for the gillnet-troll sample The low estimate is obtained in a similar fashion, but uses all vessels in the sample. Thus, older and less well-equipped vessels are used in addition those that are newer and better-equipped. The average market price per unit obtained from the sample is an estimate of the purchase price of a stock variable and as such it must be adjusted for the expected life of the asset. The flow rental price m 2 is calculated by applying a Measuring Fishery Rent Dissipation / 146 straight line depreciation rate (6) and a nominal interest rate (r) to the per unit stock price, cmval 2 as indicated in (6.8): (6.8) m 2 = c m v a l 2 ( r + 6) The annual depreciation rate is set at 4%. This is the rate suggested by Jenkins (1977) in a Economic Council of Canada Review for vessels with a gross registered tonnage in excess of 5 tons (for smaller vessels the recommended rate is 7.14%). The nominal rate of interest is taken to be 14.25%. This figure is obtained from the Bank of Canada Review (February, 1983) as a simple average of the monthly average yields in 1982 of long term Government of Canada bonds. This method is not without problems, but it seems to make the best use of the available data. The most serious problem (aside from the possibly unrepresentative nature of the small sample sizes) is that the reported market values per ton may reflect different levels of capital investment (in the form of electronic equipment, radar, sonar, etc) across vessel types. Many observers of the fishery believe that some portion of rent dissipation takes the form of excessive investment in onboard equipment. To the extent that this is true, the calculated rental prices may overestimate the true opportunity cost of a net ton of capacity in the fishery. This means that rents are underestimated. A second potential problem is that this methodology treats tonnage as if it were divisible into marketable units. This is certainly not the case for low levels of tonnage since some minimum size is likely necessary. Therefore, this type of approximation error is probably most serious for rental prices calculated for the gillnet fleet. Measuring Fishery Rent Dissipation / 147 It might also be argued that the reported survey values are not realistic and may be biased. As a check of the computed figures, actual vessel selling prices could be reviewed. Since actual selling prices are usually confidential, this strategy is not feasible. However, asking prices for commercial fishing vessels are often listed in the Classified sections of the Vancouver Sun and fisheries trade magazines, such as the Fisherman and West Coast Fisherman. There are at least two problems with these estimates. The uncontrollable problem is that, since asking prices rather than transactions prices are listed, the prices may be too high. Furthermore, the age of the offered vessel is seldom given and thus cannot be accounted for in the price. There may be a further problem. The asking price may include the value of the license to fish. Because the fishery has a limited number of licensed participants, the license to fish may be expected to acquire a positive market value. This reflects, among other things, the present value of anticipated fishery rents. This value should be netted out from the market value of the vessel, as it represents a private rather than a social opportunity cost. Advertisements over the period 1981 through 1982 are analyzed and asking prices per ton are calculated. In order to correct the vessel selling prices found in the Classified sections, I use estimates of the per ton market price of a fishing license. These values are monitored by the staff of Fisheries and Oceans Canada. An estimated value of the fishing license of $5,000 per net ton (for the seine vessel) and $4,000 per net ton for the other vessels is subtracted from the vessel asking price. They are the average values prevailing at the end of the 1981 fishing season. What remains is an estimate of the market value of the vessel alone. I find that the computed asking prices are bracketed by the high and low estimated average Measuring Fishery Rent Dissipation / 148 Table 6.1:--Estimated market rental prices and shadow prices per net ton: four vessel types Sample High Price Low Price Shadow Price Seine 5958.72 2651.26 6414.31 Gillnet 1905.02 1587.40 1350.34 Troll 3731.10 2050.35 3401.74 Gillnet-Troll 2325.67 1781.38 22494.80 Notes: The method by which these figures are calculated is discussed in the text. All figures in the table are measured in 1982 current dollars. Measuring Fishery Rent Dissipation / 149 current market values calculated from the survey data. The low and high estimates of the rental prices per net ton for each sample are given in Table 6.1.t These values are used in the next section to obtain estimates of the optimal net tonnage for each vessel type. One may compare them to the shadow values for net tonnage. The latter are obtained by differentiating the restricted profit function with respect to the tonnage variable, see equation (4.12) in Chapter 4. A numerical solution is obtained by substituting into (4.12) the means of the actual prices and the mean values of the quantities of the fixed factors, along with the estimated parameter values. The results are reported in Table 6.1 and suggest that the representative vessel in the seine and troll fleets may be at the optimal net tonnage level. However, the mean gillnet vessel has too great a tonnage, whereas the gillnet-troll vessel should invest in a larger tonnage. C. FISHERY RENT AND RENT DISSIPATION This section has two parts. The first presents results obtained from the calculation of fishery rent for the sample of 245 vessels used in Chapter 5 to estimate the harvest technologies for the four vessel types. I call the rents generated in this way within sample rents. Three cases are presented. The first generates estimates of the actual amount of rent in 1982. The second shows the potential rent that could be generated, if there were no tonnage restrictions. The third finds the amount of rent associated with a greater degree of variable input substitutability. In the second part these within sample results are extrapolated to generate total industry rents. Four tUnfortunately, only one year of data is available for the construction of these prices. Therefore, it is not known whether these price estimates are those that would prevail under normal operating conditions. Additional surveys over time would provide a means of verification. Measuring Fishery Rent Dissipation / 150 cases are simulated. The three mentioned above and a fourth that can only be done for the entire fishery. It evaluates the potential rent available from a salmon fishing fleet composed entirely of a single vessel type. 1. Within Sample Rent Rent is calculated both for the mean vessel in a sample and for each vessel in a sample. Any difference in results is illustrative of the type of aggregation bias inherent in the use of an average as representative of the entire distribution. Tables A5.1 through A5.10 in Appendix 5 present the data used to calculate the rents. The rents in Table 6.2 (obtained using the mean vessel) and Table 6.3 (obtained using all vessels) are equal to the total within sample rent per vessel type. Thus, in Table 6.2 the rent of the mean vessel is multiplied by the number of observations in the sample to obtain total within sample rent. In Table 6.3, the sum of rents for each individual vessel gives an estimate of the within sample rent. As stated earlier, rent per vessel is calculated as the seasonal profit of the vessel minus its total fixed costs. a. Case I: Actual 1982 Rent in Table 6.2, the actual rent per sample, obtained by using the rent calculated for the mean vessel of each sample, is greatest for the seine fleet, with only 21 vessels. The rent is equal to $919,000 for the low price scenario and -$741,000 for the high price scenario. At -$238,400 (low price scenario), the troll fleet, with 84 vessels, earns the lowest rents. Using the high net tonnage price increases the amount of negative rent generated by this fleet to $1,530,200. All rents are in 1982 current Canadian dollars and the data used to calculate them are given in Measuring Fishery Rent Dissipation / 151 Table 6,2:--Total within sample rents (using mean vessel): all vessel types, all cases Vessel TyjJe Seine High Price Low Price Gillnet High Price Low Price # of Vessels 21 21 80 80 Case -741.0 919.0 -279.7 -128.0 Case 520.0 484.4 Case 1,499.9 969.9 3,012.2 2,474.0 423.1 393.6 Troll High Price Low Price Gillnet-Troll High Price Low Price 84 84 60 60 -1,530.2 -238.4 282.0 510.0 -3,005.2 -117.6 782.8 1,027.7 -58,478.1 -5,392.8 578.1 849.1 Total High Price Low Price 245 245 -2,268.9 1,062.6 -202.5 4,406.7 -56,507.0 -1,676.1 Notes: Troll results are sensitive to price changes, so the high price estimate for tonnage is used for both price scenarios. All rents are expressed in thousands of 1982 dollars. The rents per sample are calculated as the product of the rent of the mean vessel (of each type) and the number of observations in the sample. Measuring Fishery Rent Dissipation / 152 tables A5.1 and A5.2 of Appendix 5. For the entire sample of 245 vessels, the low price scenario exhibits rent equal to $1,062,600 (or, -$2,268,900 for the high price scenario). This averages out to $4,337 per vessel in the low price scenario (or, -$9,261 in the high price scenario). When the rent for each vessel in the sample is calculated, as in Table 6.3, the within sample rent estimates differ slightly from those discussed above. This difference reflects the lack of homogeneity of the vessels within a sample. The gillnet-troll sample, with 60 vessels, earns the largest amount of rent, estimated at $252,800 for the low price scenario. The seine sample comes second with a rent of $56,100. Total rent for the sample of 245 vessels becomes negative for both price scenarios and is equal to -$572,000 in the low price scenario. In general, using the mean vessel only to calculate sample rents leads to an overestimation of the amount of rent. Thus, the mean is not representative of the sample. This seems to be particularly true for the troll and gillnet-troll samples. Since vessels within a sample are heterogeneous, it is possible to observe the degree of variation in the rent associated with individual vessels. For example, in the low price scenario for the seine fleet, 8 (out of 21) vessels earn positive rents. For the same price scenario, the other vessel types exhibit somewhat different rates of positive to negative rents. At the low end of the scale, only 4 out of 80 gillnet vessels earn positive rents, whereas 30 (out of 84) troll and 17 (out of 60) gillnet-troll vessels do. Overall, 59 vessels (out of the entire sample of 245) are observed to be earning positive rents. Measuring Fishery Rent Dissipation / 153 Table 6.3:--Total within sample rents (using all vessels): all vessel types, all cases Vessel Tyjje Seine High Price Low Price Gillnet High Price Low Price # of Vessels 21 21 80 80 Case -1,603.5 56.1 -803.1 -648.1 Case 17.8 1,532.3 484.7 462.0 Case III -373.0 1,132.3 440.0 413.9 Troll High Price Low Price Gillnet-Troll High Price Low Price 84 84 60 60 -1,524.0 -232.8 22.7 252.8 -10,617.5 -1,956.3 65.8 312.6 -14,331.9 -5,653.0 -122.7 148.6 Total High Price Low Price 245 245 -3,907.9 -572.0 -10,049.2 350.6 -14,387.6 -3,958.2 Notes: Troll results are sensitive to price changes, so the high price estimate for tonnage is used for both price scenarios. All rents are expressed in thousands of 1982 dollars. The rents per sample are calculated as the sum of the rent of each vessel in the sample. Measuring Fishery Rent Dissipation / 154 b. Case II: Optimal Tonnage Per Vessel As discussed earlier, this case examines the impact upon potential fishery rent, when the vessel-owner is allowed to re-optimize over a set of inputs that includes the amount of net tonnage. Each vessel-owner chooses the optimal amount according to the current market rental price of one net ton. Table 6.4 shows the actual average net tonnage per vessel and the optimal average net tonnage per vessel for each of the four vessel types. The optimal net tonnage is calculated for the two rental price scenarios and in two ways. First, the attributes of the mean vessel in each sample are used to obtain an estimate of the optimal amount of net tonnage associated with the mean vessel. Then, the optimal tonnage associated with each vessel is found and averaged over the entire sample of vessels. A comparison of the optimal and actual net tonnage values indicates that the optimally-sized seine vessel should be smaller than the actual, ie., that its net tonnage should be reduced from 23.9 to 21.3 (for the high price scenario). This is true whether the mean vessel or distribution of vessels is used to obtain the average optimal value. The gillnet vessel should also be reduced in size, ie., from 6.1 net tons to 1.2 (high price scenario). O n the other hand, both the troll and gillnet-troll vessels should be larger, ie., the troll vessel should increase to 22 net tons (when the mean vessel alone is used) or 61 net tons (when the entire distribution of vessels is used).t The discrepancy between these two sets of results is indicative of the lack of homogeneity in the troll fleet. It appears that the tAn example of the heterogeneity of the troll results is found when one compares the lowest net tonnage to the highest. The former is 6 and the latter is 250. Obviously, the troll sample results must be treated carefully given the extreme sensitivity of the optimal tonnage calculations to slight variations in the initial conditions. Measuring Fishery Rent Dissipation / 155 Table 6.4:--Sample mean net tonnage and predicted optimal mean net tonnage per vessel: all cases Vessel lY££ Seine High Price Low Price Case Actual 23.9 23.9 Case 21.3 21.5 Case Opt imal 1 Opt imal 2 Opt imal 1 Optimal' 21.3 21.5 21.2 21.4 21.2 21.4 Gillnet High Price Low Price 6.1 6.1 1.2 1.7 2.2 2.7 1.3 1.7 2.2 2.6 Tro l l 3 High Price Low Price 9.2 9.2 22.0 22.0 61.0 61.0 58.0 58.0 61.2 61.2 Gillnet-Troll High Price Low Price 7.0 7.0 7.5 7.5 7.5 7.5 8.3 8.3 8.3 8.3 Notes: 1 This is calculated by using data on the mean vessel to solve for the optimal net tonnage for that mean vessel. 2 This is calculated as the mean of the optimal net tonnage per vessel, when a solution for each vessel's optimal net tonnage is found. 3 Troll results are sensitive to price changes, so the high price estimate for net tonnage is used for both price scenarios. Measuring Fishery Rent Dissipation / 156 optimal net tonnage for some trailers is extremely large. Finally, the gillnet-troll vessel should increase it net tonnage slightly, ie., from 7.0 to 7.5. Conditions in the fishery suggest an explanation for these results. Recall the discussion in Chapter 5 which indicates that the seine and gillnet fleets are the most heavily regulated in terms of the number of available fishing days and access to the stock. In addition, these fleets usually operate at the mouths of rivers and therefore, are close to shore and processing plants. Since net tonnage is a measure of hold capacity, my findings say that additional hold capacity is not a useful characteristic for either a seiner or a gillnetter. Hence, it may be argued that net tonnage per vessel for these two fleets could be reduced. Furthermore, the fishery has evolved so that packer boats now sit by the fishing grounds used by these two vessel types. The fishing boats simply off-load their catches to the packer boats and are ready to put down another net without having to leave the fishing ground. The packer boats then deliver the catch to the processing plants. Discussions with officals at Fisheries and Oceans reveal that the latest gillnet vessels have substantially smaller net tonnages than the fleet average. These new boats are called bowpickers and are built for speed. The other two vessel types fish in very different circumstances than those described above. In particular, the troll fleet fishes off the west coast of Vancouver Island in open waters. Vessels of this type are often at sea for several weeks. Hold capacity is therefore at a premium and probably represents a profitable investment. The gillnet-troll fleet has the flexibility to respond to unanticipated area openings and owners of these vessels would profit from additional capacity. Measuring Fishery Rent Dissipation / 157 Table 6.5:--1982 salmon catch and landed value, by vessel type Vessel Type Aggregate Catch of Salmon (000's pounds) Landed Value (000's dollars) Seine Gillnet Troll Gillnet-Troll 33644 16037 23345 21009 51793.5 28137.7 43911.8 41090.9 Total Fleet 94035 164933.9 Notes: The value of the catch is expressed in 1982 current dollars. The catch and landed value are for five species: chinook, coho, sockeye, pink and chum. Source: Government of Canada, Fisheries and Oceans Canada (Pacific Region), British  Columbia Catch Statistics 1982 by Area and Type of Gear. Measuring Fishery Rent Dissipation / 158 Finally, a piece of casual evidence suggests that the results are reasonable. In calculating the net tonnage prices for the high price case I include only those vessels of recent construction. An examination of their average net tonnage figures reveals that new troll and gillnet-troll vessels are larger than the average in the existing fleet, whereas the opposite is true for the new vessels entering the seine and gillnet fleets. A comparison of potential within sample rents for Case II, calculated both for the mean vessel and for the entire distribution of vessels, reveals a pattern similar to that observed in the Case I results. (Consult Tables 6.2 and 6.3 again.) The rent obtained using the mean vessel tends to overstate the within sample rent. This is less true for the gillnet sample and is to be expected, given that this sample accepts the hypothesis of constant returns to scale (Chapter 5). The seine sample obtains the highest rent in the low price scenario, ie., it is either $3,012,200 (using the mean vessel) or $1,532,300 (using all vessels). Once again, the troll sample does not generate a positive resource rent. For the low price troll scenario (using all vessels) rent is calculated to be -$1,956,300. Overall rents for all 245 vessels are positive for the low price scenario and negative for the high price scenario. For the former scenario total sample rent is as high as $4,406,700, when the mean vessels are used, and as low as $350,600, when all vessels are used. Tables A5.3 through A5.6 in Appendix 5 give the information used to calculate these rents. It is interesting to note the distribution of inframarginal rent across vessels in each sample. In the low price scenario all seine vessels earn positive rents, whereas 14 (out of 21) do in the high price scenario. In the gillnet sample 17 (20) earn Measuring Fishery Rent Dissipation / 159 positive rents in the low price scenario (high price scenario). Rents are higher in the latter because the fixed costs associated with a larger vessel size lower the net return per vessel. Only 4 out of 84 troll vessels show positive rents in the low price scenario, whereas a single vessel does in the high price scenario. Finally, 23 gillnet-troll vessels have positive rents in the low price scenario (14 in the high price scenario). The difference between total sample rents in Cases I and II indicates the extent to which rent is dissipated through the use of tonnage restrictions. The hypothesis is that rents could be larger if vessel-owners were able to optimize over all inputs, not just a subset. An examination of the rent estimates obtained by using all vessels