Some Suggestions for the Development of a 

 Bioeconomic Theory of the Fishery' 



Russell G. Thompson^ 



ABSTRACT 



In this study, the fundamental characteristics of the Schaefer model and the 

 Thompson-George (TG) production-investment model are reviewed, and extensions of 

 the TG model are discussed. It is then indicated how a bioeconomic model for the 

 sole ownership fishery may be obtained by adjoining the Schaefer model to the TG 

 model (or any of the extensions). This leads into a discussion of the fundamental variables 

 in a dynamic analysis of the fishery problem and the limitations of published bioeconomic 

 analyses. It is further pointed out that further work needs to be directed to the 

 formulation of catch functions allowing for varying marginal returns with respect to 

 fishing effort, in particular. 



INTRODUCTION 



In 1954 Schaefer used the first-order terms of 

 the sigmoid growth law to describe the dynamics 

 of an unexploited fish population and assumed 

 the catch to be proportional to effort'' to describe 

 the exploitation by man. The catch function was 

 subtracted from the natural growth law to 

 obtain the following model (which is commonly 

 referred to as Schaefer's model): 



(1) x(t) = Tx(l) (v-xiD) - oy(t)x(t} 



where x is the fish biomass, y is fishing effort, 

 t is time, x(t) = dx(t)ldt, and the remaining 

 symbols are parameters. 



In 1968 Thompson and George formulated a 

 production-investment model for the firm in- 

 volving stocks and fiows. Less than full use of 

 the capacity was allowed for by introduction of 

 a production scale variable. Short- and long-run 

 distinctions in economics were thus possible. 

 The firm could increase the capital stock by the 



' Partially supported by the National Science Founda- 

 tion as a part of the Sea Grant Progi-am for 1970. 



2 Russell G. Thompson is Professior of Quantitative 

 Management Science, University of Houston. 



3 As indicated by Schaefer and Beverton (1963), this 

 assumption is common to the Beverton-Holt approach 

 as well. 



purchase of capacity in excess of attrition. None 

 of the capital stock could be sold within the 

 decision interval of finite length; it could only 

 be sold at the end of the interval. Therefore, the 

 problem was irreversible during the finite period. 

 Extensions to allow for increasing marginal 

 costs are straightforward and were left to the 

 reader. The decision rules for the optimal 

 production and investment controls were derived 

 by use of control theory methods. An algorithm 

 was developed by which to compute solutions to 

 the controls so that the model had practical as 

 well as theoretical value. 



In 1970 George showed that solutions to 

 the optimal controls for a cash flow form of 

 analysis (as used by Thompson and George) were 

 identical to those for a discounted form of 

 analysis. That is, in reference to the TG model, 

 the optimal controls are the same for the case 

 where 5(f)>o and D(t) s o as for the case 6(() = o 

 and D{t) is evaluated at the market rate of 

 interest i(t). George further showed that one 

 model or the other must be used (in an e.xclusive 

 sense). 



In 1971 Thompson, Hocking, and George 

 showed how the initial values for the physical 

 and money capital accounts can be derived 

 optimally as a part of the solution to the 

 investment-production problem (as well as the 

 values for the controls during the decision- 

 making period). In 1970 Proctor studied the 

 investment problem for the firm in a reversible 

 and also in an irreversible setting (where the 



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