turn relationships seem to possess a high degree 

 of variability, which suggests that a stochastic 

 specification of the problem may be more real- 

 istic than our deterministic approach. Esti- 

 mating a catch function may also be troublesome 

 since it would require a careful specification of 

 effort and of its price. 



Despite the difficulties, construction of a work- 

 ing model would be worthwhile because it would 

 provide insight into some of the unresolved math- 

 ematical problems mentioned above, and more 

 importantly it would provide further informa- 

 tion on the significance of the difference between 

 the magnitude of spawner stocks selected on the 

 basis of maximum sustained yield and stocks 

 selected on the basis of economic optimality as 

 defined in this paper. 



ACKNOWLEDGMENTS 



I wish to thank Professors B. Rothschild and 

 G. Brown for their encouragement and help in 

 preparing this manuscript. I am indebted to the 

 National Science Foundation and to the National 

 Marine Fisheries Service (Contract No. 14-17- 

 0007-1 133 A) for financial support while I was 

 undertaking research on this topic. I wish to 

 also thank two anonjonous referees for their 

 comments. 



LITERATURE CITED 



Burt, 0. R., and R. G. Cummings. 



1970. Production and investment in natural re- 

 source industries. Am. Econ. Rev. 60:576-590. 



Mathews, S. B. 



1967. The economic consequences of forecasting 

 sockeye salmon {Oncorhynchus nerka, Walbaum) 

 runs to Bristol Bay, Alaska: A computer simu- 

 lation study of the potential benefits to a salmon 

 canning industry from accurate forecasts of the 

 runs. Ph.D. Thesis, Univ. Washington, Seattle, 

 238 p. 



Quirk, J. P., and V. L. Smith. 



1970. Dynamic economic models of fishing. In A. 

 Scott (editor). Economics of fisheries manage- 

 ment: A symposium, p. 3-32. H. R. MacMillan 

 Lectures in Fisheries, University of British Co- 

 lumbia, Vancouver, Canada. 



Rothschild, B. J., and J. W. Balsiger. 



1971. A linear-programming solution to salmon 

 management. Fish. Bull. U.S. 69:117-140. 



FISHERY BULLETIN: VOL. 70, NO. 2 



APPENDIX 



A continuous time analog to the simple model 

 will now be discussed briefly. The problem is 

 to maximize. 



00 



J RtlPxfikt) —PEktJe-'^'dt [A-l] 



subject to 



St=^Rt\.l—fikt)-]—St, 



[A-2] 



where St = -rr— , and p is the continuous time 



discount rate. The appropriate Hamiltonian for 

 the maximum problem is 



H ^ e-^' (RtlP.f(kt) — PEkt} 



+ ktlRtll-fikt)] — St — St]\ [A-3] 



Assuming the existence of an interior solution, 

 the necessary conditions for a maximum are, 

 along with [A-2], 



^ - 

 ^kt - "' 



d dH _ ^H 



[A-4] and [A-5] are satisfied if 

 (Px — Xt)f = Pe, 



Xt = pXt 



- f [P. fikt) - Pe ktl ^ 



+ Xt[l -fikt)] ^ -X,) 

 where Xt = dk/dt. 



[A-4] 

 [A-5] 



[A-4'] 



[A-5'] 



If [A-4'] is used to eliminate one of the un- 

 knowns, the result is a system of two differen- 

 tial equations in two unknowns, [A-2] and 



504 