reveals that rent could increase by $922,600, if the tonnage per vessel were optimal for the given market rental prices of net tonnage. (This uses the low price estimates). Using the rent calculations for the mean vessel indicates that an even greater increase in rent, ie., $3,344,100, could be generated by the adoption of the correct amount of net tonnage. An examination of the rent differentials across the vessel types shows that the seine fleet, followed by the gillnet fleet, would experience the largest increase in rent from a relaxation in the use of tonnage restrictions. O n the other hand, rents calculated for the troll sample by using all vessels actually decrease when the tonnage restriction is removed. The optimally chosen amount of tonnage increases dramaticallyt, thereby incurring much higher fixed costs. This causes total rent to fall for this sample. In part, this result may be attributable to the sensitivity of the optimal tonnage calculations to variations in tThis is disturbing in light of the shadow value obtained for this sample and presented in Table 6.1. This gives another warning that the troll results are to regarded with some scepticism. Measuring Fishery Rent Dissipation / 160 the market price of a net ton. This is observed for the troll fleet alone. c. Case III: An Increase in Substitution Possibilities This case examines the impact of an increase in substitution possibilities among the variable inputs upon potential fishery rent The hypothesis is that an increase in the ability of the fisherman to substitute inputs leads to rent dissipation. In order to investigate this hypothesis, the following exercise is undertaken. The estimated values of the a coefficients are doubled. This is an admittedly arbitrary exercise. Optimal tonnages are recalculated, along with the predicted output supply and input demands. Tables A5.7 through A5.10 (in Appendix 5) present the quantities and associated costs for this case. As expected, fishery rent for each type of vessel and for the entire sample of 245 vessels decreases when input substitution possibilities are increased. Total within sample rent for all 245 vessels is negative for both the high and low price scenarios, whether rent is calculated by using the mean vessel or by using the entire distribution of vessels. Estimates for the latter show that rent falls from $350,600 (Case II, low price scenario) to -$3,958,200 (Case III, low price scenario), constituting a loss of potential fishery rent equal to $4,308,800. This is a much larger loss of rent than the gain in rent that could be achieved through the use of the correct amount of net tonnage, ie., the difference in rent earned in Cases I and II. Thus, input substitution appears to be an important means by which fishermen may dissipate potential resource rent. Two vessel types, in particular, are most able to take advantage of input substituting activities. They are the seine and troll vessels. Recall the dicussion in Chapter 5 of the harvest technology. It predicts Measuring Fishery Rent Dissipation / 1 Table 6.6:--Actual number of vessels and estimated minimum number of vessels  (using mean vessel): all vessel types, all cases Vessel lY££ Seine High Price Low Price Case La Actual # 539 539 Case Lb Min # 357 357 Case II Min # Case 11 Min # 359 358 359 359 Case IV Min # 1002 1002 Gillnet High Price Low Price 1331 1331 1136 1136 1196 1197 1157 1157 7012 7017 Troll High Price Low Price 1638 1638 1057 1057 623 623 390 390 2511 2511 Gillnet-Troll High Price Low Price 1020 1020 974 974 876 876 883 883 3920 3920 Total High Price Low Price 4528 4528 3524 3524 3054 3054 2789 2789 N.A. N.A. Note: Troll results are sensitive to price changes, so the high price estimate for net tonnage is used for both price scenarios. Measuring Fishery Rent Dissipation / 162 this very finding. Since these two vessels exhibit a much greater degree of input substitution than the other two types, it is to be expected that they also dissipate more resource rent. The distribution of inframarginal rent among the vessels shows that quite a few seine vessels still earn positive rents (18 out of 21 in the low price scenario), whereas all troll vessels in the same scenario earn negative rents. Results for the gillnet and gillnet-troll vessels are not different from those obtained in Case II, other than that the rents are somewhat smaller per vessel. 2. Industry Rent This section presents the results obtained for each of Cases I, II, and 111, as well as a fourth case, when the within sample rent calculations are extrapolated to generate estimates of total industry rent. For each case predicted output supply quantities are used, in conjunction with the data in Table 6.5, to find the minimum number of vessels required to take the actual 1982 catch. Table 6.5 shows the distribution of actual catch, both in quantity and value terms, among the four competing vessel types. Once the minimum number of vessels is determined, rents per vessel are appropriately scaled up to obtain an estimate of total industry rent. This is done using both the predicted output and input quantities from the mean vessel in each sample and the predicted quantities for the entire distribution of vessels in a sample. Measuring Fishery Rent Dissipation / 163 Table 6.7:--Estimated actual and optimal fleet net tonnage (using mean vessel): all Vessel Ty£e Seine High Price Low Price vessel types, all cases Case La Actual 12882.1 12882.1 Case Lb Optimal 8532.3 8532.3 Case II Optimal 7646.7 7697.0 Case III Optimal 7610.8 7682.6 Case IV Optimal 21342.6 21543.0 Gillnet High Price Low Price 8119.1 8119.1 6929.6 6929.6 1435.2 2034.9 1504.1 1966.9 8414.4 11928.9 Troll High Price Low Price 15069.6 15069.6 9724.4 9724.4 13706.0 13706.0 22620.0 22620.0 55242.0 55242.0 Gillnet-troll High Price Low Price 7140.0 7140.0 6818.0 6818.0 6570.0 6570.0 7328.9 7328.9 29400.0 29400.0 Total High Price Low Price 43210.8 43210.8 32004.3 32004.3 29357.9 30007.9 39063.8 39598.4 N.A. N.A. Note: Troll results are sensitive to price changes, so the the high price estimate of net tonnage is used for both price scenarios. Measuring Fishery Rent Dissipation / 164 a. Case /: Actual 1982 Rent Table 6.6 shows the actual number of vessels fishing in 1982, along with the estimated minimum number, not only for Case I, but for all the cases. It is now possible to distinguish two possible situations for Case I. Case I.a uses the actual number of vessels that fished in 1982 to calculate industry rent. Case l.b solves first for the minimum required number of vessels and then calculates the rent associated with a smaller fleet. Thus, rent may also be dissipated through fleet redundancy and a comparison of rent in Cases I.a and l.b gives an estimate of the extent of this phenomenon. Fleet redundancy may be expressed in many ways, including the number of excess vessels (Table 6.6 shows this when the mean vessel is used and Table 6.11 does the same using all vessels); the amount of excess net tonnage, Table 6.7 (mean vessel) and Table 6.12 (all vessels); and finally, the value of lost rent, Table 6.8 (mean vessel) and Table 6.13 (all vessels). The difference in rents obtained by using the mean vessel, as opposed to the distribution of vessels, is much smaller when results are extrapolated to the industry level than when a comparison is made of the within sample rents. Thus, only results for the distribution of vessels are presented in detail; discrepancies between these results and those obtained when the mean vessel is used are noted, as necessary. Table 6.12 gives estimates of the total net tonnage used by the entire salmon-fishing fleet for all cases. There is no current information on the actual amount of net tonnage in the salmon fleet, so it must be estimated for Case I.a. In order to do so, total net tonnage is calculated as the product of the sample mean net tonnage per vessel and the actual number of vessels. The estimated total is 43,210.8 net tons. The most recent comparable estimate comes from Sinclair Measuring Fishery Rent Dissipation / 165 Table 6.8:--Estimated total fishery rent (using mean vessel): all vessel types, all cases Vessel Type Seine High Price Low Price Gillnet High Price Low Price Troll High Price Low Price Gillnet-troll High Price Low Price Case I.a Actual # 1 -45,543 -2,935 -11,832 -9,253 -58,277 -33,086 2,670 6,579 Case l.b Min # 2 -12,676 15,544 -5,976 -3,775 -22,030 -5,775 4,403 8,135 Case II Min # 2 -7,527 . 17,673 5,870 4,944 -25,019 -2,035 11,328 14,904 Case III Min # 2 -8,127 17,215 7,114 6,492 -65,838 -27,819 8,406 12,395 Case IV Min # 2 -634 69436 34,375 28,968 -112,893 -20,256 31,750 47,751 Total High Price Low Price -112,982 -38,695 -36,279 14,129 -15,348 35,486 -58,445 8,283 N.A. N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.6. 2 Rent is calculated using the minimum required number of vessels given in Table 6.6. All entries are measured in thousands of 1982 current Canadian dollars. Measuring Fishery Rent Dissipation / 166 (1978). He reports a total fleet net tonnage in 1977 of 47,300, for a fleet population of 5084 vessels. Since 1977 about 300 vessels have left the fleet, so the estimate I present appears reasonable. Table 6.13 shows estimated fishery rents for all cases. It is noteworthy that, according to the rent calculations done for Case I.a, the British Columbia commercial salmon fishery earns in aggregate negative fishery rents in the 1982 season. For the fishery as a whole, the amount of negative rents ranges from 42.8 to 117.1 million (1982) current Canadian dollars. In contrast, only the gillnet-troll fleet appears to enjoy total rents equal to $0.8 million (low price scenario). The lowest rent is associated with the troll fleet, since it generates negative rents ranging from from $33.0 to $58.2 million. As well, substantial negative rents are attributable to the seine fleet, eg., between $2.9 and 45.5 million. Recall, however, that the within sample results suggest the presence of positive inframarginal rents earned by some 20% of the 245 vessels in the sample. Rents obtained by the mean vessel are different only for the gillnet-troll fleet (Table 6.8). In this case both price scenarios show the presence of positive resource rents. The discrepancy in these results is most likely attributable to the degree of heterogeneity among the vessels in this sample. I find that the average predicted output per vessel is smaller when all vessels are used, than when the mean vessel is used. This, of course, reduces the rent estimates per vessel, and hence, total rents for this portion of the entire salmon-fishing fleet. Although a zero rent might be expected as the worst possible outcome, there are several explanations for the appearance of negative rents in aggregate. First, and Measuring Fishery Rent Dissipation / 1 Table 6.9:--Estimated fishery rent per vessel (using mean vessel): all cases Vessel Tyjje Seine High Price Low Price Case La Actual #' -84.5 -5.4 Case l.b Min # 2 -35.5 43.5 Case II Min # 2 -21.0 49.4 Case III Min # 2 -22.6 48.0 Case IV Min # 2 -0.6 69.3 Gillnet High Price Low Price -8.9 -7.0 -5.3 -3.3 4.9 4.1 6.1 5.6 4.9 4.1 Troll High Price Low Price -35.6 -20.2 -20.8 -5.5 -40.2 -3.3 -168.8 -71.3 -45.0 -8.1 Gillnet-troll High Price Low Price 2.6 6.5 4.5 8.4 12.9 17.0 9.5 14.0 8.1 12.2 Average High Price Low Price -31.6 -6.5 -14.3 10.8 -9.8 16.8 -44.0 -0.9 N.A. N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.6. 2 Rent is calculated using the minimum required number of vessels given in Table 6.6. 3 This is a simple average of the rents of the four vessel types. All entries are measured in thousands of 1982 current Canadian dollars. Measuring Fishery Rent Dissipation / 168 perhaps most important, the estimated market rental prices for net tonnage may be too high.t Since fixed costs per vessel are given by the product of net tonnage and its rental price, they may be a substantial proportion of total cost* and may lead to the finding of negative rents. Therefore, it is interesting to determine the break-even value for the rental price of a net ton (m 2 ) . This is the rental price that would set industry rent (in Table 6.6) for each vessel type equal to zero. For Case I.a this is done using data for the mean vessel in each sample. For the seine vessel the break-even value of a net ton is $2423.42. This represents 40% of the actual high price in Table 6.1 and 91% of the actual low price. The break-even value for the gillnet fleet is $447.72 (24% of the actual high price and 28% of the actual low price). O n the other hand, the troll vessel's break-even value is very close to zero, whereas that for the gillnet-troll vessel is less than the actual prices per net ton given in Table 6.1. This is because the gillnet-troll sample earns positive rents in both the low and high price scenarios. The reasons for the high values given for m 2 are several. In addition to the value of the stripped-down vessel, the current market values used to obtain the rental prices contain the value of equipment and electronics. These items are often considered to be of a 'capital-stuffing' nature (Crutchfield 1979, Rettig 1984). Apart from the value of the vessel, there is the matter of the appropriate depreciation and interest rates to be charged against the total. For example, the depreciation rate that I use suggests a 25 year average lifetime for vessels in excess of 5 gross tSee Appendix I for a discussion of the 1982 fishing season as compared to previous seasons. tSee Appendix 5 for the calculations of total cost per vessel type. They show the importance of total fixed costs in determining total cost. The proportion of total fixed cost to total cost varies across the samples, with the highest proportion found for the seine sample. Measuring Fishery Rent Dissipation / 1 Table 6.10:--Estimated fishery rent per ton (using mean vessel): all cases Vessel IY££ Seine High Price Low Price Case I.a Actual # 1 -3.5 -0.2 Case l.b Min # 2 -1.5 1.8 Case II Min # 2 -1.0 2.3 Case III Min # 2 -1.1 2.2 Case IV Min # 2 -0.03 3.2 Gillnet High Low Price Price -1.5 -1.1 -0.9 -0.5 4.1 2.4 4.7 3.3 4.1 2.4 Troll High Price Low Price -3.9 -2.2 -2.3 -0.6 -1.8 -0.1 -2.9 -1.2 -2.0 -0.4 Gillnet-troll High Price Low Price Average 3 High Price Low Price 0.4 0.9 -2.1 -0.7 0.6 1.2 -1.0 0.5 1.7 2.3 0.8 1.7 1.1 1.7 0.5 1.5 1.1 1.6 N.A. N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.6. 2 Rent is calculated using the minimum required number of vessels given in Table 6.6. 3 This is a simple average of the rents of the four vessel types. All entries are measured in thousands of 1982 current Canadian dollars. Measuring Fishery Rent Dissipation / 170 tons and 17 years for smaller vessels. However, the newest vessels introduced into the fleet are built of aluminum and have a longer expected lifetime than wooden hulled boats. The concerns expressed above may also be applicable to the costs of gear since the rental price per unit of gear is based on assumed depreciation and interest rates. The maintained service life estimates may understate the true life of the gear. This is especially true when the fishery is going through a difficult time. Fishermen respond by endogenizing the rate of depreciation through an increased amount of maintenance. Along the same lines it is often argued that the social opportunity cost of fishery labour is zero, especially in remote communities. Thus, the estimated price of labour may overstate the true social costs of labour. O n the other hand, the cost of labour computed in this thesis is calculated only for the elapsed number of fishing weeks. This assumes that unemployment in the fishery is only frictional and that fishermen immediately take up another form of employment at the end of the fishing season. If, however, the lag time between jobs is substantial, then the labour costs that I calculate underestimate the true social costs of fishery labour. O n the other hand, it is possible that the revenue estimates are low. This may be attributable to unrecorded catches. That is, fishermen deliberately understate their catches for tax evasion purposes. No data are available on the extent of this phenomenon. However, as suggested in Appendix I this does not appear to be a serious problem for the British Columbia salmon fishery because it has built-in incentives to encourage complete reporting. Measuring Fishery Rent Dissipation / 171 Table 6.11:—Actual number of vessels and estimated minimum number of vessels (using all vessels): all vessel types, all cases Vessel Tyjje Seine High Price Low Price Gillnet High Price Low Price Troll High Price Low Price Gillnet-Troll High Price Low Price Total High Price Low Price Case La Actual # 539 539 Case l.b Min # 1331 1331 1638 1638 1020 1020 4528 4528 356 356 1096 1096 1056 1056 942 942 3450 3450 Case II Min # 229 227 1153 1153 302 302 1002 1002 2686 2684 Case II Min # 256 254 1201 1201 370 370 1002 1002 2829 2827 Case IV Min # 640 634 6774 6560 1221 1221 4500 4500 N.A. N.A. Note: Troll results are sensitive to price changes, so the high price estimate for net tonnage is used for both price scenarios. Measuring Fishery Rent Dissipation / 172 There are two more reasons for the finding of negative rents that are not data-based. The first may be due to regulatory error. That is, the harvest level chosen is suboptimal because the permitted escapement is too high. This implies a smaller landed value of fish or revenue to the fishermen. Second, the regulators may have non-economic objectives, such as the need to preserve the stock at some biologically critical level, or the desire to achieve some form of income distribution within the fishery. Both of these actions impose costs upon the fishery and may lead to the generation of low or negative static rents in a given period. A comparison of Cases l.a and l.b reveals the extent of the fleet redundancy problem and its impact upon the loss of potential fishery rent. The gain in total rent from reducing the number of vessels fishing is shown to be $55,255,000 (Table 6.13) in the low price scenario or $79,393,000 (high price scenario). However, only the seine and gillnet-troll fleets earn positive rents in Case l.b. For the seine fleet, they are equal to $15,602,000 (low price scenario). The seine and troll fleets would gain the most rent by a reduction in the number of vessels. From Table 6.11 (or 6.6 for the mean vessel) it is possible to determine the number of excess vessels by subtracting the minimum required number (Case l.b) from the actual number (Case l.a). For the seine fleet the difference is 183; gill, 235; troll, 582; and gillnet-troll, 78. The number of redundant vessels in the industry is 1078. This means that the number of vessels could be reduced by 24% without a reduction in the actual 1982 harvest. The amount of excess net tonnage associated with this fleet redundancy is calculated to be 11715.4 or 37%. Pearse (1982) calls for the entire fleet to be halved. The Fleet Rationalization Committee Measuring Fishery Rent Dissipation / 1 Table 6,12:--Estimated actual and optimal fleet net tonnage (using all vessels): all Vessel l Y £ e Seine High Price Low Price vessel types, all cases Case I.a Actual 12882.1 12882.1 Case l.b Optimal 8518.7 8518.7 Case II Optimal 4917.1 4914.6 Case III Optimal 5475.2 5464.7 Case IV Optimal 13743.9 13736.5 Gillnet High Low Price Price 8119.1 8119.1 6685.6 6685.6 2494.1 3019.7 2607.0 3168.0 14670.0 17719.7 Troll High Price Low Price 15069.6 15069.6 9656.3 9656.3 18511.9 18511.9 22633.2 22633.2 74716.2 74716.2 Gillnet-troll Price Price High Low Total High Price Low Price 7140.0 7140.0 43210.8 43210.8 6634.8 6634.8 31495.4 31495.4 7595.2 7595.2 33518.3 34041.4 8330.0 8330.0 39045.4 39595.9 34110.0 34110.0 N.A. N.A. Notes: Troll results are sensitive to price changes, so the the high price estimate of net tonnage is used for both price scenarios. Measuring Fishery Rent Dissipation / 174 (1982) suggests that the seine fleet be cut to 400 vessels, which is very close to my findings, the troll to 1467 vessels, and the combined gillnet and gillnet-troll fleet to 2208. With estimates of the number of redundant vessels, it is now possible to estimate the amount of seasonal fleet redundancy deadweight loss (Munro and Scott 1985).