FISHERY BULLETIN; VOL. 69, NO. 1 



could be caught, and which we might, similarly, 

 wish to minimize. Thus, for example, we simul- 

 taneously derive, by virtue of the LP model, as 

 we have formulated it, both the escapement and 

 the catch. 



Now, in the dual, we can place unit values 

 on the scarce resources rather than on the levels 

 of the activities, as in the primal. It is thus 

 helpful, in making management decisions, to 

 know the imputed unit value of a unit of cannery 

 capacity, a large male fish, an egg, etc. These 

 imputed values are commonly known as shadow 

 prices and correspond to the optimal values of 

 the Yj's in (15) ; they will be discussed briefly 

 in our interpretation of the salmon model. It 

 should also be mentioned that we have slack 

 variables in the dual formulation just as we had 

 slack variables in the primal. 



The dual slack variables can be viewed as 

 opportunity costs in the sense that if we fail 

 to meet a constraint, this is an opportunity 

 foregone ; and the dual slack variable then gives 

 the value foregone by the "bad" management 

 either of nature (that is, the vagaries induced 

 in the system which are uncontrollable by the 

 management agency) or of the management 

 agency. 



Finally, it is worth noting a feature of the 

 shadow prices vis-a'-vis the relation of the right- 

 and left-hand side of the constraint equations. 

 If in some solution of a particular problem, the 

 right-hand side becomes equal to the left-hand 

 side, then we say the constraint is binding. If 

 the constraint is binding, then the shadow price 

 has some positive value, namely the imputed 

 value of an additional unit of scarce resource; 

 but if, on the other hand, the constraint is not 

 binding, then an additional unit of the resource 

 is "free" within the bounds of the problem for- 

 mulation — consequently the shadow price of the 

 free resource is zero. 



EXAMPLES BASED ON THE 

 NAKNEK-KVICHAK RUN 



As an example, we have decided to consider 

 the implications of a LP ajipi'oach to implement- 

 ing the management framewoik of one of the 

 most important salmon runs in North America, 



the sockeye salmon run to the Naknek-Kvichak 

 system of Bristol Bay, Alaska. Our approach 

 was to use the LP model described in the pre- 

 vious section employing actual data where avail- 

 able for the constraints and the objective func- 

 tion. Although we examined behavior of the 

 model for several of the years for which we 

 had data, we are presenting in this paper our 

 partial analysis for the 1960 run only. Initially, 

 we indicate how we assigned values to the var- 

 ious coefficients in the problem and then we give 

 the actual examples. 



First, we assigned values to the objective 

 function (1) which is to be maximized. With 

 respect to the number of entities in the objective 

 function, there is a relatively large number of 

 ocean-age groups represented in the Naknek- 

 Kvichak run, but the very great majority are 

 either the relatively large .3 ocean-age fish or 

 the relatively small .2 ocean-age fish. Because, 

 as we will see in subsequent paragraphs, the 

 male fish are valuated diff'erently than the fe- 

 male fish, we used four entities: male or fe- 

 male, .2 or .3 ocean-age fish. 



Next, in order to assign values c,-; to each 

 entity in the objective function according to the 

 conventional per-fish management unit, we used 

 the aforementioned observation that the male 

 fish in each age group tend to be larger and 

 hence more valuable than the female fish. On 

 the other hand, the eggs which are contained 

 in the females are processed into a caviar-type 

 product, "sujiko." by Japanese firms in the 

 Bristol Bay canneries. Thus, the females, be- 

 cause of the eggs which they contain, are more 

 valuable than males of the same weight. Taking 

 these factors into consideration and using an 

 average ex-vessel value of $0.25, pound, we have 

 computed the average value for each entity. 

 These calculations are set forth in Table 2 which 

 shows, among other things, that the added value 

 of eggs tends to offset the reduced value of fe- 

 males relative to males of the same age class. 



As indicated previously, the $0.25 pound is 

 an average value and it should be emphasized 

 at this point that it is not a computed average 

 since generally speaking a fixed price is paid 

 for fish throughout the season. But as we in- 

 dicated earlier, some fish are certainly more 



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