FISHERY BULLETIN: VOL. 78, NO. 1 



.56 1.12 168 2.24 2.80 3 36 3.92 4 48 5.04 5,60 616 672 



X 

 Figure 2.— Continued. 



research to determine guidelines for when fine 

 tuning would be expected to produce such "trim- 

 ming" of the tails of the ergodic (long-run probabil- 

 ity) distribution. 



Including smoothing costs also tells us a great 

 deal about traditional concepts of fisheries man- 

 agement, such as MSY. It is clear from Figures 2 

 and 3 that anything close to an MSY policy is 

 optimal only if the smoothing costs exceed the per 

 unit value of the harvest. As whole systems of laws 

 for regulating fisheries have been constructed 

 around the idea of smooth, constant harvests, it is 

 clear that this imputes lower average catches, and 

 a significant preference for constancy of the har- 

 vest over total amount harvested. 



The analysis has assumed that Equation (1.1) or 

 similar equations are available, and that the 

 parameter estimates are accurate (in this case, 

 estimates of R^, R^, and cr^). In the latter case, 

 management measures would seem more reason- 



able if they were known to be robust against mis- 

 specifying the parameters. This involves knowing 

 how an optimal policy and total expected value 

 would vary if the true underlying parameter val- 

 ues differ from those specified, and also how the 

 estimate of the long-run probability distribution 

 differs from the true one. 



Walters and Hilbom (1976) have examined a 

 similar question of trying to solve the Bayes model 

 of this problem, i.e., where there is an original 

 prior probability given to each value of the 

 parameter, and this probability is updated each 

 period using Bayes theorem and the observed val- 

 ues during the period. However, they could not 

 obtain a solution, and Walters and Hilborn ( 1978) 

 raised questions as to the validity of some of their 

 numerical approximations. 



Fortunately, qualitative results are possible for 

 this particular class of Bayes problems. Let 6 be 

 the parameter (or vector of parameters) under 

 consideration. Let q^iQ) be the initial prior dis- 

 tribution on 6, and let q^ ( B) be the updated prior 

 distribution after n period has elapsed. Let Cl be 

 the set of all possible prior distributions. Then it is 

 proven in van Hee (1977a) that if the state of the 

 system is expanded to {Xf,q^), the resulting optimi- 

 zation problem is Markovian. Following argu- 

 ments similar to those in Scarf (1959) and van Hee 

 (1977a) it follows that an optimal Bayes policy 

 takes the form: 



For each element q e Ci, there is an x{q) such 

 that: 



do not harvest ifx^^xiq) 



harvest x^ - x{q) ifXf>x(q). 



For example, if cr^ in the distribution of c? is itself a 

 random variable, then each possible probability 

 distribution of cr^ yields a possibly unique base 

 stock size policy. 



Table 2. — Vital statistics for the nine policies approximating the smoothing cost policies for Wood and Branch Rivers. 



44 