+ The number of superfluous vessels in each sample is multiplied by the non-salvageable capital costs per vessel type. If it is assumed that the fishing vessel has no value outside of the fishery, then its entire capital cost is nonsalvageable.t- Thus, the tonnage flow price multiplied by the amount of surplus tonnage equals the deadweight loss. This is done for both price scenarios. Using the high tonnage price the deadweight loss associated with the surplus seine vessels is $26.0 million; using the low price, the loss is estimated at $11.6 million. The gillnet fleet has 235 surplus vessels and the high and low price costs associated with this are $2.7 and $2.2 million. For the troll fleet, the deadweight costs of surplus vessels are $19.8 and $10.9 million (high and low price scenarios). The gillnet-troll fleet has the smallest number of redundant vessels and their tonnage costs are $1.3 million (high price) and $1.0 million (low price). The deadweight loss for the entire fleet is estimated to be $49.9 million for the high price scenario and $25.7 million for the low. The former represents about one half of the total amount of negative fishery rent in Case l.a, and the latter about 60%. tObviously, this is an estimate that applies to the 1982 fishing season. A larger fish stock would probably require more vessels. So, this does not claim to be the final word on the optimal size of the salmon fishing fleet. An answer to this question would require a longer term analysis of the fishery. t-This would be the case if the vessel could not be used in any other production process such as pleasure boating or recreational charters. Measuring Fishery Rent Dissipation / 1 Table 6.13:--Estimated fishery rents (using all vessels): all vessel types, all cases Vessel Case I.a Case l.b Case II Case III Case IV Type Actual # 1 Min # 2 Min # 2 Min # 2 Min # 2 Seine High Price -45,522 -12,570 14,846 -5,221 28,735 Low Price -2,926 15,602 31,001 15,097 73,090 Gillnet High Price -10,228 -3,454 5,051 4,641 29,186 Low Price -7,650 -1,331 4,670 4,217 27,198 Troll High Price -58,195 -21,908 -40,861 -65,941 -177,382 Low Price -33,017 -5,678 -9,547 -27,559 -50,830 Gillnet-troll High Price -3,107 273 1,104 -2,197 -14,650 Low Price 804 3,884 5,263 2,367 4,030 Total High Price -117,052 -37,659 -19,860 -68,718 N.A. Low Price -42,789 12,477 31,387 -5,878 N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.11. 2 Rent is calculated using the minimum required number of vessels given in Table 6.11. All entries in the Table are measured in thousands of 1982 dollars. Measuring Fishery Rent Dissipation / 176 An interesting comparison of actual rents in Case l.a is given by an analysis of the difference in rents obtained by the two different price scenarios. It is suggested in an earlier discussion that the difference may reflect the adoption of an excessive amount of electronic and other equipment on newer vessels. If this is indeed the case, then it is possible to obtain some idea of the amount of rent dissipated in this manner. This is caculated as the difference between the rents in the high and low price cases. For Case l.a this rent differential amounts to $42.6 million in the seine fleet alone. For the troll fleet, the difference is $25.9 million. As expected very little variation in low and high price rents is found for the gillnet and gillnet-troll fleets, ie., $2.6 million and $3.9 million. However, if the difference in these rents is a true measure of the "capital-stuffing" phenomenon, then the use of electronic equipment contributes in large measure to the total amount of dissipated rent. b. Case II: Optimal Tonnage Per Vessel Optimal quantites and rent for each vessel are calculated when the optimal net tonnage per vessel is used in place of the actual. In this case, the minimum number of vessels required to take the 1982 catch is much lower than in Case l.b, ie., 2686 rather than 3450 (Table 6.11). The results obtained by using the mean vessel (Table 6.6) are not much different from those obtained using all vessels (Table 6.11). In fact, as noted earlier, there is less discrepancy than that found for the within sample rents. Results for the troll and seine samples show the greatest variation in results. In particular, 359 seine vessels of mean size are required to take the 1982 seine catch, as compared to 229 when all vessels are used. By using the entire distribution of vessels, the presence of larger vessels substantially Measuring Fishery Rent Dissipation / 177 Table 6.14:-Estimated fishery rent per vessel (using all vessels): all cases Vessel Type Case I.a Actual # 1 Case l.b Min # ' Case II Min # 2 Case III Min # 2 Case IV Min # 2 Seine High Price Low Price -84.5 -5.4 -35.3 43.8 64.8 136.6 -20.4 59.4 44.9 115.3 Gillnet High Price Low Price -7.7 -5.7 -3.2 -1.2 4.3 4.1 3.9 3.5 4.3 4.1 Troll High Price Low Price -35.5 -20.2 -20.7 -5.4 -135.3 -31.6 -178.2 -74.5 -145.3 -41.6 Gillnet-troll High Price Low Price -3.0 0.8 0.3 4.1 1.1 5.3 -2.2 2.4 -3.3 0.9 Average High Price Low Price -25.9 -9.4 -10.9 3.6 -7.4 11.7 -24.3 -2.1 N.A. N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.11. 2 Rent is calculated using the minimum required number of vessels given in Table 6.11. 3 This is a simple average of the rents of the four vessel types. All entries are measured in thousands of 1982 current Canadian dollars. Measuring Fishery Rent Dissipation / 178 increases the average predicted output per vessel. This is also true for the troll fleet, where 302 vessels versus 623 (of mean size) are needed. Obviously, the number of vessels affects the rent calculations. Rent is much higher for the seine sample, since the output per vessel is larger. O n the other hand, the larger sizes of the troll vessels increase fixed costs per vessel, thereby lowering rents. The gain in total industry rent possible through the use of optimal tonnage is $18,910,000 (Table 6.13). It is not very different from the gain indicated when the mean vessel is used, $21,357,000 (Table 6.8). It is interesting to note the degree of gain across the samples. For example, the seine fleet could increase its rents by $15,399,000 (low price scenario, using all vessels, Table 6.13), as compared to a gain of only $2,100,000 (when using the mean vessel alone, Table 6.8). The gillnet fleet could gain $6,001,000 (Table 6.13) and the gillnet-troll fleet, $1,379,000, both low price scenarios. On the one hand, according to the calculations done using all vessels, the troll fleet would lose rent equal to $3,869,000 (Table 6.13). O n the other hand, by using the mean vessel the troll fleet rent increases when going from Case l.b to Case II by $3,740,000. The discrepancy in these two results may be explained by examining the optimal net tonnage values obtained when the mean vessel is used, as opposed to that obtained when all vessels are used. Since the mean net tonnage, obtained by averaging over the entire distribution of optimal net tonnage for each vessel, is 61.2, this means that a large number of vessels have a much greater net tonnage. Given the high market rental prices of a net ton, fixed costs are much higher when all vessels are used than when the mean vessel is used, since its mean net tonnage is 58. However, predicted output does not increase in proportion to an increase in net tonnage. As a result, the rent per Measuring Fishery Rent Dissipation / 179 vessel must fall. This phenomenon is also manifested in the gillnet-troll results. According to the rent calculations done using the mean vessel, rent is as high as $14.9 million for this fleet in the low price scenario, compared to $5.3 million when all vessels are used. Overall; it is clear that the fishery could generate a substantial amount of resource rent, ie., as much as $31,387,000, if the optimal net tonnage per vessel were used in conjuction with the minimum number of vessels. This represents 19% of the total landed value of fish in 1982 Of the two forms of rent dissipation considered so far fleet redundancy contributes most to the loss in potential fishery rent. Static 1982 total fishery rent could be $30,312,000 larger than the actual, if the fleet had been of minimum size. A relaxation of tonnage restrictions would increase rent a further $18,910,000. c. Case ///: An Increase in Substitution Possibilites The a priori expectation is that industry rent is reduced when the vessel-owner is able to exploit a greater degree of input substitution, than when he is not. As Tables 6.8 (using mean vessel) and 6.13 (using all vessels) show, total industry rent falls when this is the case. For example, comparing Case II rents (actual degree of input substitution) to Case III rents (double the degree of input substitution), it is clear that potential fishery rents are higher in the former case. The low price scenario total rent, obtained by using all vessels, is calculated to be $31,387,000 for Case II and -$5,878,000 for Case III. A comparison of the latter to the former indicates that total rent falls by $37,265,000. Thus, input substitution is of serious concern to the regulator who wishes to generate a substantial resource rent from Measuring Fishery Rent Dissipation / 1 Table 6.15:~Estimated fishery rent per ton (using all vessls): all cases Vessel Type Seine High Price Low Price Case La Actual # 1 -3.5 -0.2 Case l.b Min # 2 -1.5 1.8 Case II Min # 2 3.0 6.3 Case III Min # 2 -1.0 2.8 Case IV Min # 2 2.1 5.3 Gillnet High Price Low Price -1.3 -0.9 -0.5 -0.2 2.0 1.5 1.8 1.3 1.5 1.5 Troll High Price Low Price -3.9 -2.2 -2.3 -0.6 -2.2 -0.5 -2.9 -1.2 -2.4 -0.7 Gillnet-troll High Price Low Price Average 3 High Price Low Price 0.4 0.1 -2.7 -1.0 0.04 0.6 -1.2 0.4 0.1 0.7 -0.6 0.9 -0.3 0.3 -1.8 -0.1 -0.4 0.1 N.A. N.A. Notes: 1 Rent is calculated using the actual number of vessels given in Table 6.11. 2 Rent is calculated using the minimum required number of vessels given in Table 6.11. 3 This is a simple average of the rents of the four vessel types. All entries are measured in thousands of 1982 current Canadian dollars. Measuring Fishery Rent Dissipation / 181 the fishery. It is as serious a problem as that of fleet redundancy and more serious than that of inefficient net tonnage restrictions.+ A comparison of rent per vessel type reveals that the troll fleet is one of the worst offenders of rent dissipation that takes the form of input-substituting activities. Using either set of rent calculations, ie., that obtained either from the mean vessel or from all vessels, it is observed that the potential rent of Case II falls by $18,012,000 (all vessels) or $25,784,000 (mean vessel) in the low price scenario. The size of this fleet's component of total rent is large enough to make industry rents become negative in both price scenarios. The seine fleet, when the entire distribution of vessels is used, earns much higher rents in Case II, than when the mean vessel is used. Thus, the reduction in rent from Case II to Case III (low price scenario) is very large for the former, ie., $15,904,000 and quite small for the latter, ie., $458,000. Rent from the two other fleets does not fall by much, merely $453,000 (gillnet) and $2,896,000 (gillnet-troll). This is because the harvest technology of these two vessel types does not exhibit much substitution in the first instance. Therefore, a doubling of the elasticities does little to alter the optimizing decisions of the vessel-owner. Once again the gillnet fleet shows the least discrepancy in results when a comparison is made between rents generated by using the mean vessel and by using the entire distribution of vessels. tThe rents calculated in Case ill are done for the optimal net tonnage associated with the greater degree of input substitution, so the difference in rents is attributable solely to an increase in input substitution possibilities. Measuring Fisher)' Rent Dissipation / 182 d. Case IV: Single Vessel Type Harvesting The final case performs the following experiment. Take the optimal vessel tonnage/input configuration from Case II for each of the four samples. Now relax the assumption regarding the distribution of the total catch across vessel types. Then, calculate the rents associated with permitting each of the four fleets to take the entire 1982 catch alone. The rent for each fleet becomes total fishery rent from single vessel type harvesting. Tables 6.6 and 6.11 present estimates of the minimum number of vessels needed and Tables 6.7 and 6.12 show the associated total net tonnage for each vessel type. Thus, the figures in all tables, 6.6 through 6.15, are fleet totals for Case IV. They show that, in keeping with the expert's view about the fishery, the seine fleet could catch the total harvest at least cost. For the low price scenario, total fishery rent (using all vessels) is estimated to be $73.1 million. This represents approximately 44% of the gross value of the total 1982 landed catch. The increase in industry rents from Case II to Case IV is approximately 60%. However, in order to take the entire 1982 harvest, my results suggest that the seine fleet must be increased to 640 vessels. In terms of total net tonnage, the total seine tonnage in Case IV is about half of the current fleet tonnage. The results show that the gillnet fleet (using all vessels, Table 6.13) generates the second largest amount of fishery rents ($29.2 million) by using 6774 vessels in the high price scenario. In fact, this fleet produces more rent in the high price scenario than does the seine. This is due to the higher fixed costs associated with the seine fleet and also due to the fact that the seine sample exhibits decreasing Measuring Fishery Rent Dissipation / 183 returns to scale, whereas the gillnet fleet accepts constant returns to scale (Chapter 5). The gillnet-troll fleet produces the third largest rents ($4.0 million in the low price case). However, according to the results for this sample when the mean vessel is used, total fishery rent is much larger, ie., $47.8 million in the low price scenario (Table 6.8). The difference in rents merely serves to highlight the importance of using the entire distribution of vessels, since the mean vessel may not be representative. Results for the troll fleet are consistent, however, it always produces negative rents. They range from -$50.8 to -$177.4 million. Once again this result is not unexpected. Many observers of the fishery believe that this segment of the fleet is the least efficient. In fact, it functions best when selectively harvesting the high value fish (chinook and coho) for the fresh or frozen market. It does not do so well at the wholesale catch of the other low valued species destined for the cannery. D. CONCLUSIONS Tables 6.9-6.1 Q and 6.14-6.15 show the rents per vessel and per net ton for each of the cases, Cases I through IV. Tables 6.9 and 6.14 show that, although each individual vessel earns a small amount of negative rent, total industry rent may be very large and negative. These tables are merely summary statistics of the results already discussed in Tables 6.6-6.8 (using the mean vessel) and Tables 6.11-6.13 (using all vessels). However, it is interesting to observe the rents per net ton since they measure the marginal shadow value of one unit of the restricted input, Tables 6.10 and 6.15. These values may be compared to estimates of the market price of the license to fish. Strictly speaking the license is attached to the vessel and may not be sold independently. An informal market for licenses has grown up since the Measuring Fishery Rent Dissipation / 184 inception of the license program with values expressed per one unit of licensed net tonnage. Fisheries and Oceans Canada monitors these sales in an informal way and gives the following data for the end of the 1982 season. Licenses for the gillnet, troll, and gillnet-troll fleets average $2,700 to $3,000 per ton, while the per ton price of a license for the seine fleet averages $4,000. These prices are much lower than those prevailing at the start of the 1982 season. For example, the per ton seine license price is $5,000 and for the other vessels, $4,000. Since 1982 is not as good a year as the boom years of 1977-1979 it seems that the tonnage prices reflect this fact, but with a lag. Thus, expectations are changed slowly in the fishery. The British Columbia commercial salmon fishery is found to be earning rents equal to -$42.8 million in 1982. The total landed value of the catch is $164.9 million. An estimate of the amount of seasonal fleet redundancy deadweight loss shows that it is a substantial proportion of the negative rent, ie., $25.7 million, or 60%. (Munro and Scott 1985) However, it only measures the excess amount of non-salvageable costs associated with excess fleet tonnage. If the minimum number of vessels were used to take the 1982 catch, the fishery rent could become positive, ie., $12.5 million. This represents a gain of $55,266,000 brought about by the use of fewer vessels, less net tonnage per vessel, and fewer variable inputs. If, in addition, the optimal tonnage were used, total industry rents would increase an additional $18.9 million, bringing the total to $31.4 million or 19% of the total landed value. The loss of rent equal to $18.9 million is a measure of the deleterious effect of inefficient tonnage restrictions. Input substitution allows a further $37.3 million in total rent to be dissipated. This represents about 23% of the gross value of the Measuring Fishery Rent Dissipation / 185 1982 catch. If single vessel harvesting is permitted, the seine fleet has the potential to generate the largest fishery rents, $73.1 million. The gillnet and gillnet-troll fleets could also produce positive, but smaller, amounts of fishery rents, in contrast, the troll fleet is unable to generate postive rents for the cases examined in the thesis. Thus, the inefficient distribution of catch among the four vessel types contributes a further $41.7 million in potential rent lost to the fishery. The British Columbia commercial salmon fishery is capable of generating a great deal of fishery rent, possibly as much as 44% of the current landed value of $164.9 million. This thesis demonstrates that static resource rent is dissipated in, at least, four ways: fleet redundancy, inefficent tonnage restrictions, input substitution, and an inefficient distribution of the total catch among vessel types. The greatest amount of rent appears to be dissipated through the government's use of an inefficient catch distribution among competing user groups, although this probably reflects an implicit efficiency/equity trade-off. However, each of the other forms of rent dissipation have the ability to substantially reduce the potential rent that the fishery is capable of generating. VII. CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH In this thesis I use micro-economic production theory to describe the within-season input demand and output supply allocation decisions of a fishing firm that operates in a regulated environment. Appealing to duality permits a description of the harvest technology by means of a restricted profit function. Micro-level data are collected from the 1982 British Columbia commercial salmon fishery and are