BOOTH: MODEL FOR OPTIMAL SALMON MANAGEMENT 



T-1 



Figure 1. — The Ricker spawner-return relationship for 



salmon. 



jective functions are mathematically feasible, 

 but it is not obvious what they might be without 

 going beyond the scope of this paper and under- 

 taking a political analysis of salmon fishery man- 

 agement. Specifying a terminal value function 

 is a mathematical necessity and will be discussed 

 later. 



Assumption (iv) implies, among other things, 

 that the catch Xt from the salmon stock being 

 analyzed is not large enough to influence the total 

 industry price structure for salmon and that the 

 factor supply market for effort, Et, is competi- 

 tive. 



The problem of gear congestion on the fishing 

 ground is adequately dealt with elsewhere and 

 is avoided here by assuming that the fishery 

 management authority undertakes appropriate 

 policies to insure that only efficient levels of ef- 

 fort are employed (see footnote 3; Quirk and 

 Smith, 1970). It is also assumed that salmon 

 are not caught on the high seas but are harvested 

 as they return to the river to spawn. 



X (1 I ^y [Rt fikt) Px - RthtPE^^ 



+ (1 ^ ^)T G(Rt), 



subject to: 



St = Rt — Rtf(kt),St^O,kt ^0,R(0) = Ro, 



and 



Rt = a St-i e^-^^t-i\ 



The appropriate Lagrangian for the maximi- 

 zation problem is: 



L (k, S, X) = 

 T-l 



X (1 I ^y {^t (Px fikt) - Pe kt)) 



+ 



(TTTp 



G(Rt) 



T-l 







[1] 



where k = (ko, . . . , kT-i),S = (So, ... , St~i), 

 and X = (Xo, . . . M-i), and Ro is given. 



The appropriate Kuhn-Tucker necessary condi- 

 tions for a maximum of L (k, S, k) are as fol- 

 lows: 



A SIMPLE MODEL 



In this section the simplest type of spawner- 

 return relationship is examined where salmon 

 spawned in time period t — 1 return to their 

 spawning ground in time period t. Given the 

 assumptions of the above section, the problem 

 is to maximize 



dL/-dkt < 0; kt dL/dkt 

 t = 0, ..., T—1; 



dL/dSt ^ 0; St dL/dSt 

 t = 0, ..., T—1; 



dL/dh = 0, 



t = 0, ..., T—1. 



= 0, 



= 0, 



[2] 



[3] 



[4] 



499 



