firm may buy and sell its capital stock during 

 the period as well as at the end). He further 

 derived the demand functions for capital in each 

 case and deduced their economic characteristics. 



CONCEPTUAL MODIFICATIONS 



By adjoining the Schaefer model to any one 

 of these formulations, a production-investment 

 model for the sole ownership fishery is obtained. ^ 

 Such a formulation has a number of distinct 

 advantages: First, the inherently dynamic prob- 

 lem of the fishery is formulated accordingly in a 

 mathematical sense, second, the model (since it 

 encompasses the economic and biological rela- 

 tions) is bioeconomic in form; third, given mean- 

 ingful expressions for the functions involved, 

 decision rules for the production and investment 

 controls (and hence the basis for a bioeconomic 

 theory) may be derived by the straightfoi'ward 

 use of published mathematical methods. 



Lack of such a methodology may be the reason 

 for the historical development of the bio- 

 economic theory for the fishery. For example, 

 virtually all economists who have published 

 in the professional journals (or by the way of 

 Resources for the Future) have commonly as- 

 sumed the inherently dynamic problem of the 

 fishery to be static at the outset of their 

 analyses (cf. Smith 1969), Christy and Scott 

 (1965), Gordon (1954), and Crutchfield and 

 Pontecorvo (1968). 



Another example is provided by the form of 

 the catch function used. Until recently, econo- 

 mists have not seriously questioned the form of 

 the catch function introduced by Schaefer, oyx. 

 This formulation implies constant marginal 

 returns with respect (w.r.) to effort and in- 

 creasing returns to scale. 



Crutchfield and Zellner (1962) made static 

 and dynamic analyses of the fishery problem 

 (with this catch function) and found different 

 constant solutions! They failed to note that a 

 capacity limitation must be imposed on fishing 

 effort. The problem is similar to maximizing 

 the function y = .v in which the domain must be 



■* Any of these forms of the problem are consistent 

 with Turvey's formulation (1964). Variations in mesh 

 size would be associated with different capital character- 

 istics, and require the introduction of more than one 

 capacity variable and possibly functions relating vessel 

 types and mesh size. 



bounded from above for the problem to have 

 finite solution. 



Following this analysis, Cnitchfield and Zell- 

 ner introduced a Cobb-Douglas form for the 

 catch function and made a partial analysis of 

 this case. This problem also requires a capacity 

 limitation on effort to be well posed. In addition, 

 increasing returns to scale in capacity for 

 sufficiently small expenditures may be neces- 

 sary as well as decreasing returns beyond some 

 point. This is particularly relevant when the 

 competitive model is desired for a reference 

 framework. Decreasing returns everywhere are 

 inconsistent with the market requirements for 

 a competitive structure (Proctor, 1970). 



Still another example of the unusual approach 

 used to date is the specification of an infinite 

 horizon for the completely irreversible invest- 

 ment problem. The optimal length of the horizon 

 in a common property resource problem might 

 well be one of the fundamental results being 

 sought in the analysis, and not an input to the 

 analysis, as specified by Crutchfield and Zellner. 

 There are no transferable rights to the fishery 

 resource; and hence, the entrepreneur might 

 desire to take all of the resource within a finite 

 period of time. Thus, the optimal solutions to 

 the investment and production controls and the 

 length of the decision horizon would be expected 

 to be the fundamental variables for a bioeconomic 

 theory of the fishery. 



For the case of the Schaefer model, the decision 

 rules for the production-investment controls 

 follow immediately from the TG model. The 

 necessai'y condition for the optimal length of 

 the decision interval, if one exists, follows as 

 an immediate extension of their results. In 

 fact, the decision rules for investment and 

 production are particularly straightfoi-ward and 

 easy to state. Let v = investment, m = the 

 investment upper-bound, 7 = the fish price, 

 d = production cost per unit of effort, ? = in- 

 vestment cost per unit of capacity, = the 

 discount function, z = fishing capacity, pi = 

 the marginal value of the fish per unit weight, 

 and Pi = the marginal value of capacity. Then 

 the decision rules are: 



(2) 



i\i = 



if pj 

 m if p: 



0? <0. 

 - 0r >0, 



93 



