Appendix A. — Technical Description of Bioeconomic Model 



SPECIFICATION OF THE GENERAL 

 RESOURCE USE MODEL 



Before we are able to evaluate the economic impact 

 of various management strategies, it is necessary to 

 develop a general bioeconomic model of how a fishery 

 functions. The following general model has been de- 

 veloped by Fullenbaum, Carlson, and Bell (1971): 



or 



In the above system, X is the biomass; K equals the 

 number of homogeneous operating units or vessels; .v 

 is the catch rate per vessel; C is total industry cost (in 

 constant dollars) or total annual cost per vessel muhi- 

 plied by the number of vessels; ir is equal to total 

 annual cost per vessel (in constant dollars) or oppor- 

 tunity cost'^; 77- is industry profit in excess of oppor- 

 tunity cost,p is the real ex-vessel price; andgp 8., rep- 

 resent the rates of entry and exit of vessels, respec- 

 tively. 



Equation (1) represents the biological growth func- 

 tion in which the natural yield or net change in the 

 biomass (X) is dependent upon the size of the biomass, 

 X, and the harvest rate, Kx. X reflects the influence of 

 environmental factors such as available space or food, 

 which constrain the growth in the biomass as the latter 

 increases. The harvest rate or annual catch, Kx, sum- 

 marizes all growth factors induced by fishing activity. 

 Equation (2) presents the industry and firm production 

 fiinction for which it is normally assumed that — 



8x 



pi>0 and M. 

 hK 



g2<0.'5 



In other words, catch per vessel increases when the 

 biomass increases and declines when the number of 

 vessels increases. Equations (3) and (4) are the indus- 

 try total cost and total profit function, respectively. 

 Equation (5) is very important since it indicates that 

 vessels will enter the industry when excess industrial 

 profits are greater than zero (i.e., greater than that rate 

 of return necessary to hold vessels in the fishery, or 

 the opportunity cost) and will leave the fishery when 

 excess industrial profits are less than zero (i.e., below 

 opportunity cost). 



'■•Opportunity cost is defined as the necessary payment to fisher- 

 men and owners of capital to keep them employed in the industry or 

 fishery compared to alternative employment or uses of capital. 



'^In some developing fisheries, it is possible that g.. > 0. For 

 example, in the Japanese Pacific tuna fishery, intercommunication 

 between vessels may increase the catch rate as more vessels enter 

 the fishing grounds. 



The equilibrium condition for the industry (tt = 0) 

 may be formulated as shown below: 



P = 



TT 



g(X,K) 



(6) 



Equation (6) merely stipulates that ex-vessel price is 

 equal to average cost per pound offish landed (i.e., no 

 excess profits). 



There are two important properties of the system 

 outlined in equations (1) to (5). First, the optimum size 

 of the firm is given and may be indexed by ir. Thus, 

 the firm is predefined as a bundle of inputs.'" Sec- 

 ondly, the long-run catch rate per vessel per unit of 

 time is beyond the individual firm's control.'" It is, in 

 effect, determined by stock or technological 

 externalities.'* Finally, we are assuming that the 

 number of homogeneous vessels is a good proxy for 

 fishing effort. Alternatively, we may employ fishing 

 effort directly in our system by determining the 

 number of units of fishing effort applied to the re- 

 source per vessel. This will be discussed below. 



A QUADRATIC EXAMPLE OF 

 THE RESOURCE USE MODEL 



By combining the more traditional theories depict- 

 ing the dynamics of a living marine resource, with 

 some commonly used economic relations, we may de- 

 rive a quadratic example of the general model 

 specified above. This example effectively abstracts 

 from complications such as ecological interdepen- 

 dence and age-distribution-dependent growth of the 

 biomass on the biological side and, furthermore, as- 

 sumes the absence of crowding externalities (i.e., 

 ^2 - 0) in the production function on the economic 

 side. 



The dynamics of a fish stock may be depicted by the 

 logistic growth function (Lotka, 1956): 



X(t) 



where L>0,C>0,A>0, (7) 



1 + Ce: 



where L and A' are assumed to be environmental con- 

 stants. Differentiating (7) and substituting we obtain. 



■^In other words, because we are dealing with a long-run theory of 

 the industry, we are assuming that variations in output result from 

 the entry or exit of optimum sized homogeneous vessels. 



"We have implicitly assumed that such short-run changes as 

 longer fishing seasons, etc. , are all subsumed in a long-run context. 

 Normally longer fishing seasons, for example, do not change catch 

 rates per unit of time fished; nor do they change costs per unit of 

 time fished. They do. however, change the effective level of K. 



"A technological externality exists when the input into the pro- 

 ductive process of one firm affects the output of another firm. In the 

 context of fishing, an additional firm or vessel entering the fishery 

 will utilize the biomass (as an input) and, as a result, in the long run 

 will reduce the level of output for other vessels in the fleet. (See 

 Worcester, 1969.) 



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