FISHERY BULLETIN: VOL. 70, NO. 3 



Table 2. — Optimal allocation of skipjack and yellow^n 

 tuna in tons of fish to various size classes of fishing boats. 



500-ton boats, 429^ ) to obtain the yellowfin: skip- 

 jack ratios of 1.32, 0.94, and 0.72, respectively, 

 thus yielding the technological constraints: 



Table 2 gives the maximization of the objective 

 function which yielded $1,248,835 in tons of fish. 

 The optimal solution then indicates that in the 

 process of production, the entire capacity of the 

 vessels was utilized. Because the catchable por- 

 tion of the stocks was greater than this capacity, 

 22,070 tons of yellowfin and 47,489 tons of skip- 

 jack were unused by the fishery (slack vari- 

 ables). Note also that the catch of skipjack is 

 greater than that of the more valuable species, 

 yellowfin, because of the technological con- 

 straints enforcing the lower yellowfin: skipjack 

 ratios in the more numerous larger boats. The 

 imputed marginal values, the so-called shadow 

 prices of a ton of yellowfin and skipjack are, of 

 course, zero because the capacity of the stock to 

 produce these quantities of fish was not reached; 

 but, however, the capacity of the vessels was 

 reached and, therefore, the marginal value of 

 an extra ton capacity on the 300-, 400-, and 500- 

 ton boats is imputed to be $8.08, $9.81, and $7.04, 

 respectively. These shadow prices are simply 

 the weighted average profits for each size class, 

 e.g.: 



$8.08 



10,112 

 23,459 



7.32 + 



13,347 

 23,459 



8.65 



Perhaps of even greater interest is the way 

 in which the various production inputs interact 

 with one another. For example, in this partic- 

 ular problem, we could increase the yellowfin and 

 skipjack catchable i)opulation constraints ad in- 

 finitum without changing the nature of the op- 



timal solution. But if we were to reduce the 

 catchable population of yellowfin tuna from 

 90,000 tons to 67,930 tons or skipjack from 

 120,000 tons to 72,510 tons, we would eliminate 

 the yellowfin and skipjack slack variables, re- 

 spectively, and these would no longer be in the 

 optimal solution. Putting it another way, in- 

 sofar as this particular problem is concerned, the 

 nature of the solution, in terms of, for example, 

 those variables to which some monetary value 

 greater than would be imputed, would not 

 change until the catchable population of yellow- 

 fin dropped below 67,929 tons or skipjack to be- 

 low 72,510 tons. The point of this is that (again 

 insofar as this particular problem is concerned) 

 we are not going to change the nature of our op- 

 timal solution for any catchable populations of 

 yellowfin >67,929 tons or of skipjack >72,510 

 tons. This means that it may not be necessary 

 to be concerned with precise estimates of the 

 catchable population if the catchable population 

 is, as in this case, much larger than the lower 

 bounds for changing the solution. This reflects, 

 within the scope of the model, the bounds within 

 which changes in the catchable population will 

 have no efl["ects upon the components of the ob- 

 jective function. This demonstrates, in an an- 

 alytical way, that population dynamics theory 

 may offer solutions that are, in some instances, 

 apparently more precise than that which is 

 needed. In other words, we frequently postpone 

 resource decisions to obtain a certitude in our es- 

 timate, which would not change the optimal so- 

 lution of the input-output process. This post- 

 ponement is almost never without social costs 

 which may be substantial. 



Now with respect to modifying the fleet capa- 

 city, we can, given the stock constraints, increase 

 the capacity of the small boats to 62,250 tons or 

 decrease it to tons. If we exceed the upper 

 bound then this means that we need to catch at 

 least 38,790 additional tons of fish and, of these, 

 579^ must be yellowfin amounting to an addi- 

 tional catch of 22,110 tons of yellowfin. But if 

 we catch this additional quantity of yellowfin, 

 we will use up our 90,000 tons of yellowfin, drop- 

 ping the yellowfin slack variable from our solu- 

 tion. At the lower bound, it is obvious that if 

 we constrain the catch of small boats to be 0, 



674 



