FISHERY BULLETIN: VOL. 70, NO. 2 



It is assumed that the solution to the maximizing 

 problem is interior; i.e., kt > 0, St > 0, Xt > 0. 

 Then [2] and [3] are satisfied if [5], [6], and 

 [7] are true: 



(P,_X,)/'(^-t) =Pe, t = 0,...,T-l; [5] 



Xt-i = 



 ^ ^ ^^ (Px fikt) — Pe k!j dRt/dSt-i 



+ 



(1 + r) 



Xt [1 — fikt)] dRt/dSt-i, 



t 



1 r— 1- 



_ 1 dG dRr 



^T-i - (1 + r) "MT dSr-i ' 



[6] 

 [7] 



and [4] is satisfied if [8] is true. 



St = Rt — Rtfikt), i = 0,...,r— 1. [8] 



For t = 1 equation [6] can be written more 

 simply as 



Xo = 1 + , [Px f(h) - Pe k,l 4|- 



+ 



1 + r 



Xi 



dSo ' 



[6'] 



By substituting for Xi in [6'], an expression of 

 Xo results with Xo in it, and by substituting for 

 Xi an expression of Xo results with Xs in it, and 

 so on until an expression of Xo results with Xr-i 

 in it. Equation [7] can then be substituted for 

 Xt-i." The resulting expression is: 



Xo = 



V 1 



^ (1 + ry 



P.fikt) —PEkt 



dRt 

 9^ 



+ 



dG dJ^T 

 "3^ 



(1 + r)^ dRr 

 Equation [5] can be rewritten for t 



Xo = Px — PE/f'iko). 



[9] 



1 as 



[5'] 



Equation [5'] suggests that Xo can be inter- 

 preted as the marginal profitability of catching 

 an additional fish in period 0, while equation [9] 

 suggests that Xo can be interpreted as the present 

 value of the marginal profitability of adding an 

 additional spawner to the escapement level in 

 period 0; i.e., the level of escapement should be 

 selected in time period such that the incre- 

 mental profitability of an additional fish caught 

 today is just equal to the present value of the 

 profitability of the future return resulting from 

 an incremental spawner. In order to attain the 

 desired level of escapement it is necessary to 

 select an appropriate level of ko, since escape- 

 ment is equal to the run size in period minus 

 the catch. Note that this analysis can be applied 

 to any time period t, not just to t = Q. 



In order for the analysis to be valid in the 

 general form presented here, it is necessary to 

 prove the existence of values for A; = (ko, . . . , 

 kr-i), S = (So, . . . , St-i), and X = (Xo, . . . , 

 Xt-i), which satisfy equations [5], [6], [7], and 

 [8] given Ro = R{0). This is not an easy task 

 if T is finite, so it is assumed that T -» oo.' In 

 this situation an equilibrium, or steady state 

 solution is possible, where 



k* — ko = ki = . . . , = kr 

 S = So = Si = . . . , ^ St 



X = Xo = Xi = . . . , = Xt. 



If T is finite this kind of a solution makes no 

 sense. If 7" -» oo and an equilibrium solution 

 exists, then equation [9] can be rewritten as 



' This procedure is discussed in Burt and Cummings 

 (1970). 



' For a discussion of the mathematical problems in- 

 volved in the case of an infinite time horizon (Burt and 



Cummings, 1970). 



500 



