(3) y„ = 



if Q^aXo — Pi ox„ — (pO <0, 

 Zc, if 07a.v„ — Pi ox,, — (pO >0. 



with the subscript on v, y, x and z denoting 

 optimum values. 



method may be further enhanced considerably 

 by the development and estimation of more 

 robust forms of the catch function. 



The sole owner firm invests the maximum pos- 

 sible amount if the marginal value of capacity 

 is greater than the discounted marginal cost of 

 capacity and does not invest at all if the opposite 

 is the case. The firm uses all of its capacity if 

 the discounted net marginal revenues from 

 fishing effort, 0(7a.r,,— 0), are greater than the 

 marginal value of the fish resource, p/ o.x,., and 

 the firm does not fish at all if tht marginal value 

 of the fish resource is greater than the net 

 marginal revenue from fishing. 



The difference between a sole owner firm and 

 a competitive firm is immediate. In the latter 

 case, the effects of fishing on the resource are 

 ignored; and hence, the marginal value of the 

 fish resource is always zero (since pi (t) = o). It 

 can further be shown that pi^(t) for all f. Thus, 

 the marginal value of the fish resource reduces 

 the value of the decision rule for fishing effort. 



If the Schaefer model is augmented to allow 

 for a Cobb-Douglas type of catch function, for 

 example, then an interior solution (in the 

 interval [o, ^o]) for fishing effort is possible. 

 Similarly, an interior solution (in the interval 

 [o, m]) for investment costs is possible if 

 increasing marginal costs of capacity are 

 specified. 



The main difficulty in applying the TG model 

 (as first developed) is specification of the invest- 

 ment upper-bound. It is clearly a proxy for 

 various limitations on investment. For instance, 

 there might be borrowing limitations imposed 

 by the financial community. If so, Rahman's 

 extension (1970) of the TG model may be ap- 

 propriate. On the other hand, the investment 

 upper-bound may be superfluous if the catch 

 function is of a traditional production function 

 form. Few serious efforts have been directed to 

 investigations of alternative forms for the catch 

 function. Further efforts of the type being 

 pursued by Carlson (1969) surely need to be 

 given top priority in fishery research. 



In summary, an operational methodology for 

 the management of a fishery is available by 

 adjoining the Schaefer model to the TG model, 

 or one of its extensions. The potential for this 



LITERATURE CITED 



CARLSON, ERNEST W. 1969. A Bio-economic Model 

 of a Fishery. Working Paper No. 12, Division of 

 Economic Research, National Marine Fisheries Service, 

 U.S. Department of Commerce. 



CHRISTY, F. T., JR., and A. SCOTT. 1965. The Common 

 Wealth in Ocean Fisheries. Published for Resources 

 for the F'uture, Inc.. by the Johns Hopkins Press, 

 Baltimore, 281 pp. 



CRUTCHFIELD, J. A., and G. PONTECORVO. 1969. 

 The Pacific Salmon Fisheries, A Study of Irrational 

 Conservation. Published for Resources for the Future, 

 Inc., by the Johns Hopkins Press, Baltimore. 



CRUTCHFIELD. J. A., and A. ZELLNER. 1962. 

 Economic Aspects of the Pacific Halibut Industiy. 

 Fishery Industrial Research. Vol. 1, No. 1. 



GEORGE, M. D. 1970. Discounting and Cash Flow- 

 Analysis in Investment Problems. Unpublished manu- 

 script available from author on request. 



GORDON, H. S. 1954. The Economic Theory of a 

 Common Property Resource: The Fishery. Journal of 

 Political Economy, 62(2): 124-142. 



PROCTOR, M. S. 1970. Investment Theory for the 

 Firm: Deterministic and Stochastic Models. Unpub- 

 lished Ph.D. dissertation, Texas A&M University. 



RAHMAN, QUAZl MD. MAPIZUR. 1970. An Optimal 

 Investment and Financial Control Model: Theoretical 

 Solutions and an Application. Unpublished Ph.D. 

 dissertation, Te.xas A&M University. 



SCHAEFER, M. B. 1954. Some Aspects of the Dynamics 

 of Populations Important to the Management of the 

 Commercial Marine Fisheries. Inter-American Tropical 

 Tuna Commission, Bulletin, 1(2): 26-56, La Jolla, 

 California. 



SCHAEFER, M. B., and R, J. H. BEVERTON. 1963. 

 Fishing Dynamics — Their .\nalysis and Interpretation. 

 In M. N. Hill (editor) The Sea, pp. 464-483. Interscience, 

 Vol. 2, New York. 



SMITH. V. L. 1969. On Models of Commercial Fishing. 

 Journal of Political Economy. 77(2): 181-198. 



THOMPSON, R. G.. and M. D. GEORGE. 1968. Optimal 



94 