used to estimate a normalized, quadratic, restricted profit function. This methodology allows for an integration of the issues of input substitution and rent dissipation. The approach is a fruitful one in another way as it permits the role of input restrictions to be evaluated fully. The shadow value of one of the restricted inputs, net tonnage, is calculated and compared to the market price. In addition, the optimal level of this input is determined. A comparison of the actual and optimal levels enables me to obtain an estimate of the cost associated with regulatory-induced inefficiency. The normalized, quadratic, restricted profit function is found to perform very well. It has several features of use to this study. First, it allows for the imposition of convexity in prices. This procedure leads to estimated output supply and input demands which perform in acccordance with microeconomic principles. Since it is desirable to obtain measures of rent under alternative senarios using the predicted output and input levels, it is necessary to have well-behaved functional estimates. A second advantage, gained through use of this functional form, is that the equation for the optimal tonnage is linear in parameters. This contrasts with the equation derived from the translog form. It is nonlinear and does not have a closed form 186 Conclusions and Directions for Future Research / 187 solution (Brown and Christensen 1979). Estimation proceeds in two stages. First, a linear set of the input demand and output supply equations are estimated using an iterative Zeller or seemingly unrelated regressions technique. For those samples that do not accept price convexity in the linear case, a second estimation is undertaken. The estimating equations impose convexity by the use of nonlinear parameters. These new equations require a nonlinear maximum likelihood estimation technique. Own- and cross-price elasticities, both output-variable and output-constant, are computed for the four vessel types active in the fishery, ie., the seine, gillnet, gillnet-troll and troll vessels. In addition, non-normalized elasticities of intensity between variable and fixed factors are computed. In general, the results suggest that the fishing firm has more potential for the substitution of inputs than is commonly believed. The largest vessel in operation is the seine vessel. Many observers point to it as the most important source of rent dissipating behaviour. The calulated elasticities seem to bear out this a priori expectation. This conclusion is not altered by an examination of the fishery rents obtained under alternative scenarios. However, the second largest vessel, the troll, is also found to contribute greatly to the loss in potential fishery rent. The methodology used to obtain estimates of fishery rent is similar to that adopted by Crutchfield and Pontecorvo (1969). However, it improves on their work by establishing an economically efficient tonnage/input configuration as the benchmark characteristics of the representative vessel. Furthermore, this economically efficient vessel is also allowed to choose its profit-maximizing level of output. This contrasts with the assumption made by Crutchfield and Pontecorvo of a technically efficient Conclusions and Directions for Future Research / 188 output level per vessel. The new procedure developed in this thesis allows prices to enter the decision making of the fisherman in an integrated and consistent manner. Since rent dissipation behaviour is inherently related to economic incentives, this is seen as an important step towards an increase in our understanding of this process. It is found that the Class II rent dissipation manifests itself in at least four different ways: input substitution, inefficient tonnage restrictions, inefficient distribution of catch among vessel types, and fleet redundancy. These findings are of significance for the future management of the fishery. O n the one hand, evidence of input substitution in the harvest technology suggests that a quota system might be a valuable tool for preventing rent dissipating behaviour. Under a quota system the onus is upon the fisherman to use the least cost combination of inputs to harvest a given catch. Furthermore, if quota trading is allowed, more efficient boats may take a greater share of the catch. Less efficent boats will have incentives to sell their quota and leave the fishery. Used in conjunction with a royalty tax, this scheme could provide revenues to the resource owner, the Canadian public. O n the other hand, the estimated elasticities indicate that fishermen are sensitive to prices, so that a royalty scheme on its own might reduce the excessive amount of inputs used to take the given harvest. The model developed in this thesis provides a useful starting point for a further investigation of the role of regulations in a fishery. Several extensions and related questions that have generated interest during the course of the thesis research are given. They are presented briefly, not only as a suitable ending to this thesis, but also as a starting point for future research. Conclusions and Directions for Future Research / 189 Using the framework developed in this thesis, it is possible to evaluate the income distribution and efficiency effects of alternative regulatory schemes. For example, it has been proposed that the Pacific Coast salmon fish stock be increased through a salmonid enhancement program (SEP). In the context of this model this means an increase in the level of a fixed factor. It is possible to predict the output supply and input demand responses under alternative scenarios concerning the state of the harvesting fleet. The effect of this policy upon total harvesting costs can be evaluated. The success of the program has usually been predicated upon there being no increase in harvesting costs. If costs are found to increase, the viability and usefulness of the program must be called into question. Going further, it is possible to examine the effects of such a policy upon the incomes of fishermen. One of the stated objectives of the limited entry program is to increase fishermen's incomes in an equitable fashion. Using a standard Lorenz curve approach the new income distribution can be compared with the old. Other scenarios, which can be simulated, are an increase in the number of fishing days and the imposition of a royalty tax. With respect to the latter management tool, of interest is the incidence of this policy (Devoretz and and S.chwindt 1985). Again the income distributional and efficiency impacts of these regulations may be examined. Another interesting question is the following. What are the optimal levels of all three fixed factors, when they are permitted to vary freely? The answer involves the solution of a system of three equations in three unknowns. The optimal amount of net tonnage obtained in this manner could be compared to that already calculated Conclusions and Directions for Future Research / 190 in the thesis. Of particular interest, is the interaction between the net tonnage and the fishing days restrictions. The model employed in this thesis assumes that the available stock of fish is known with certainty. In reality, the stock is known only imperfectly and may fluctuate from season to season. As well, there are a host of other relevant factors in the fishery environment which are not understood perfectly (Gates 1984). Furthermore, the number of allowed fishing days is correlated with the regulator's estimate of the size of the stock and, thus, it may also fluctuate. Extending the model to incorporate the effects of stock uncertainty would be a useful direction for research. The effects of output price uncertainty also bear investigation. This is usually handled by assuming different processes of expectations formation. Because of the implicit intertemporal nature of these two extensions it would be necessary to recast the problem as a dynamic or multi-period problem. This might involve assuming the existence of a finite, eg., two or three year, planning horizon. In fact, a shortcoming of the thesis is that the input demand for gear, and other multi-lived inputs, is assumed to depend upon current prices only. It would be interesting to compare the optimal input allocation decisions obtained from the static and dynamic cases. One alternative, which exploits the fundamental uncertainty surrounding the operation of the fishery, is to allow for strategic behaviour by the participants. For example, the effects of fishermen forming coalitions could be analyzed. Also, the imperfect ability of regulators to monitor the actions of the fishing vessels could be endogenized into the design of optimal regulatory schemes (Wilen 1985). Conclusions and Directions for Future Research / 191 Another extension along these lines is a full investigation of the role of electronic equipment in the dissipation of resource rent. This would require a much better data set than is currently available, in particular, a vessel-specific time series of costs and output would be needed. Returning to the static problem, one obvious area of research is the role of multiple outputs in determining the efficiency of harvesting. It might be possible to disaggregate the salmon catch into its five components (Squires 1984, 1987a; Kirkiey 1986). However, the advantages to this are probably not so many as those gained by keeping salmon as an aggregate output and adding the roe herring catch. A large number of vessels, especially seiners and gillnetters, participate in both fisheries. There may be economies of scope in harvesting these two distinct species, as their fisheries take place at different times in the year. In fact, some observers believe that the salmon seine plus herring vessels are more efficient and certainly more profitable than the single salmon seiners. This hypothesis could be tested in an extended model. However, doing this requires an evaluation of the impacts of two very different license limitation programs. BIBLIOGRAPHY Adasiak, A. "Alaska's Experience with Limited Entry" Journal of the Fisheries Research  Board of Canada 36(7) (1979): 770-782 Anderson, F.J. Natural Resources in Canada: Economic Theory and Policy. Toronto: Metheun, 1985. Anderson, L.G. "The Relationship Between Firm and Fishery in C o m m o n Property Fisheries" Land Economics 52(2) (1976): 179-191. Anderson, L.G. The Economics of Fisheries Management. 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APPENDIX 1 : DATA CONSTRUCTION This type of micro level production study has never been attempted for this fishery, due largely to the lack of expenditure and price information at the vessel level. Therefore, the construction of a suitable data base is a significant component of this thesis research. There is no single source of data, although Fisheries and Oceans Canada is the major one. Economists in the Program, Planning and Economics Branch have been most generous in providing access to survey data and other background material. The Economics and Statistics Branch has been instrumental in permitting the use of the output data. A. DATA SOURCES The data have been made available by the (Federal) Department of Fisheries and Oceans, Vancouver. There are two major data sources. The first is a cross-sectional survey of Pacific Coast fishermen for 1982. It has expenditure information by vessel. The second is the 1982 Sales Slips file which supplements the first data set by providing revenue and output information. An important part of this work is the linking of these two data sources in a manner that preserves the confidential' nature of each data set. Each data set is described in detail below. This discussion includes an analysis of the choice of the four samples. Following the descriptions of the basic data is a list of the types of variables used in a normalized, quadratic, restricted profit function. The appendix concludes with a detailed description of the origin and subsequent transformations of each variable. 209 / 210 1. The 1982 Survey of Pacific Coast Vessel Owners This data set provides the first detailed cost and input data at the level of the individual vessel owner in the commercial fisheries of British Columbia.* The lack of this type of data has hampered previous efforts to examine fishing behaviour at the micro level. Furthermore, these data have never been used for the objectives described in this thesis. The survey sample size is 762 cases. A case represents one vessel owner who may own more than one vessel up to a maximum of three.* Of the total cases 560 fished salmon with or without other fish. This represents 13.6% of the salmon-licensed fishing population in 1982. The data set includes information on each vessel's characteristics: crew size, length, gross tonnage, hull type, engine type, horsepower, age, home port, gear types used, and licenses held. In Tables A1.1 through A1.4 descriptive statistics are presented for selected variables of interest to this research. Another component of the data set consists of information on major equipment purchases such as the original cost and year of purchase, current market value, net book value, replacement value, and estimated license value. This component of the data refers to the vessel. A fourth section in the data set provides a gear inventory for 1982. It provides detail on the number and value of gears purchased, sold, destroyed, and repaired. tThe survey is not designed to elicit truthtelling from the participants, nor does it provide a means of cross-checking the answers. However, there is no way of knowing what biases, if any, exist in the data. *Only 80 cases indicated ownership of more than one vessel. / 211 Fishing activities are described in the fifth section, eg., the number and duration of fishing trips, the number of days fished, the number of deliveries. Finally, a breakdown of revenues, including the value of bonuses to the vessel-owner, and expenditures, including the amount of money spent on fuel, is included. 2. Sales Slip Data for 1982 Whereas the survey described above proves to be an excellent source of input expenditure and quantity information, it does not have the required detail on the output side. This information is contained in the so-called Catch Statistics collected by Fisheries and Oceans Canada. The vessel-owner is required to record the sale of every catch to a registered buyer. These buyers are generally the processing companies that receive the catch, but may also be so-called packer boats. The packer boats are registered with Fisheries and Oceans Canada. They wait on the fishing grounds and collect catches from the vessels. In either case the buyers must fill out a sales slip that records the transaction date and location of the catch, along with the units of each fish type and the prevailing price per unit. The price varies by species, as well as within the season and according to location. It is possible to generate two important variables from this data set. The first is the output supplied by each vessel and the associated vessel-specific price. The second is information by vessel about which fishing grounds are frequented and at what time periods. This is used, along with biomass data, to generate an estimate of the fish stock encountered by an individual vessel. / 212 B. VESSEL SELECTION AND DATA TRANSFORMATION Given the basic data sources two tasks must be performed. The first involves the choice of which vessels to include in the analysis. The second is the generation of a complete set of data for each vessel. 1. Vessel Selection Three different gear types are used to harvest salmon: seine nets, gill nets; and troll lines with baited hooks. Most vessels participating in the fishery employ only one of the three and are referred to as seiners, gillnetters, or trailers. However, some vessels are called combination vessels; gillnet and troll gears are the dominant combinations. Since the scale of operations of salmon fishing vessels varies a great deal according to gear used, four separate samples are developed. There is a sample for each of seiners, gillnetters, and trailers, as well as the combination of gillnet and troll gears. A second reason for this classification scheme is uncertainty about the uniformity of the production structure exhibited by the different gear types. For example, seiners are the biggest vessels with largest crewsizes that move swiftly from one area to the next. Gillnet vessels tend to congregate at one river mouth and are less mobile. Trailers operate both in open seas, on the west coast of Vancouver Island, for long periods of time before unloading their catch, or in Georgia Strait where they come in to port frequently. For each gear type the objective is to obtain a sample of vessels that uses that gear type (predominantly) to fish salmon as a major species. In particular, I do not / 213 want to include vessels that fish both herring and salmon as major species. The reason is that I cannot apportion the total costs given in the survey to the two fisheries in a predetermined way. One possible solution is to estimate a multi-output model with salmon and roe herring as the two outputs and test for jointness in the inputs (Squires 1984 and 1987a). Since my objective is to concentrate on linking the harvest technology to rent dissipation in the salmon fishery, which is the major fishery on the West Coast, I do not do this. A complicating factor is that the fisheries operate under very different licensing systems. In the salmon fishery the vessel and its net tonnage are the licensed factors, whereas licenses issued in the roe herring fishery are person-specific. However, the license-holder must designate a vessel for the season. Ideally, I would like the herring catch of one vessel to be combined with the salmon catch of the same vessel. However, in many cases this is impossible given the informal and unrecorded leasing arrangements made with respect to the use of herring licenses. My starting point is the vessels included in the 1982 Survey of Pacific Coast Vessel Owners. Of the 762 vessels surveyed only 560 permanantly-licensed vessels fished salmon as a major species, with or without other minor species. In 1982 the total number of vessels licensed to fish salmon is 4638, however, only 4528 vessels actually fished in 1982 (See Chapter 6). Of the licensed vessels 4112 are permanant, non-Indian licenses; 358 are Indian licenses, and 168, temporary licenses. When the licensing system was first adopted special provisions were made for hardship cases. The owners were issued so-called "B" or temporary licenses. These licenses were to last for ten years only. In the ensuing years extensions have been granted to the holders of these licenses, but their numbers are rapidly declining. / 214 They are distinguished from the regular salmon or "A" licenses, which do not specify the length of tenure. Instead, they are meant to be held in perpetuity. After dividing the 560 vessels into one of four categories and deleting those that have caught herring roe I am left with the following number of observations per sample. Seine vessels number 21; gillnet, 80; troll, 84; and gillnet-troll, 60. According to a recent review of the fishery by an economist at the Department of Fisheries and Oceans, the number of active salmon vessels, ie., those that reported landings, for 1982 is 539 in the seine fleet, 1331 in the gillnet fleet, 1638 in the troll fleet, and 1020 in the gillnet-troll combination fleet. It appears that each of my samples constitutes about 5-6% of the active fleets. This is a somewhat greater percentage of coverage than Squires (1984, 1987a). His study of the New England Otter Trawl fleet uses 2 years of observations on 21 vessels in a total population of 776. 