are discussed in the concluding section. In the 

 Appendix the continuous time analog to the 

 simple model is presented. 



NOTATION AND ASSUMPTIONS 



The notation to be used is as follows: 



Rt = the size of a run of salmon into a given 



river in time period t; 

 St = the number of spawners allov^ed to 



escape up the river in time period t ; 

 Xt = the catch of salmon from a given river 



in time period t; 

 Et = the amount of effort used to catch Xt 



salmon in time period t; 

 Pe = the price per unit of effort; 

 Px = the price per unit of fish; 

 r = the appropriate discount rate; 

 T = the total number of years. 



The assumptions of the analysis are as fol- 

 lows: 



(i) The industry catch for a given river, Xt, 

 is a linear homogeneous function of the amount 

 of effort employed Et and the size of the run Rt'. 



Xt = FiEt, Rt) 



= Rt F{Et/Rt, 1) 



= Rt f{kt); where kt = Et/Rt." 



(ii) The biological spawner-return relation- 

 ship is of the form developed by Ricker (Math- 

 ews, 1967): 



Rt 



a St-i e'-''^t-i\ 



A graph of this function for a simple first order 

 model appears in Figure 1. If a policy of max- 

 imum sustained yield is followed, the escapement 

 in year t — 1, S°t-i, occurs where Rt — St-i is a 

 maximum, or where 



d (St-i a e^-^^t-P — St-i) 



d St-i ~ 



o ef-''St-i)(l — b St-i) —1 = 0. 



FISHERY BULLETIN: VOL. 70, NO. 2 



The escapement in year t is: 



St = Rt — Rt f (kt). 



(iii) The appropriate objective function to 

 maximize is assumed to be 



r-i 



V 1 



^ (1 + r)' 

 fc=0 



Rtfikt) Pr—RtktPE 



+ (1 ^ ^)r <j(-Rt) 



The first derivative of / will be denoted /'. 



where the expression on the left is the present 

 worth of industry profits over T — 1 years, and 

 the second expression is the present worth of a 

 value function for the terminal stock of fish. 



(iv) The price of fish Px and the price of effort 

 Pe are assumed to remain constant for all time 

 periods. 



For some readers the purpose of making these 

 assumptions may at this point appear unclear. 

 Hopefully, the comments to follow will clarify 

 any ambiguities. 



In assumption (i) a linear homogeneous ag- 

 gregate production function is selected for its 

 convenient mathematical properties, and because 

 it has an important economic property, constant 

 returns to scale. In most industry aggregate 

 production function studies, the assumption of 

 constant returns has been found to be reasonably 

 realistic. However, in the case of the salmon 

 industry, this hypothesis has yet to be tested. 



The spawner-return relationship specified in 

 assumption (ii) has the usual properties of fish- 

 ery recruitment functions. It is clear from the 

 graph_that spawner stocks to the right of the 

 point St- 1, where Rt is a maximum as a function 

 of St ~i, are irrelevant from a policy standpoint, 

 since for any feasible run size Rt there corres- 

 ponds a spawner stock St~i with St~i ^ St-i. 

 Note that no species interaction is implicitly 

 assumed. 



The assumption that the present worth of in- 

 dustry profits is the appropriate objective func- 

 tion to maximize is an assumption commonly 

 made in economic analysis. Other types of ob- 



498 