2. Data Generation To estimate the input demand equations derived from the normalized, quadratic, restricted profit function I need price and quantity information for the following variable inputs: labour, fuel, and gear services. In addition, it is necessary to have information on the three fixed or restricted inputs, vessel tonnage, fishing days and the relative stock of fish available.* To estimate the output supply equation I need output price and quantity information. These data must be vessel-specific. Since the data are cross-sectional, the input/output prices are either region- or tThe justification for including these variables is given later in this appendix. / 215 vessel-specific. Regional prices have been used successfully in studies of other fisheries (Squires 1984, 1987a). Furthermore, the geography of British Columbia, in particular, the vast amount of non-uniform coastal area, is a sufficient justification for the use of regional prices. Homeports are scattered across the province from the west coast of Vancouver Island to the northern area of Prince Rupert and down to Vancouver and Victoria. Seven major regions, with their major centre(s) are listed below. 1. the North Coast (Prince Rupert) 2. the Central Coast (Namu, Klemtu) 3. East Georgia Strait (Powell River, Madeira Park) 4. West Georgia Strait (Campbell River, Nanaimo) 5. Lower Mainland (Vancouver) 6. Capital Area (Victoria) 7. West Island (Port Alberni, Ucleulet). A detailed description of all data generated and subsequent transformations begins with the variable inputs. Next, a discussion of the output variable is undertaken. Finally, the restricted inputs are described. a. The Labour Variable: Price and Quantity The quantity of labour input is obtained from the Survey data. Two types of labour services are used on the vessel: crew and skipper. The number of crew members varies from 0 to 5 across the entire sample. It is assumed that the services of one skipper are used per vessel. / 216 The fishing industry is unlike other primary industries in that a share system is often adopted as a means of remunerating the crew. That is, the crew gets a share of the value of the catch.t In this way the returns to labour vary from year to year in accordance with stock or price fluctuations. This presents a potential simultaneity bias. That is, one of the so-called independent variables may be correlated with the error terms. One solution that is frequently adopted is to find a relevant price for labour that is taken to be exogenous by the vessel-owner (Anderson 1977). Hannesson (1981) argues that "...as a first approximation it is not unreasonable to set the opportunity cost (of labour) equal to the "wage" rate outside the fishery..."* In practice, most fisheries researchers employ the wage in the manufacturing industry (Roy, Schrank, and Tsoa 1982, Hannesson 1983, Squires 1984 and 1987a; Bj0rndal 1984). Following the work of Squires (1984, 1987a) and Roy, Schrank, and Tsoa (1982), in particular, this research adopts the notion of opportunity cost wages to construct a unit price for labour. The relevant wage for the British Columbia commerical fishery is assumed to be average weekly earnings (including overtime) for all employees in the industrial composite category, Statistics Canada catalogue 72-002, Employment,  Earnings, and Hours. Data are collected for five cities in British Columbia corresponding to the regions mentioned already and for British Columbia on average. Before explaining the method used to calculate the relevant opportunity cost wage for each vessel a brief discussion of the suitability of this data is given. tAccording to one study of the British Columbia salmon fishery, the share varies by fleet and even by vessel (Cislason 1978). *R. Hannesson, O . Hansen, and S. A. Dale "A Frontier Production Function for the Norwegian C o d Fisheries" in Applied Operations Research jn_ Fisheries edited by K.B. Haley, 338-350. Plenum Press, 1981. p.338. / 217 There are two matters to consider. The first is the choice of alternative type of employment for the labour used in the fishery; the second is the use of regional wage data. With regard to the first matter it must be noted that the means of the two labour price series should be similar. Two pieces of evidence reveal this to be the case. The first is shown in research by Gislason (1979) and Gardner (1980) on the British Columbia salmon fishery. The latter quotes discussions with officials from the Department of Fisheries and Oceans and a report on fisherman's incomes (Hunter 1971). Both sources identify agriculture and forestry as the most popular forms of alternative employment for the fisherman. In his empirical study, Gardner uses the average weekly wage in the forest industry. Gislason, on the other hand, tries to calculate weekly returns to labour as a residual, after assuming a competitive return on the capital employed in the fishery. He finds that they vary in all four fleet types over a range that includes the average weekly wage in the manufacturing industry. The second piece of evidence comes from Revenue Canada, Department of National Revenue, Taxation Division. Every year this division publishes a volume, entitled Tax  Statistics, which presents summary personal income statistics based on taxable returns. Taking the information in Table 8 or 8A or 9 (the table number changes over the years) I am able to calculate the average weekly income earned by fishermen in British Columbia. Over time this closely follows incomes in the manufacturing sector, Table A1.5. The second matter for discussion concerns the use of regional wage information. In Table A1.6 the average weekly earnings for British Columbia on average and five / 218 regional centres are presented for 1982. The data show a great deal of variation which would appear to suggest a lack of mobility in the labour force. This evidence accords with the assumption made in this thesis that each vessel owner has a homeport for his vessel and draws upon the regional market for his crew. This is what Squires (1984, 1987a) also assumes as relevant for his fishery study. Starting with the March 1983 catalogue, Statistics Canada publishes the necessary data for the following urban areas: Vancouver, Victoria, Nanaimo, Port Alberni, and Courtney, and for British Columbia as a whole. For 1982 the data are not as disaggregated. To obtain estimates for these urban areas for 1982 the 1983 differential between the B.C. average and the respective urban areas is applied to the B.C. average for 1982. This generates average weekly earnings in 1982 for the 5 urban areas mentioned above. A comparison of the 1983 differential to the 1984 and 1985 differentials reveals little variation. Given the relatively high unemployment rates in 1982 it is necessary for the opportunity cost wage to reflect the difficulty of obtaining alternative employment. The opportunity cost wage I calculate for each vessel is taken to be the expected average weekly earnings. This is defined as: (A1.1) EAWE = (1-U)*AWE + (U)*AWUIB AWE is the average weekly earnings found in the Statistics Canada catalogue mentioned earlier, Table 16. U is the unemployment rate and comes from Cansim Series D771624 to D771633, a monthly series on unemployment rates in different economic regions of British Columbia. The title of this data series is British  Columbia. Basic Labour Force Characteristics by Economic Region and Metropolitan / 219 Area. The areas chosen are Economic Regions 950, 960, and 980-990, in addition to the two metropolitan areas, Vancouver and Victoria. The employment rate is determined as one minus the unemployment rate. These rates then give probabilities to the two states of the world. AWL) IB is the average weekly unemployment insurance benefit and comes from the Statistics Canada catalogue 73-001, Statistical  Report on the Operation of the Unemployment Insurance Act (quarterly). The data for benefits paid to British Columbia fishermen are used. They come from Table 12, Average Weekly Payments by Province, Month and Type. The weekly benefits are averaged over the year to create an average weekly benefit payment rate. The data generated by the equation (A1.1) represent expected average weekly earnings. These figures need to reflect the opportunity costs relevant to the salmon fishing season. To obtain opportunity cost wages for the salmon fishing season the weekly earnings are multiplied by the number of weeks each vessel fished; in all cases they are much less than 52 weeks. Information on the number of weeks fished comes from the sales slip data and is determined as the elapsed number of weeks from first to last delivery. This may miss out a week or two at the beginning and end of the fishing season if nothing is caught, but the figures may be viewed as reasonable estimates. The work done on the opportunity cost wages for this fishery differs from that of Squires (1984, 1987a) and Roy, Schrank, and Tsoa (1982). They calculate annual opportunity cost wages on the assumption that fishing is a full-time occupation. This is not a valid assumption for the British Columbia commercial salmon fishery. Although, the fishery is the most important of two major fisheries on the West / 220 Coast, the five species of salmon are available only from late March to early November (MacDonald 1982). A study of the 1976 fishing season shows that the average number of weeks fished is quite small. For seiners it is 14; for gillnetters, 12.6; for trailers, 15.8; and for gillnet-trollers, 18 (Gislason 1979). The data used for this thesis generates the following statistics: 15 weeks for seiners, 12 for gillnetters, 18 for trailers, and 16 for gillnet-trollers. It is a reasonable assumption to view the labour employed on salmon vessels as seasonal. Hence, the relevant opportunity cost wage should reflect the number of weeks a vessel fished, since this is the number of weeks the crew could not work in an alternative job. The assumption used to generate the opportunity cost wages is that the labour market exhibits only frictional unemployment, Thus, crew members are assumed to find new jobs immediately at the end of the fishing season. To the extent that this assumption underestimates the amount of time between jobs, the opportunity costs wages may underestimate the true opportunity cost of fishermen. An alternative methodology is to increase the number of weeks used to account for the fishing season. O n the other hand, some observers believe that labour used in the fishery has a zero opportunity cost. This may be especially true in remote communities. Thus, it seems that the choice made for an estimate of opportunity cost wages in this thesis strikes the middle ground. So far the discussion has concerned only the crew members. Skippers are treated in a slightly different fashion, since many of them are owners who do spend time in the off-season repairing and maintaining the vessel. Discussions with officials at the Department of Fisheries and Oceans indicate that 6 weeks on average are spent / 221 on maintenance of vessels owned by processing companies. This number is taken to be an estimate for the extra number of weeks that independent skippers also spend on such activities. Therefore, it is determined that the number of weeks attributed to the skipper are equal to the vessel-specific number of weeks fished plus 6 more weeks of repairs and maintenance. The notion that skippers differ from crew members may also enter into the determination of the relevant amount of average weekly earnings. This is the question of whether skippers should be viewed as managers and accorded a higher wage rate than the crew. Both papers mentioned earlier take this view. Squires adjusts the opportunity cost wage for skippers upwards by an arbitrary 20%. Roy, Schrank, and Tsoa choose the figure of 10% for independent skippers and 50% for those employed by processing companies. It is not clear that skippers face higher opportunity costs, ie., that their fisheries management skills are not fishery specific. Therefore, the average weekly earnings chosen for the crew are also used for the skipper. Indeed, this is not a bad approximation, since the average weekly earnings I use are an average over various categories of skilled, unskilled, part-time, and full-time employees. In order to obtain an aggregate measure of the wage to both types of labour (crew and skipper) a divisia index is generated (Diewert 1978, 1986). This is a standard procedure used to create appropriate aggregates. The unit price of labour is set equal to 1.00 for the first vessel in each sample. This indexing procedure is followed for all other prices and for the fixed inputs. The implicit aggregate index of labour quantity is determined by dividing total expenditure on labour by the / 222 aggregate price index. b. The Fuel Variable: Price and Quantity The cross-sectional survey gives information on total fuel expenditures per vessel, as well as the type of engine, eg., diesel or gas. In addition, it provides information as to the nature of vessel ownership, eg., independent or owned by a processing company. The breakdown between these two types of ownership is as follows. Of the 21 seine vessels four are owned by processing companies. In the gillnet sample 16 out of 80 vessels are owned by processing companies, whereas 14 out of 60 gillnet-troll vessels are not independent. The entire troll sample is independently owned. Regional fuel prices are obtained from Esso Petroleum Canada. They are found in Table A1.7. Esso is chosen because of its involvement in marine fuel retailing. Two sets of prices are available: one for gasoline and the other for diesel. Fuel used for commercial fishing purposes is subject to two types of exemptions. The customs clearance exemption applies to diesel fuel used for deep sea fishing by a Canadian fishing ship and is valued at 2.9 cents per litre in 1982. The family fishing exemption, valued at 1.1 cents per litre, applies both to gas and diesel purchases. All vessels with diesel engines qualify for the first exemption. Only those vessels independent of processing companies are able to take advantage of the second. For each vessel the homeport is used as an indicator of where fuel is purchased. This is a reasonable approximation, for vessels usually come back to their homeports to deliver their catches and refuel. Prices are available for the following / 223 centres: Prince Rupert, Namu, Klemtu, Madeira Park, Port Hardy, Campbell River, Vancouver/Steveston, Nanaimo, Bamfield/Ucleulet, Tofino, and Kyuquot. Fuel prices change once during the 1982 season, at the beginning of October. Therefore, a share-weighted average fuel price is calculated. The share for the pre-October price is determined as the number of deliveries prior to October divided by the total number of deliveries. Since the shares sum to one, the post-September fuel prices are weighted by one minus the pre-October share. The weighted fuel prices are summed to generate an aggregate fuel price. Fuel prices vary about 20% over the samples; it is likely that the variation reflects transportation costs. The quantity of fuel is determined by dividing expenditures on fuel by the vessel's own fuel price. c. The Gear Variable: Price and Quantity The gear variable is treated as a malleable capital good whose services are not entirely exhausted in one year. The component parts are the nets, traps, lines, etc., that are used by each vessel. It is appropriate to view this type of equipment as being variable within a season, since well-defined markets exist for used gear. That is, gear can be easily taken on board or removed from a vessel. The survey data provides detailed information about the gear onboard each vessel. In particular, it gives an inventory of each type of gear on board the vessel at the start of the 1982 season (the old gear units), as well as the loss or addition of gear over the season. Unfortunately, given the data limitations, it is not possible to include electronic equipment in the measure used for this input. / 224 The first step involves the generation of the stock of gear of each type. This is calculated by a simple addition of the number of units of each gear type. This is justified by the following assumptions. It is assumed that both old and new units of gear are treated equivalently in production. That is, gear units are assumed not to deteriorate over time in terms of their productive ability. This is because repairs and maintenance are done in the off-season period to assure a constant flow of service. This is a reasonable assumption given the nature of the fishing operation, as gear maintenance and repair is an important part of pre-season preparations. As well, second-hand or rental markets for gear appear to be well established, as classifieds in the trade magazines attest.t There is, however, a depreciation effect to be dealt with. This is a price phenomenon; that is, the price of a unit of gear decays over time. In part, this is attributable to the shorter life expectancy of an old gear unit. As well, maintenance expenditures decline over time for a unit of gear. Thus, for each vessel and unit of gear, price depreciation is an endogenous phenomenon. Given the data limitation, in particular the age structure of each gear unit is not known, it is not possible to estimate a relationship between the age of the gear per vessel and its price depreciation. Therefore, a straight line depreciation is assumed for convenience. It is taken to be a linear approximation of the true price depreciation. The next step is the calculation of suitable rental prices for the different gear types. In light of the assumed constant flow of services, it is assumed that the asset price depreciates over time. For the 1982 season, the appropriate method of tThey include The Fisherman, Pacific Fishing and Westcoast Fisherman. / 225 calculating the rental price is to use a modified version (Schworm 1977) of the standard Jorgenson type of capital services price Oorgenson 1963). In its usual guise, this method has components that include the current (1982) purchase price of the gear, the economic depreciation rate, the discount rate, and parameters that reflect aspects of the corporate tax structure. The gear rental price for each type of gear is chosen to be the following: (A1.2) Pg = Q*(R + D) + RM. In this equation Pg represents the unit user or rental cost of a given gear type. Q is the current (1982) unit price of the gear type. This is specific to each vessel and gear type. O n the other hand, the discount rate, R, and the depreciation rate, D, are taken to be constant and are industry averages. RM is the value of maintenance or repair work done in 1982 and is vessel- and gear-specific. The basis for this equation is work done by Schworm (1977). It seems that the corporate tax structure is not relevant to the fisherman's decision making, since most vessels are privately owned and their owners are not incorporated. However, I assume the personal income tax structure does not affect the gear rental price calcuation. Data used to calculate the unit rental rates for each gear type and for each vessel originate from several sources. The current purchase prices are available from the inventory data found in the Survey of vessel owners. In addition, the inventory provides detailed repair and maintenance expenditures. It is assumed that the different gear types all follow straight-line depreciation rates, but that they vary according to gear type. Estimates of depreciation rates are obtained from industry / 226 sources and provide ranges for each gear type.t For the seine nets these rates vary between 20% and 33.3%, whereas for the gill and troll gears the range is from 33.3% to 50%. Two other gear types used on board the vessels are handlines and traps. The depreciation rates for handlines are 25% to 33.3% and for traps, 33.3% to 50%. They do not differ from rates used by Gardner (1980) and Gislason (1979) to study this fishery. These ranges provide a natural way to test for the robustness of results. The programs are first run with high depreciation rates, then, rerun with low rates. No appreciable difference is found in the estimates. The results report only the estimates obtained by using the low rates. The nominal interest rate is assumed to be the same for all gear types. A discussion with a representative of the Gulf and Fraser Credit Union, which has many fishermen as clients, elicited the information that loan rates to fishermen have usually been calculated as the prime rate plus 1%. The prime rate chosen is the average of monthly rates on business loans in 1982. The rates are obtained from the Bank of Canada Review, February 1983. The rate is 16.8125%. In some cases, two or more gear types are kept on board a vessel. This requires the construction of an aggregate gear quantity over all gear types and an associated rental price. A divisia price index is calculated using the quantity and unit rental price information. The aggregate quantity index associated with the aggregate rental price is obtained by dividing the total value of gear by the price index. tSources include gillnet and troll fishermen and a marine supply store. / 227 d. The Output Variable: Price and Quantity Raw data come from the 1982 sales slips, also known as the Catch Statistics. They record every sale made officially to a processing company or a packer boat. The data include the unit price (per pound) and quantity landed of each type of fish over the season, as well as the type of gear used to catch the fish. Prices vary by type of fish, by area, and by week. Prices taken from the sales slips are called "book prices". That is, the processing companies record the catches at the stated book prices as the season progresses. No payments are made until the end of the season. At that time, skippers are given bonuses according to the species caught. The per species bonuses are calculated as percentage increases in the book prices. That is, the bonus rates per species effectively raise the per unit prices obtained per species. The bonus rates vary across species and gear types and are also subject to negotiations between processing companies and individual skippers. Unfortunately, these rates are not available per vessel. Instead, the Department of Fisheries and Oceans has information on the 1983 rates.t These rates are used to obtain end of the season unit prices. The rates are given in Table A1.8. Total revenues are calculated as the landed value of the catch from the Sales Slips data plus the total value of bonuses given in the survey data. Pearse (1982) complains about the usefulness of the Catch Statistics data because they do not record over the side sales. He claims that these unofficial sales might be a significant portion of all sales because fishermen wish to underreport their tPersonal Communication from Heather Fletcher, economist in the Planning and Economics Branch, Department of Fisheries and Oceans (Pacific Region), Vancouver. / 228 official catches for tax evasion purposes. A discussion with an employee of the Department of Fisheries and Oceans provides reasons why Pearse's criticisms might not be too troublesome. The employee suggests that it is in the best interest of the fisherman to report all landings for two reasons. The first has to do with the way in which unemployment benefits are paid, ie., benefits are based upon the number of weeks worked and may be paid to unincorporated skippers, as well as crew members. In addition, vessels that do not report landings for two years have their license revoked. Using the price and quantity information per species a divisia index of aggregate output price is calculated per vessel. The aggregate output quantity is obtained by dividing the aggregate price into the total value of sales plus bonuses. e. Restricted or Fixed Inputs The Survey gives information per vessel on the gross tonnage, not the net tonnage. The latter is obtained from license registry files kept by the Department of Fisheries and Oceans. The Survey also has data on the number of fishing days by each vessel. Net tonnage is restricted by the government, although within a season the gross tonnage is also a fixed factor per vessel. It is reasonable to assume the number of fishing days per vessel to be fixed a some level which is less than that desired by the vessel-owner. This is because of the way the fishery is managed. The regulator may declare an area closed to all fishing or open only to certain gear types. In addition, nature has control over the maximum number of possible fishing days. Between these two forces, the fishing vessel finds its fishing days to be restricted within the season. / 229 The third fixed factor is the fish stock encountered by the vessel within a fishing season. It seems reasonable to view this as fixed from the vessel's point of view; that is, nature provides only a certain number of fish each year. Furthermore, although a vessel-owner can try to control the amount of fish encountered by choosing areas of great abundance, he cannot control the actions of the other vessels that serve to reduce the available fish. It is important to include this factor, since it may have an impact on the degree of input substitutability. I use a measure of the fish stock available in an area at the beginning of the season.t For each species I construct an estimate of the stock after the fact by adding together escapement (the number of fish that spawn), the commercial catch, and for some areas and species the sports catch and the Native food fishery catches.* The stock generated in the manner described above is measured in terms of the number of fish. In order to obtain an estimate of the total weight of fish, these stock values are multiplied by the average weight of fish caught in 1982. This is be done because catches are usually described in terms of their weight. Thus, the underlying harvest production function should be thought of as explaining the weight, not the number, of fish caught. For stock management purposes thirty areas have been designated by DFO. A t A more complex method would generate weekly stock estimates by subtracting the previous week's catches from the beginning total stock. However, most fish pass through an area quickly, so that most fishing takes place within a short time (ESSA 1982). As well, total catch is generally a small proportion of the total stock. *They are important only for certain species and for specific areas. / 230 management area is the basic data unit that I use to compile stock estimates. For 1982, both commercial catch data and escapement data are obtained by species. Unfortuntely, the area-specific stocks calculated in the manner described above may not reflect the number of salmon available at any given time to the fishermen. The reason is that the Department of Fisheries and Oceans collects its escapement data according to the spawning destination. This means, for example, that fish heading for the Fraser River are counted as escaping in that area. However, these fish migrate through a number of other management areas, where some of their number are actually caught. Hence, Fraser River catches are small, even though escapement is high. Likewise, other areas may have small escapements and very large catches. The fish stock variable is meant to proxy the average amount of fish available to the vessel owner over the season. A suitable measure for the individual vessel should be the relative abundance of salmon, not the absolute abundance. That is, the fisherman chooses a particular area not only on the basis of the relative prices of fish, but also relying upon a subjective estimate of the relative abundance. Lane (1986) has recently done some work in this area. Thus, for each area a measure of relative abundance is created by dividing that area's stock estimate by the total biomass. This is done for each species. Next, the relative stock measures per area are aggregated into a single quantity index. This is accomplished by weighting the 5 relative stock measures per area by the relative prices that prevail in that area. These weighted stocks are then added together. Finally, each vessel's fishing patterns should reflect in the final aggregate stock measure. Thus, for each vessel the appropriate area aggregate stocks are weighted by the relative number of weeks that the vessel fished in a given area. Within each sample there is a fair degree of / 231 variation in this variable as shown by the summary statistics in Tables A1.1 through A1.4. C. IS 1982 A REPRESENTATIVE YEAR? A final question remains to be answered. Given variability in fishing conditions from year to year, the 1982 fishing season must be examined for its suitability to be considered as represenatative of past and future fishing seasons. Three fishery-specific items are considered. These are the fish biomass, the prices per species, and the landed catch, including its distribution among the gear types. The fourth and final item is an economy-wide phenomenon, namely, the interest rate. In general, the 1982 fishing season does not appear too different from previous years, t With regard to the 1982 stocks of salmon it appears that availability of the biomass varies widely across the species. Two of the five stocks are of average size (sockeye and coho); one exhibits an unexpectedly large return (chum) One stock, which has been severely depressed in the previous years, continues in this state (chinook) and one stock has an unexpectedly bad year (pink). tit was subject to a very brief strike in August by members of the United Fisherman and Allied Workers Union (UFAWU). Most observers do not believe that the strike had a detrimental effect upon the fishery due to its short duration. This union, which includes processing workers as members, went on strike to complain about the. minimum landed prices offered to them by the processing companies. Every year bargaining takes place for minimum prices for the various species. In the 1970's actual prices were often the minimum negotiated prices, but this is no longer the case. Actual current day prices are typically much larger than the minimums. / 232 Real prices in 1982 for the salmon species show different trends. In a comparison with 1975, it is found that sockeye and coho prices have not changed.* O n the other hand, pink and chum prices are substantially lower than their 1975 counterparts. Chinook prices are much higher. However, the relative position of species prices has not changed over the last few years. Data on the value of the landed catch per species and on the distribution of the catch of each species are available for three vessel types only. They are the seine, the gillnet, and the troll fleets.* For 1982, the data show that the seiners caught a larger constant dollar value of sockeye and chum than in previous years, but a much smaller value of pink salmon. Landed values for chinook and coho do not differ from their counterparts in earlier years. Overall, the total salmon seine catch is lower than the four previous years, although it is not too different from the four years preceeding the boom years, 1977-1979. Troll-caught chinook and pink are of lower value than in previous years, but troll-caught coho, chum, and sockeye, especially, exhibit larger values. The total landed value in constant dollars for this fleet is more or less the same as the previous years, with the exception of the two boom years, 1978 and 1979. Cillnetters catch much the same landed value in constant dollars of chinook and sockeye, much less of pink and coho, and somewhat more of chum. It has been suggested that the unusual decline of the pink stocks is particularly detrimental to the gillnet fleet. The total landed value of their catch appears to reflect this possibility. tData on prices come from a draft of the 1985 Commercial Salmon Fishery Season  Update. This is written by members of the Regional Planning and Economics Branch of the Department of Fisheries and Oceans, Pacific Region. *This information is from the 1984 Commercial Fishing Guide, Pacific Region and from "Review of the Financial Status of the Salmon Fishery" by Heather Fletcher of the Department of Fisheries and Oceans, October, 1986. / 233 The catch distribution by species reflects observations that experts have made about the fishery, ie., trailers and seiners have been enjoying increasing shares of most species at the expense of the gillnetters. However, it may also be true that gillnet-troll combination vessels have been claiming a larger portion of the entire troll catch over time. The catch statistics may hide this fact, since catches are typically recorded as taken by either troll or gillnet gear. These gears may be used by either troll-only equipped vessels, gillnet-only equipped vessels, or by the combination gillnet-troll vessels. In 1982 the Canadian economy experienced the highest interest rates for many years. The British Columbia commercial salmon fishery has not been exempted from the detrimental impacts of high borrowing costs. Some people have argued that the few years of high interest rates preceeding 1982 have added large amounts of debt-servicing payments to the fisherman's costs. Given these high interest rates it is possible that the rental prices calculated, both for gear and net tonnage, may be larger than is normal for the fishery. This means that rents in this year might be lower than otherwise. However, they do reflect the real conditions under which the fishery must operate. Furthermore, the estimation results are robust in the face of large changes in the depreciation rates. Therefore, whereas relatively small variations in interest rates do not pose serious problems for the econometric analysis, higher-than-normal interest rates might explain the preponderance of negative rents. / 234 Table A1.1:--Vessel characteristics and expenditures: seine Characteristic Mean St. Dev Minimum Maximum Crew Size 5.1 0.44 4.0 6.0 GRT 35.1 22.8 14.0 123.0 NRT 23.9 15.4 9.9 83.7 Length 56.5 8.6 44.8 86.0 HP 214.5 81.9 n/a 365.0 Age 25.8 18.3 4.0 56.0 Days Fished 32.0 15.8 12.0 79.0 Stock Index 1.11 0.27 1.00 1.83 Expenditure Labour 27524.0 6581.5 15199.0 38214.0 Fuel 8480.7 4819.9 2800.0 22500.0 Gear 20768.0 15183.0 6509.7 61059.0 Revenue 93579.0 43424.0 31231.0 173620.0 Notes: Sources: The Survey Data and the Sales Slip Data. The number of observations is 21. Crewsize includes the skipper. GRT and NRT are gross and net registered tonnage. They are measured in imperial units. Length is measured in feet. HP is the horsepower of the vessel. Age represents the vessel's age in 1982. Stock index represents the index of stock abundance and is indexed with reference to the first observation set at 1.00. Revenue is the sum of the value of landed catch and bonuses. Gear expenses are calculated using low depreciation rates. Labour expenses are calculated as the opportunity cost wage bill. Expenditures and revenue are in constant 1982 dollars. / 235 Table A1.2:-Vessel characteristics and expenditures: gillnet Characteristic Mean St. Dev Minimum Maximum Crew Size 1.2 0.43 1.0 2.0 CRT 6.8 3.1 1.0 23.0 NRT 6.1 1.9 0.7 12.4 Length 33.0 3.8 15.0 38.0 HP 178.7 86.1 n/a 440.0 Age 14.1 8.3 1.0 32.0 Days Fished 22.0 12.5 6.0 91.0 Stock Index 1.28 0.50 0.33 2.39 Expenditure Labour 7520.3 2710.8 4084.4 17133.0 Fuel 1647.0 1146.8 200.0 5750.0 Gear 5978.5 4356.7 601.4 19613.0 Revenue 17982.0 10220.0 5027.4 67199.0 Notes: Sources: The Survey Data and the Sales Slip Data. The number of observations is 80. Crewsize includes the skipper. GRT and NRT are gross and net registered tonnage. They are measured in imperial units. Length is measured in feet. HP is the horsepower of the vessel. Age represents the vessel's age in 1982. Stock index represents the index of stock abundance and is indexed with reference to the first observation set at 1.00. Revenue is the sum of the value of landed catch and bonuses. Gear expenses are calculated using low depreciation rates. Labour expenses are calculated as the opportunity cost wage bill. Expenditures and revenue are in constant 1982 dollars. / 236 Table A1.3:--Vessel characteristics and expenditures: troll Characteristic Mean St. Dev Minimum Maximum Crew Size 1.9 0.60 1.0 3.0 GRT 11.2 6.5 3.0 60.0 NRT 9.1 2.4 2.8 17.7 Length 39.4 4.5 27.0 58.0 HP 130.6 85.8 n/a 471.0 Age 21.6 13.2 1.0 54.0 Days Fished 97.1 30.3 30.0 165.0 Stock Index 0.45 0.27 0.07 1.00 Expenditure Labour 9773.6 3816.8 3352.1 21394.0 Fuel 3542.3 1975.6 950.0 1000.0.0 Gear 4624.7 5594.3 5.2 32409.0 Revenue 42659.0 23254.0 10154.0 108630.0 Notes: Sources: The Survey Data and the Sales Slip Data. The number of observations is 84. Crewsize includes the skipper. GRT and NRT are gross and net registered tonnage. They are measured in imperial units. Length is measured in feet. HP is the horsepower of the vessel. Age represents the vessel's age in 1982. Stock index represents the index of stock abundance and is indexed with reference to the first observation set at 1.00. Revenue is the sum of the value of landed catch and bonuses. Gear expenses are calculated using low depreciation rates. Labour expenses are calculated as the opportunity cost wage bill. Expenditures and revenue are in constant 1982 dollars. / 237 Table A1.4:--Vessel characteristics and expenditures: gil Inet-troll Characteristic Mean St. Dev Minimum Maximum Crew Size 1.6 0.58 1.0 3.0 GRT 7.6 1.8 5.0 14.0 NRT 7.0 1.4 3.9 10.6 Length 35.7 2.2 32.0 42.0 HP 171.3 97.1 n/a 671.0 Age 15.6 11.1 1.0 50.0 Days Fished 57.5 36.0 13.0 150.0 Stock index 1.11 0.51 0.37 2.66 Expenditure Labour 11447.0 4880.2 5428.2 30903.0 Fuel 3043.9 1981.8 800.0 12000.0 Gear 6803.6 5468.2 36.8 22438.0 Revenue 32015.0 22413.0 10910.0 135900.0 Notes: Sources: The Survey Data and the Sales Slip Data. The number of observations is 60. Crewsize includes the skipper. GRT and NRT are gross and net registered tonnage. They are measured in imperial units. Length is measured in metric feet. HP is the horsepower of the vessel. Age represents the vessel's age in 1982. Stock index represents the index of stock abundance and is indexed with reference to the first observation set at 1.00. Revenue is the sum of the value of landed catch and bonuses. Gear expenses are calculated using low depreciation rates. Labour expenses are calculated as the opportunity cost wage bill. Expenditures and revenue are in constant 1982 dollars. / 238 Table Al.5:--Average weekly earnings, British Columbia, current year dollars Year Fishermen Manufacturing  Employee Forestry  Operator 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 168.46 278.51 223.33 178.41 240.03 265.36 417.82 399.54 336.27 289.46 304.60 369.03 158.22 163.79 176.63 197.23 171.28 168.03 267.30 194.16 343.19 320.61 300.82 503.69 188.47 219.74 227.56 234.79 262.03 268.30 362.54 319.45 423.87 396.29 404.40 408.67 Notes Sources: Revenue Canada, Department of National Revenue Canada, Taxation Division, Taxation Statistics, 1974-1985 (Table 8, 8A or 9, depending upon the edition). Average annual wages are obtained by dividing total reported taxable income by the number of returns; weekly wages are obtained by dividing the average annual wages by 52 weeks. / 239 Table A l . 6:--Average weekly earnings and unemployment rate by region Region Average Weekly Earnings Unemployment Rate (Current 1982 $) (%) North Coast Central Coast East Georgia Strait West Georgia Strait 400.726 (B.C. average wage) 400.726 (B.C. average wage) 286.207 (Courtney wage rate) 359.816 (Nanaimo wage rate) 393.533 (Metro Vancouver wage) 11.758 11.758 10.358 13.800 13.800 9.825 11.117 13.800 Lower Mainland Capital Region West Island 393.533 (Metro Vancouver wage) 345.807 (Victoria wage rate) 430.965 (Port Alberni wage rate) Notes: The average weekly unemployment insurance benefit is calculated to be $178.97. Sources: Statistics Canada 72-002 Employment, Earnings and Hours, 1983-1985, Table 16. CANSIM, British Columbia Basic Labour Force Characteristics by Economic Region and  Metropolitan Area, (D771624 to D771633). Statistics Canada 73-001 Average Weekly Payments by Province, Month and Type, 1982, (Table 12). Table A1.7:--Fuel prices by region / 240 Region North Coast -Prince Rupert Central Coast -Namu -Klemtu Gasoline Price ($/Litre)  (1982$ per litre) 0.428 0.450 0.470 Diesel Price ($/Litre)  (1982$ per litre) 0.386 0.411 0.436 East Georgia Strait -Madeira Park 0.437 0.397 West Georgia Strait -Port Hardy 0.430 -Campbell River 0.437 Lower Mainland -Vancouver 0.410 -Steveston 0.410 0.391 0.397 0.371 0.371 Capital Region -Nanaimo 0.430 0.391 West Island -Bamfield -Ucleulet -Torino -Kyuquot 0.440 0.440 0.440 0.466 0.411 0.411 0.401 0.426 Notes: Source: Esso Canada, Ltd. These prices are for the period October 1, 1982, prices are $0.015/litre less than $0.029/litre fishing exemption. Neither the $0.011/litre family fishing exemption. through December, 1982. Prior to October those reported. Diesel prices include the gasoline nor the diesel prices include the / 241 Table A l ^--Representative bonus rates(%) by species and gear type Gear Type Species Chinook Sockeye C o h o Pink Chum Seine 11.4 13.5 10.8 13.4 13.0 Gillnet 8.2 12.2 9.4 11.7 10.8 Troll 3.2 3.2 3.2 3.6 3.4 Notes: Source:The Department of Fisheries and Oceans, Pacific Region. These rates are representative of the 1983 rates paid to fishermen for their landed catch. Personnel at the Department of Fisheries and Oceans believe that the average 1982 rates are not very different from these. APPENDIX 2 : PARAMETER ESTIMATES - NONLINEAR CASE In this appendix the parameter estimates obtained using a nonlinear estimation technique are presented for five samples. They are the seine sample without constant returns to scale imposed, the gillnet sample, both with and without constant returns to scale, and the gillnet-troll sample, both with and without constant returns to scale. The single output supply and three input demand functions estimated are described in equations (5.8) through (5.11) in chapter 5. The technique is nonlinear maximum likelihood and details may be found in chapter 5. Testing of the results is also given in chapter 5, along with a discussion of the elasticities obtained using the parameter values. The equations for these elasticities, in the case of the normalized, quadratic, restricted profit function, are given in Appendix 4. 242 Table A2.1:—Nonlinear parameter estimates: seine / 243 Variable Coefficient Standard Name Value Error ex -0.13898T 0.06017 e 2 -0.27236+ 0.15059 e 3 -0.10149E-05 0.30752 e 4 0.32342t 0.10552 e 5 0.66494E-06 0.23151 e 6 0.60085E-07 0.12445 b 2 2 3.4979 2.6956 b 2 3 -3.7283+ 2.2831 b 3 3 -1.4241+ 0.70353 b 0 -3.5805 3.0012 b 2 -0.41799 3.0397 b 3 5.4111t 1.1033 c 1 2 -0.01292 0.70759 c 1 2 1.6776 1.5384 LOG-LIKELIHOOD Variable Coefficient Standard Name Value Error c 1 3 -1.6698T 0.72191 c1 1.2204 1.9528 c 2 1 0.19913 0.66133 c 2 2 1.4732 1.3651 c 2 3 -2.4131 + 0.61377 c 2 -2.4782 1.9881 c 3 1 0.18851 1.1531 c 3 2 * -1.4136 2.0234 c 3 3 -2.9992t 0.94912 c 3 -1.0295 2.6248 c 4 1 13.7110+ 7.7443 c 4 2 19.7710+ 10.8770 c 4 3 -9.4196+ 5.9794 c 4 -31.1710+ 13.7130 FUNCTION = -95.612 Notes: The symbol t indicates that the estimated coefficient value is significantly different from zero at a = 0.10. Table A2.2:--Nonlinear parameter estimates: gillnet(crs) / 244 Variable Coefficient Standard Name Value Error ex -0.30848t 0.07642 e 2 0.02347t 0.00768 e 3 0.24644E-06 0.01804 b 2 2 0.34610t 0.25899 b 2 3 -0.05924t 0.05655 b 2 4 -0.25907 0.29721 b 3 3 -0.03166t 0.01712 b 3 4 0.00855 0.08770 b 4 4 0.28517 0.28500 c x l 0.10269 0.12499 c 1 2 0.19981 0.23837 LOG-LIKELIHOOD Variable Coefficient Standard Name Value Error c 1 3 0.24806t 0.05250 c 1 4 0.27135 0.31932 c 2 1 0.05437 0.07420 c 2 2 -0.09300 0.14759 c 2 3 -0.17623t 0.03784 c 2 4 -0.41892t 0.20449 c 3 1 0.01004 0.05840 c 3 2 0.01958 0.12812 c 3 3 -0.04878t 0.02873 c 3 4 -0.36074t 0.17667 = -50.3367 Notes: The symbol t indicates that the estimated coefficient value is significantly different from zero at a = 0.10. / 245 Table A2.3:--Nonlinear parameter estimates: gillnet(non-crs) Variable Name e 3 b 2 2 b 2 3 b 2 4 b 3 3 b 3 4 b , « b 0 b 2 b 3 b 4 c, , Coefficient Value -0.25528t 0.02188+ 0.60436E-07 -0.02156 0.04651 -1.0575T 0.08117t 0.14710t 0.90405 0.89043 0.47506 -0.31174t .0.45444 -0.06858 Standard Error 0.07556 0.00704 0.01799 0.40886 0.07844 0.63005 0.02678 0.10497 1.0564 0.97200 0.59378 0.15388 0.94111 0.14927 Variable Coefficient Standard Name Value Error 0.17764+ 0.20204+ 0.18694 0.47888 -0.01576 0.04361 -0.18205T -0.22369 -0.04301 -0.06439 0.17657 -0.6123+ -0.13477 -0.11300 0.26625 0.05691 0.41075 0.47115 0.08914 0.16310 0.03615 0.26293 0.29480 0.07789 0.15109 0.03033 0.24993 0.26552 LOG-LIKELIHOOD FUNCTION =• -46.319 Notes: The symbol t indicates that the estimated coefficient value is significantly different from zero at a = 0.10. 3 2 2 — e i / ' a 2 3 — e i e 2 ; a , s = e l + e | ; / 246 Table A2.4:--Nonlinear parameter estimates: gillnet-troll(crs) Variable Name Coefficient Value 0.16337t -0.05490 0.36502t -0.39102E-02 -0.01254 -0.30223E-08 -0.26218 0.11032 -0.35911E-02 0.21649 2.4660t Standard Error 0.10405 0.24739 0.08334 0.03313 0.02534 0.04324 0.38829 0.16203 0.05527 0.53457 0.72924 Variable Name Coefficient Value -0.40717t -0.95225E-03 -0.95825t -0.30314T -0.10147 -0.54612t -0.41800t 38.6840t -93.9910t 2.5369 Standard Error 0.0.2300 0.19577 0.35617 0.10273 0.20390 0.36021 0.11250 24.349 30.672 10.4550 LOG-LIKELIHOOD FUNCTION = -536.7018 Notes: The symbol t indicates that the estimated coefficient value is significantly different from zero at a = 0.10! a 2 4 = e 1 * e 4 ; a , , - e l + e § a 3 4 = ( e 2 * e 4 + e 3 * e s ) ; a 4 4 = e f . + e § + e 2 . / 247 Table A2.5:--Nonlinear parameter estimates: gillnet-troll(non-crs) Variable Name b. b 3 b, Coefficient Value 0.12799 -0.12676 -0.29299+ 0.59117E-02 0.57212E-02 0.14491 E-08 -2.6812+ 0.45804t 0.07340t -3.4896T 3.0380t -0.56747T 0.27289 2.6970t Standard Error 0.11095 0.24470 0.12682 0.02761 0.02876 0.03954 1.5572 0.20406 0.05269 1.9096 1.6183 0.21560 0.55843 1.2666 Variable Name Coefficient Value 0.45318+ -0.37025 -0.03259 -1.5662+ -0.26009T 0.79165 -0.15556 -1.479+ -0.36415T 1.1607T 40.5510t -80.1440+ 3.2792+ -18.4010 Standard Error 0.23642 1.4360 0.18662 0.56867 0.09589 0.66303 0.19664 0.59394 0.10388 0.68181 22.1390 61.7270 10.9660 68.5170 LOG-LIKELIHOOD FUNCTION = -526.621 Notes: The symbol t indicates that the estimated coefficient value is significantly different from zero at a = 0.10. APPENDIX 3 : PARAMETER ESTIMATES, TESTS, AND RESULTS - LINEAR CASE This appendix presents the linear parameter estimates, tests, and elasticity results for the seine, gillnet, troll, and gillnet-troll samples. The estimation technique is described in detail in Chapter 5 and the four equations to be estimated are given in equations (5.3) and (5.4). Whereas the discussion in Chapter 5 concentrates upon the nonlinear results (with the exception of the troll sample, since it does not require estimation with a nonlinear set of equations), this appendix discusses the linear results in some detail. A brief comparison is made between results of the two cases, where applicable. The definitions for the elasticitiy measures are given in Chapter 4 and the forms specific to the normalized, quadratic, restricted profit function may be found in Appendix 4. The parameter estimates obtained using the linear estimation technique described in Chapter 5 are given for all fleets in the set of tables numbered from A3.1 through A3.5. Each of the seine, troll, and gillnet-troll fleets have one set of parameter estimates, whereas the gillnet fleet has two, both with and without constant returns to scale imposed. A test for the hypothesis of returns to scale is made and only the gillnet fleet cannot reject the possibility of constant returns to scale in the fixed factors. Table A3.7 presents the relevant x2 values. The parameter estimates for the troll fleet are included with those of the other fleets for completeness, although these results and the elasticities derived from them are discussed in Chapter 5. It is interesting to compare the linear parameter estimates and equation-specific R 2 values with their nonlinear counterparts. A complete set of information, including 248 / 249 equation-specific R 2 values, for the linear case is found in this appendix in tables A3.1 through A3.5. Parameter estimates for the nonlinear case are given in appendix 2, tables A2.1 through A2.5. However, the nonlinear estimation uses a maximum likelihood technique, therefore it is necessary to calcuate a goodness-of-fit measure for each of the output supply and input demands that is similar to the linear R 2 . This is done by calculating the squared correlation between actual and predicted values. Estimates of goodness-of-fit for the nonlinear case are found in Chapter 5 in Table 5.5. In general, more than half of the linear parameters are significantly different from zero at a 90% confidence level. A greater number of parameters are significant than in the corresponding nonlinear cases. Furthermore, the R 2 values for each linear equation are somewhat larger than those calculated for the nonlinear equations, in particular, the output and labour equations for the seine sample have much higher R 2 values. It is likely that this reflects the fact that the seine sample is the furthest from accepting convexity. A comparison of the eigenvalues in Table 5.1 of Chapter 5 bears out this assertion. A test is made to examine the hypothesis of cross-price symmetry in the linear estimates and results are given in Table A3.6. All samples reject symmetry at the a = 0.010% level of significance. However, at a = 0.005% symmetry is accepted in the seine and gillnet-troll cases. This contrasts strongly with the nonlinear results in chapter 5. There is it shown that both gillnet samples, that with constant returns to scale and that without, accept symmetry for the two levels of significance specified. The other samples reject the hypothesis. A regularity condition called monotonjcity must be checked, although verification of / 250 its acceptance cannot be made with a statistical test. Monotonicity requires that predicted output for each observation is positive and that predicted input is negative. The troll sample accepts this condition. This is not an unexpected result, since the troll fleet is the only one to accept convexity. The other samples do not perform as well, but since they reject convexity, this is not surprising. The gillnet sample, for both the constant and non-constant returns to scale cases, predicts output supply as positive and fuel demand as negative for all observations. However, it predicts the gear input as positive for a single observation out of a total of 80. Similar results obtain in the gillnet-troll sample, where output supply and the labour and fuel input demands are all the correct signs. Once again, the gear input is predicted as positive for 3 out of 60 observations. The nonlinear results are found to be identical to these. The situation is different for the seine fleet; once again, this reflects the fact that this fleet is least likely to accept convexity. The output supply, labour and gear demands are all predicted to be of the correct signs for all observations. However, fuel demand is predicted as positive for all observations. Several explanations exist for this phenomenon. First, the seine sample has the least number of observations, ie., 21. Second, most vessels register Vancouver as their homeport. Vancouver is some distance from most fishing grounds and therefore more fuel is required by these vessels than others with homeports closer to the grounds. O n the other hand, fuel prices in Vancouver are generally lower than elsewhere. Combining these two factors it is possible to understand why a positive fuel demand is predicted. This has ramifications for the elasticities that may be generated using the parameter / 251 estimates. In particular, the elasticity estimate for the own-price of output is adversely affected. These results are discussed next for each sample and compared with their nonlinear counterparts, beginning with the seine sample. In Chapter 5 Table 5.6 presents estimates of the own- and cross-price elasticities for the nonlinear seine case. Table A3.8 in this appendix has the comparable linear estimates. Of the own-price elasticities only that for gear demand does not change either in sign or magnitude. However, the other own price elasticities differ substantially in the linear case from the nonlinear case, both with respect to magnitude and sign. In particular, the labour and fuel elasticities are large, positive and significantly different from zero in the linear case. O n the other hand, the own elasticity of output supply is very large, negative, and significant. These results run counter to the accepted theory; as such, they are disturbing. Estimates of a negative output supply are not new to studies of fisheries. Squires studies two New England fisheries, the multi-output otter trawl fishery (1984, 1987a) and the single output sea scallop fishery (1985), and obtains this result. In addition, he finds the own-price elasticity of the demand for labour to be positive. These results may reflect, in part, the cross-sectional nature of the data which may offer insufficient price or quantity variation to obtain good parameter estimates. A small number of observations merely exacerbates these problems. O n the other hand, conditions in the fishery itself may contribute to the generation of a negative supply elasticity. In the British Columbia commercial salmon fishery, for example, landed prices for the 5 species of salmon vary widely from $0.35 per pound for pinks to $2.50 for chinooks. However, the fish are not available in similar quantities. / 252 In fact, the regulator tries to control the seine fleet's access to the higher-priced chinook and coho stocks, since these stocks are not very large. This may contribute to a negative own-price supply elasticity. A comparison of the cross-price elasticities of demand for the seine fleet in the linear and nonlinear cases reveals them to be of the same sign and order of magnitude. This is very encouraging for the analysis of the degree of substitution possibilities between the variable inputs, for it implies that the results discussed in Chapter 5 are invariant to the imposition of convexity in prices. The elasticities of output and input use have different signs in the nonlinear and linear cases for the labour and fuel variables. This follows from the sign changes obtained for the own-price elasticities of demand for labour and fuel. However, the nonlinear results have the correct, ie., theoretically expected, results. Finally, as regards the elasticities of intensity, the appropriate tables to compare are 5.10 in Chapter 5 and Table A3.12 in this appendix. The values are similar and the signs do not change when a comparison is made between the linear and nonlinear estimates. In particular, significant elasticities hardly change at all in size. Once again, these results are encouraging since they support the analysis discussed in Chapter 5. Thus, the discussion of the relationships between the variable inputs and the fixed factors need not be changed for the case when convexity does not obtain. Moving on to a comparison of the linear and nonlinear results for the other / 253 samples, much less variation is found. For the gillnet, with and without constant returns to scale, and the gillnet-troll, with and without constant returns to scale, the magnitudes and signs of all sets of elasticities show very little variation. The nonlinear parameter estimates are found in Chapter 5, tables 5.7, 5.9, 5.11, 5.13, 5.15, 5.17, 5.18, 5.19, 5.22, and 5.23, whereas the linear estimates are given in this appendix in tables A3.9-A3.11 and A3.13-A3.15. The lack of variation most likely reflects how "close" the linear estimates come to accepting convexity, ie., Table 5.1 in Chapter 5. For these samples the only own price elasticity to change sign in going from the linear to the nonlinear case is that for the gear demand. It is positive for the linear case, but imposing convexity makes it negative. This is the desired result. In general, this elasticity is not significantly different from zero which suggests that the demand for gear for these fleets is not at all responsive to price changes, at least in the short run. Thus, as suggested in Chapter 5, the gear input might be specified more appropriately as a fixed factor in the short run. This is not done for this study, since it is desirable to obtain individual elasticities of intensity between a complete set of variable factors and the more obviously fixed factors, in particular, the net tonnage of the vessel. A discussion of the other elasticities need only be brief. In short, neither the signs nor the magnitudes change much when a comparison of the linear and nonlinear results is made. In general, the nonlinear estimates obtain slightly larger elasticity values, especially for the elasticities of intensity. Only in the gillnet-troll fleet does an elasticity of intensity change sign. It happens twice, once between output and stock and then between fuel and stock. In the linear case they are both negative and only the output-stock elasticity is significantly different from zero. In the / 254 nonlinear case they are positive and significant. Since the parameter estimates obtained for this fleet show the least number of significant values and the R 2 values for each equation are the lowest of all sample, this is not a surprising result. Table A3.1:--Linear VARIABLE ESTIMATE ST. ERR. Output Equation a 2 2 -0.01116t 0.04535 a 2 3 -0.12979t 0.02805 a 2 4 -0.05065t 0.01732 a 3 3 -0.69575t 0.15418 a 3 4 -0.09576t 0.06724 a 4 4 0.05610 0.04975 b 2 2 7.8018t 2.8185 b 2 3 -5.8896t 1.9953 b 3 3 -0.79829t 0.50318 b 0 -3.4013t 2.2167 b 2 -1.7465 2.5932 b 3 4.5025t 0.88513 Cn -0.75789t 0.53745 2.2684t 1.3256 -1.2064t 0.57820 Ci 1.0942 1.3857 R 2 = 0.7961 SEE = 0.2037 LLF = -81.449 Labour Equation a 2 2 -0.11164t 0.04535 a 2 3 0.12979t 0.06390 a 2 4 -0.05065t 0.01732 b 2 2 7.8018t 2.8185 b 2 3 -5.8896t 1.9553 b 3 3 -0.79829t 0.50318 bo -3.4013t 2.2167 b 2 -1.7465 2.5932 b 3 4.5025t 0.88513 c 2 1 -0.29856 0.52118 C 2 2 2.5519t 1.2493 c 2 3 -1.7211t 0.55973 c 2 -1.7144 1.3550 R 2 =0.8416 SEE = 0.08367 / 255 parameter estimates: seine VARIABLE ESTIMATE ST. ERR. Fuel Equation a 2 3 0.12979t 0.06370 a 3 3 -0.69575t 0.15418 a 3 4 -0.095761- 0.06724 b 2 2 7.8018t 2.8185 b 2 3 -5.8896t 1.9553 b 3 3 -0.79829t 0.50318 b 0 -3.4013t 2.2167 b 2 -1.7465 2.5932 b 3 4.5025t 0.88513 c 3 l 0.45981 1.1755 c 3 2 0.29436 2.0388 C33 -1.8811t 0.94328 c 3 0.44078 2.3536 R 2 =0.2091 SEE=1.1617 Gear Equation a24 -0.05065t 0.01732 a34 -0.095761- 0.06724 a 4 * 0.05610 0.04975 b 2 2 7.8018t 2.8185 b 2 3 -5.8896t 1.9553 b 3 3 -0.798291- 0.50318 b. -3.40131- 2.2167 b 2 -1.7465 2.5932 b 3 4.5025t 0.88513 c 4 a 13.166t 8.2836 c 4 2 21.046t 12.027 c 4 3 -8.9233t 6.2620 c 4 -30.280t 14.872 R 2 = 0.2280 SEE = 9.1659 Note: The symbol t denotes significance at the level of a = 0.10 / 256 Table A3.2: --Linear parameter estimates: gillnet(crs) VARIABLE ESTIMATE ST. ERR. VARIABLE ESTIMATE ST. ERR. Output Equation Fuel Equation a 2 2 0.09276T 0.03859 3 2 2 0.09276t 0.03859 a 2 3 -0.8006E-02t 0.0026 a 2 3 -0.8006E-02+ 0.0026 a3 3 -0.9605E-04 0.00063 b 2 2 0.35368t 0.26311 b 2 2 0.35368T 0.26311 b 2 3 -0.06567t 0.04905 b 2 3 -0.06567T 0.04905 b 2 4 -0.25275 0.28983 b 2 4 -0.25275 0.28983 b 3 3 0.03112t 0.01580 b 3 3 0.03112t 0.01580 b 3 4 0.01367 0.07829 b 3 4 0.01367 0.07829 b44 0.27220 0.26513 b 4 4 0.27220 0.26513 C»i 0.05652 0.06901 C n 0.08367 0.12261 C 2 2 -0.08279 0.13368 Cia 0.19165 0.22621 C 2 3 -0.17322T 0.03506 C l 3 0.2528T 0.0539 C 24 -0.4205+ 0.1872 0.26866 0.30659 R 2 =0.3001 SEE = 0.2547 R 2 =0.2258 SEE = 0.5800 Gear Equation . LLF = -49.79 3 2 3 -0.8006E-02+ 0.0026 a3 3 -0.9605E-04 0.0006 b 2 2 0.35368+ 0.26111 b 2 3 -0.06567t 0.04905 b 2 4 -0.25275 0.28983 b 3 3 0.03112t 0.01580 b 3 4 0.01367 0.07829 b 4 4 0.27220 0.26513 c 3 i 0.01043 0.05679 c 3 2 0.02802 0.12144 C 3 3 -0.0466+ 0.0288 C 3 4 -0.3634t 0.1701 R 2=0.1401 SEE = 0.2049 Note: The symbol t denotes signiciance at the level of a = 0.10 / 257 Table A3.3:-Linear parameter estimates: troll VARIABLE ESTIMATE ST. ERR. VARIABLE ESTIMATE ST. ERR. Output Equation Fuel Equation a 2 2 0.13337T 0.02577 a 2 3 0.02285 0.03634 a 2 3 0.02285 0.03634 a 3 3 0.43442T 0.13651 a 2 4 -0.8553E-03 0.0029 a 34 0.2179E-02 0.0034 33 3 0.43443T 0.13651 b 2 2 -0.02060 0.10338 a 3 4 0.21790E-02 0.0034 b 2 3 -0.07768 0.10280 a 44 0.68019E-03 0.00128 b 3 3 -0.07855 0.19672 b 2 2 -0.02060 0.10338 b 0 0.12539E-2 0.28867 b 2 3 -0.07768 0.10280 b 2 0.10897E-02 0.17349 b 3 3 -0.07855 0.19672 b 3 0.15106 0.19392 b 0 0.12539E-02 0.28867 C31 -0.56070T 0.26129 b 2 0.10897E-03 0.17349 C3 2 -0.91479T 0.33269 b 3 0.15106 0.19392 C3 3 -0.73352T 0.29133 C n 2.6153T 0.70872 c 3 -0.64344t 0.45529 Cia 2.2514T 0.72877 R 2 =0.3814 SEE = 0.4416 Cl3 3.1010+ 0.75726 Gear Equation Ci -1.4410T 0.97165 a 2 4 -0.85529E-03 0.00294 R 2 =0.2477 SEE = 1.655 a 3 4 0.217909E-02 0.00340 LLF = -447.37 a 44 0.68019E-03 0.00128 Labour Equation b 2 2 -0.2060 0.10338 a 2 2 0.1337+ 0.02577 b 2 3 -0.07768 0.10280 a 2 3 0.02285 0.03634 b 3 3 -0.07855 0.19672 a 24 -0.85529E-03 0.00294 b 0 0.12539E-02 0.28867 b 2 2 -0.02060 0.10338 b 2 0.10897E-3 0.17349 b 2 3 -0.07768 0.10280 b 3 0.15106 0.19392 b 3 3 -0.07855 0.19672 C41 -0.15098 1.1481 b 0 0.12539E-02 0.28867 1.0612 1.1552 b 2 0.10897E-03 0.17349 C4 3 -0.32061 1.2458 b 3 0.15106 0.19392 C4 -1.6436 1.5730 C 2 i -0.64257+ 0.22326 R 2 = 0.0084 SEE = 2.7845 C 2 2 0.40369 0.32139 C 23 -0.17596 0.25897 c 2 -1.6061 + 0.39391 R 2 =0.3961 SEE = 0.3850 Note: The symbol t denotes significance at the level of a = 0.10 / 258 Table A3.4: --Linear parameter estimates: gillnet-troll(non-crs) VARIABLE ESTIMATE ST. ERR. VARIABLE ESTIMATE ST. ERR. Output Equation Fuel Equation a 2 2 0.01165 0.03084 a 2 3 -0.02086 0.0281 a 2 3 -0.02086 0.02805 3 3 3 0.09146t 0.0434 a 2 4 0.62659E-02 0.0067 a 3 * 0.4131E-02 0.0074 a 3 3 0.09146t 0.04342 b 2 2 -4.1940t 1.363 a 3 4 -0.41309E-02 0.0074 b 2 3 0.058971t 0.1796 a * 4 -0.05144t 0.01008 b 3 3 .04515 0.0473 b 2 2 -4.1940t 1.3625 b 0 -3.4504t 1.6921 b 2 3 0.58971 0.17958 b 2 : 3.767l t 1.3957 b 3 3 0.04515 0.04731 b 3 -0.61733t 0.1834 b 0 -3.4504t 1.6921 c 3 1 -0.07268 0.17938 b 2 3.7671t 1.3957 C 3 2 -0.85500T 0.5363 b 3 -0.61733T 0.18335 -0.36538t 0.09798 C n -0.05562 0.47116 c 3 0.45883 0.6442 Cx 2 4.6402t 1.2292 R 2 =0.3709 SEE = 0.4013 C l 3 0.11122 0.22481 Gear Equation Ci -1.8207t 1.3895 a 2 4 0.62659E-02 0.0067 R 2 =0.4032 SEE = 1.697 a 3 4 -0.4131E-02 0.0074 LLF = -516.18 a 4 4 0.5144t 0.0101 Labour Equation b 2 2 -0.2060 0.1034 a 2 2 0.01165 0.03084 b 2 3 -4.T940I 1.3625 a 2 3 -0.02086 0.02805 b 3 3 0.04515 0.0473 a 2 4 0.62659E-02 0.00665 b 0 -3.4504t 1.6921 b 2 2 -4.1940t 1.3625 b 2 3.7671t 1.3957 b 2 3 0.58971t 0.17958 b 3 -0.61733t 0.1834 b 3 3 -0.04515 0.04731 40.7390t 25.770 b 0 -3.4504t 1.6921 c 4 2 -79.895 64.185 b 2 3.767Tt 1.3957 c 4 3 3.3870 11.508 b 3 -0.61733t 0.18335 c 4 -18.964 73.354 C j i 0.02468 0.16993 R 2 = 0.0563 SEE = 99.358 c 2 2 -0.9000t 0.51974 c 2 3 -0.28604t 0.09015 c 2 0.06548 0.62701 R 2 =0.3169 SEE = 0.3512 Note: The symbol t denotes significance at the level of a = 0.10 / 259 Table A3.5 :--Linear parameter estimates: gillnet(non-crs) VARIABLE ESTIMATE ST. ERR. VARIABLE ESTIMATE ST. ERR. Output Equation Fuel Equation a 2 2 0.06035T 0.03346 a 2 2 0.06035+ 0.03346 a 2 3 -0.609E-02t 0.0021 a 2 3 -0.609E-02.T 0.0021 a 3 3 -0.45965E-05 0.00052 b 2 2 0.02273 0.35950 b 2 2 0.02273 0.35950 b 2 3 0.03954 0.07586 b 2 3 0.03954 0.07586 b 2 4 -1.0393t 0.54753 b 2 4 -1.0393+ 0.54753 b 3 3 0.08066+ 0.02595 b 3 3 0.08066t 0.02595 b 3 4 0.14703t 0.09536 b 3 4 0.14703t 0.09536 b 4 4 0.76686 1.0226 b 4 * 0.76686 1.0226 b„ 0.78726 0.92508 b 0 0.78726 0.92508 b 2 0.44082 0.48656 b 2 0.44082 0.48656 b 3 -0.30656+ 0.14471 b 3 -0.30656+ 0.14471 b 4 -0.33665 0.94009 b 4 -0.33665 0.94009 C 21 -0.02055 0.08469 -0.09031 0.14476 C 2 2 0.04452 0.15895 c 2 2 0.17278 0.26524 C 23 -0.17876t 0.03563 Cl3 0.2063+ 0.0585 C 24 -0.23590 0.25311 Cl'4 0.1871 0.3973 c 2 -0.229E-03 0.0008 c x 0.48198 0.46245 R 2 =0.3178 SEE = 0.2514 R 2 =0.2495 SEE = 0.5710 Gear Equation LLF = -45.86 3 2 3 -0.609E-02t 0.0021 a 3 3 -0.4596E-05 0.0005 b 2 2 0.02273 0.35950 b 2 3 0.03954 0.07586 b 2 4 -1.0393+ 0.54753 b 3 3 0.08066t 0.02595 b 3 4 Q. 14703+ 0.09536 b 4 4 0.76686 1.0226 b 0 0.78726 0.92508 b 2 0.44082 0.48656 b 3 -0.30656t 0.14471 c 3 i -0.07211 0.07675 c 3 2 0.17700 0.14566 c 3 3 -0.06111t 0.03134 c 3 4 -0.14744 0.23637 C3 -0.07943 0.25719 R 2 =0.1872 SEE = 0.1992 Note: The symbol t denotes significance at the level of a = 0.10 / 260 Table A3.6:--Testing for symmetry: all samples (linear estimates) Sample LLF(R) LLF(U) -2LOC(u) x 2 Value Decision (a = 0.010) Seine -81.449 -75.230 12.438 11.345 Reject Gillnet -non-crs -45.858 -33.155 25.406 6.635 Reject -CIS -48.037 -35.948 24.178 6.635 Reject Troll -446.365 -436.577 19.576 11.345 Reject Gillnet-Troll -516.184 -510.262 11.844 11.345 Reject Note: The null hypothesis of symmetry in cross price terms cannot be rejected if the calculated value of -2LOG(M) is less than the critical value. The number of degrees of freedom used to determine the critical value of x 2 is given by the number of restrictions. For the gillnet case this number is 1, for all other cases this number is 3. It should also be noted that at a = 0.005, the critical value is x 2 = 12.838. In this case symmetry is accepted in the seine and gillnet-troll cases. Table A3.7:--Testing for constant returns to scale: all samples (linear estimates) Sample LLF(R) LLF(U) -2LQG(M) X 2 Value Decision (o=0.010) Seine -106.085 -81.449 49.272 18.475 Reject Gillnet -49.790 -45.858 7.8646 18.475 Accept Troll -460.418 -447.365 26.106 18.475 Reject Gillnet-Troll -528.890 -516.184 25.412 18.475 Reject Note: The null hypothesis of constant returns to scale cannot be rejected if the calculated value of -2LOG(M) is less than the critical value. The number of degrees of freedom used to determine the critical value of X 2 is given by the number of restrictions. For each sample this number is 7. / 261 Table A3.E 1:--Linear estimates of output-variable own- and cross-price elasticities: Quantity/ Output Labour Fuel Gear Price Output -2.546+ 0.0005 2.351 + 0.195+ (0.457) (0.171) (0.474) (0.076) Labour -0.0002 0.143+ -0.170+ 0.028+ (0.060) (0.058) (0.084) (0.009) Fuel -0.913+ -0.190+ 1.044+ 0.059+ (0.184) (0.093) (0.231) (0.042) Gear -0.035t 0.014+ 0.027t -0.007+ (0.014) (0.005) (0.019) (0.006) Note: Asymptotic standard errors are in parentheses and the symbol "+" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Table A3.9:—Linear estimates of output-variable own- and cross-price elasticities: gillnet(crs) Quantity/ Output Fuel Gear Price Output 0.077+ -0.118+ 0.041 + (0.058) (0.060) (0.014) Fuel 0.491 + -0.655+ 0.163+ (0.251) (0.272) (0.052) Gear -0.079+ 0.076+ 0.003 (0.028) (0.024) (0.018) Note: Asymptotic standard errors are in parentheses and the symbol "+" signifies that the estimated elasticity is significantly different from zero at a = 0.10. / 262 Table A3.10:--Linear estimates of output-variable own- and cross-price elasticities: gillnet-troll(non-crs) Quantity/ Price Output Labour Fuel Gear Output 0.096 (0.111) 0.012 (0.046) -0.134T (0.084) 0.026t (0.011) Labour -0.034 (0.129) -0.031 (0.083) 0.077 (0.104) -0.012 (0.013) Fuel 0.494t (0.309) 0.101 (0.136) -0.610t (0.290) 0.015 (0.026) Gear -0.028t (0.001) -0.0005 (0.0005) 0.0004 (0.0008) 0.003t (0.0006) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Table A3.11:--Linear estimates of output-variable own- and cross-price gillnet(non-crs) Quantity/ Output Fuel Gear Price Output 0.051 -0.086t 0.036t (0.060) (0.062) (0.014) Fuel 0.360t -0.509t 0.149t (0.260) (0.282) (0.052) Gear -0.070T 0.070t 0.0001 (0.027) (0.024) (0.017) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. / 263 Table A3.12:-Linear estimates of elasticities of intensity: seine Quantity/ Stock Net Registered Fishing Fixed Factor of Fish Tonnage Days Output -0.536t 0.021 0.548t (0.320) (0.102) (0.146) Labour 0.130t -0.004 0.033 (0.053) (0.023) (0.037) Fuel 0.069 0.351t 0.272 (0.642) (0.190) (0.243) Gear -1.517t -0.427t 0.423 (0.958) (0.278) (0.354) Note: Asymptotic standard errors are in parentheses and the symbol "f" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Table A3.13:-Linear estimates of elasticities of intensity: gillnet(crs) Quantity/ Stock Net Fishing Labour Fixed of Registered Factor Fish Tonnage Days Output 0.036 0.101 0.545+ 0.196 (0.140) (0.171) (0.104) (0.225) Fuel -0.259t 0.134 0.588+ 0.732+ (0.157) (0.202) (0.128) (0.263) Gear 0.217 0.005 0.237+ 0.962+ (0.236) (0.230) (0.142) (0.293) Note: Asymptotic standard errors are in parentheses and the symbol "+" signifies that the estimated elasticity is significantly different from zero at a=0.10. / 264 Table A3.14: -Linear estimates of elasticities of intensity: gillnet-trollfnon-Quantity/ Stock Net Registered Fishing Fixed Factor Fish Tonnage Days Output -0.028T 1.622T 0.086 (0.154) (0.369) (0.116) Labour -0.015t 0.379+ 0.274+ (0.075) (0.173) (0.062) Fuel -0.017 0.507+ 0.469+ (0.152) (0.353) (0.123) Gear -0.875+ 1.615 -0.119 (0.554) (1.303) (0.400) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. Table A3.15:—Linear estimates of elasticities of intensity: gillnet(non-crs) Quantity/ Stock Net Fishing Labour Fixed of Registered Factor Fish Tonnage Days Output -0.074 -0.036 0.448T -0.077 (0.160) (0.197) (0.114) (0.291) Fuel -0.182 0.242 0.718t 0.935T (0.186) (0.235) (0.149) (0.368) Gear 0.626 0.046 0.356T 1.104T (0.725) (0.261) (0.162) (0.428) Note: Asymptotic standard errors are in parentheses and the symbol "t" signifies that the estimated elasticity is significantly different from zero at a = 0.10. APPENDIX 4 ELASTICITY AND SHADOW VALUE FORMULAE This appendix gives the elasticity formulae for the own- and cross-price elasticities of supply and demand and for the elasticities of intensity. As well, the formula used to generate an estimate of the shadow value of net tonnage is given. For completeness, the shadow value formulae for the two other restricted factors are included. These formulae are calculated from the normalized, quadratic, restricted profit function defined in chapter 5 using equations (5.1) and (5.2). The formulae are given for the case of four variable quantities, with output price as the normalizing price, and three fixed quantities, with the stock of fish as the normalizing factor. They are presented for the linear case. To obtain the elasticities using the nonlinear parameter estimates one must use the relationships between the a . , and the e , , etc., eiven in the tables found in i k Appendix 2. These elasticities are used to obtain the values for the seine, troll, and gillnet-troll fleets as given in chapter 5 and appendix 3. For the gillnet fleet slight modifications to the formulae are needed, since its parameters are estimated using a system of three variable quantities and four fixed quantities as discussed in chapter 5. S Own-Price Elasticity of Supply: e (A4.1) = (3 2ir/3P 2) -(P^X,) = OX,/BP,) .(P,/X,) = [2* i = 2 k=2 l k • ( P . P k / p 3 ) ] .(P,/X,) 265 / 266 Output-Variable Own-Price Elasticities Of Input Demand: e . . (A4.2) 0 2 7 r / 3 2 P . ) . (P./X.) = (3X./3P.•(P./X.) = ( a i i - ( z j = i a j Z j / P ^ J ^ P ^ X . ) f o r i = 2,3,4 c Output Elasticity With Respect to Input Price: e . (A4.3) = ( 3 27r/3P 13P. ) = ( 3 X , / 3 P . ) • ( P . / X , ) - - [ ( ( I . a . Z . ) / P 2 ) . Z U a . k P k M P . / X , ) f o r i , k = 2,3,4 and i * k Input Elasticity With Respect to Output Price: e ^ (A4.4) S e., = (3 2TT/3P. 3P, ) = ( 3 X . / 3 P , ) • ( P , / X . ) =-[((!. a.Z.)/ P 2 ) - I < = 2 a i k P k ].(P,/X.) f o r i , k = 2,3,4 and i * k / 267 Elasticities of Intensity Between Fixed and Variable Factors (A4.5) 8 2TT/3P 13Z 1 = ( 3 X 1 / 3 Z 1 ) • (Z,/X, ) = [ ( - ( a l / 2 P ? ) . ( Z 1 = 2 L U a . k P.P K) - ( ^ / 2 Z ? ) . ( Z ^ E i = 2 b j l Z j Z l ) +c -(/3 n/Z i ) . ( Z 3 b.Z. + l / 2 b o M ] -(Z./X,) (A4.6) 3 2 j r / 9 P . 3 Z i = (3X./3Z, ) • (Z,/X. ) 1 1 l 1 1 l + c . = 1 - ( / 3 . / Z 2 ) . ( 2 » = 2 b j Z j + 1/2 b 0 / 3 . ) ] .(Z,/X.) f o r i = 2,3,4 (A4.7) a 27r/9P, = O x r / 3 Z j ) • ( Z j / X , ) - [ ( - ( a i / 2 P 2 ) . ( E i - 2 Z « - 2 a . k P.P k) + ( / 3 , / Z 1 ) . ( b j j Z j + b j l Z 1 + b . ) + c . j ] . ( Z j / X 1 ) f o r l , j = 2,3 and l * j (A4.8) 3 2TT/3P.3Z. = (3X./3Z .) • (Z ./X. ) " [ ( a j Z K = 2 a i k P k ) / P ' *ifi./ZO'(b..Z. - b . ^ ^ b . ) +C..MZ./X.) f o r l , j = 2,3 and i = 2,3,4 Shadow Prices of Fixed Factors (R, and R . ) : (A4.9) 37r/3Z, = R, = ^ 2 i = 2 g = 2 a i k ( P i V P ^ [ 1 / 2 5=2 ?=2 b j ^ Z . Z ^ Z ? ) + £ = 2 ( b . Z . / Z 2 ) + 1 / 2 S = 2 ( a . P . b o / Z 2 ) ] +2 * c .P. i=l i=1 I (A4.10) 3TT/3Z . = R . 3 D ^ j L i=2 £=2 a i k ( P i V P i ) + f > i a i p i - [ f=2 w z ' > + ( V Z i )  + ? = i c i j p i APPENDIX 5 : CALCULATIONS FOR CHAPTER 6 Table A5.1:--Mean predicted quantities and expenditures per vessel (using mean vessel), all samples: Case I Quantity Seine Gillnet Troll Gillnet-Troll Y L F G 94350.1 5.6 1817.3 0.5 14113.8 I. 2375 586.1 II. 3 22095.8 3.9 2467.1 10.9 21573.7 1.6 1775.2 20.0 Costs TVC TFC-High 38172.9 142413.4 63365.1 180586.3 101538.0 10723.0 19306.7 17369.2 30029.7 28092.2 28246.5 34139.6 18760.7 62386.1 47007.2 21294.5 16372.7 12540.9 37667.2 33835.4 -Low TC-High -Low Notes: The quantity terms are defined as follows: Y is output (pounds of fish), L is labour (persons), F is fuel (gallons), and C is gear (nets, lines, etc.) For the gillnet fleet, labour is taken as a fixed fator. All cost figures are expressed in 1982 current Canadian dollars. Total Variable Costs (TVC) for the gillnet fleet include expenditures only on fuel and gear. Total Fixed Costs (TFC) for the gillnet fleet include the cost of labour. High refers to the high rental price of a net ton and low to the low rental price of a net ton. These notes refer to all tables in this appendix. 269 / 270 Table A5.2: -Mean predicted quantities and expenditures per vessel (using all vessels), all samples: Case I Quantity Seine Gillnet Troll Gillnet-Troll Y L F C 94373.1 5.6 1818.0 0.5 14642.0 1.2375 562.2 10.0 22107.9 3.9 2465.7 10.8 22301.0 1.5 1775.2 51.7 Costs TVC TFC-High -Low TC-High -Low 38170.0 142375.6 63348.3 180545.6 101518.3 9518.6 19306.7 17368.9 28824.9 26887.5 28214.3 34121.7 18750.9 62336.0 46965.2 26950.0 16381.0 12547.3 43331.0 39497.3 / 271 Table A5.3:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), seine: Case II Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y L F C 93837.9 5.6 1760.8 0.5 93872.0 5.6 1765.5 0.5 146908.4 4.8 2671.1 0.3 148223.7 4.8 2677.0 0.3 Costs T V C ' 38316.3 38305.5 33388.7 33265.0 TFC-High 126920.7 - 127951.3 -Low - 57002.1 - 57407.5 TC-High 165237.0 - 161340.0 --Low - 95307.6 - 90672.5 Note: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 272 Table A5.4:-Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet: Case II Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net, tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y F C 13411.4 476.3 9.1 13401.9 490.6 9.5 13882.6 481.9 8.6 13907.8 494.1 9.0 Costs TVC 8590.1 8976.2 8230.3 8523.3 TFC-High 10029.2 - 11810.5 --Low - 10400.5 - 11847.9 TC-High 18619.3 - 20040.7 -Low - 19376.7 - 20371.2 Note: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 273 Table A5.5:-Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), troll: Case II Quantity Y L F C Costs TVC TFC-High -Low TC-High -Low Notes: 1 Sample totals expenditures. Using mean Vessel Predicted with high price net tonnage 37452.6 3.8 3973.7 6.3 Predicted with low price net tonnage 37452.6 3.8 3973.7 3.8 Using all Vessels 1 Predicted with high price net tonnage 77005.4 4.2 8310.6 54.6 Predicted with low price net tonnage 77005.4 4.2 8310.6 54.6 28746.3 81897.6 110643.9 28746.3 45005.2 73751.4 51925.2 228406.9 280332.1 51925.2 124660.8 176586.0 are divided by the number of vessels to obtain mean quantities and / 274 Table A5.6:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet-troll: Case II Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y L F G 23990.6 1.6 1133.2 3.3 23990.6 1.6 1133.2 3.3 20920.3 2.2 1492.8 4.0 20901.7 2.2 1492.2 4.0 Costs TVC 16533.1 16533.1 22278.3 22265.3 TFC-High 17442.5 - 17629.2 -Low - 13360.4 - 13491.1 TC-High 33975.6 - 39907.5 -Low - 29893.5 - 35756.5 Note: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 275 Table A5.7:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), seine: Case III Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y L F C 99687.5 5.7 1766.0 0.5 99724.9 5.7 1770.5 0.5 131267.1 5.1 2863.4 0.3 132501.8 5.0 2868.7 0.3 Costs TVC TFC-High -Low TC-High -Low Note: 38797.2 128112.5 166909.7 38787.7 57532.3 96319.5 35404.1 127277.4 162681.5 35297.2 57066.5 92363.8 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 276 Table A5.8:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet: Case III Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y F C 13860.8 443.5 8.4 13859.5 455.0 8.8 13340.9 604.0 7.9 13355.3 619.0 8.2 Costs TVC 8008.4 8323.6 7756.2 8057.2 TFC-High 10162.5 - 11824.4 -Low - 10384.6 - 11876.7 TC-High . 18170.9 - 19580.6 -Low - 18708.2 - 19933.9 Note: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 277 Table A5.9:--Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), troll: Case III Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y L F C 59877.9 6.7 10352.2 42.2 59877.9 6.7 10352.2 42.2 63333.4 6.6 10758.0 54.3 63333.4 6.6 10758.0 54.3 Costs TVC 65006.0 65006.0 68738.0 63738.0 TFC-High 216403.8 - 228482.8 -Low - 118920.3 - 124634.7 TC-High 281409.8 - 297220.8 -Low - 183926.3 - 193110.7 Notes: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. / 278 Table A5.10:—Mean predicted quantities and expenditures per vessel (using mean vessel and using all vessels), gillnet-troll: Case III Quantity Using mean Vessel Predicted with high price net tonnage Predicted with low price net tonnage Using all Vessels 1 Predicted with high price net tonnage Predicted with low price net tonnage Y L F G 23801.5 1.6 1476.8 3.6 23801.5 1.6 1476.8 3.6 20997.8 2.3 1864.1 4.3 20980.9 2.3 1863.2 4.3 Costs TVC 17713.0 17713.0 23865.8 23851.0 TFC-High 19303.1 - 19335.5 -Low - 14785.5 - 14795.6 TC-High 37016.0 - 43201.3 -Low - 32498.4 - 38646.6 Note: 1 Sample totals are divided by the number of vessels to obtain mean quantities and expenditures. 

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