FISHERY BULLETIN: VOL 78, NO. 1 



u, and then finding a policy that minimizes 



'^ This methodology de- 



lim 



e\; k^ {Zt-uY 



pends on the values of u chosen. It also determines 

 the policy that minimizes the approximate long- 

 run variance for a given long-run mean harvest. 

 This is not equivalent to reducing the size of the 

 year-to-year fluctuations. 



A second method is to include "smoothing costs" 

 into the one-period return. This approach has been 

 studied analytically by Mendelssohn.'* Let y be the 

 cost of a unit decrease in the harvest from year to 

 year, and let e be the cost of a unit increase in the 

 harvest from year to year. 



If 2 was harvested last year, then net revenues 

 this year, for any harvest 2, , are decreased by 



1(2 -Zt) 







e{Zt-z) 



if 2 >2^ 



Z <2i 



Amended to Equation (1.2), this would imply a 

 one-period net benefit of 



pix^ - y^) - y[z - (x^ - yi)V - e[(x^ - y^) -zV 



where (a)^ denotes the positive part of a. An alter- 

 nate form is to let e = (y-e)/2andc = (y+e)/2.Then 

 the one-period return is: 



pix^ -y^) + e(x^-y^)-c\{Xf-yi)-z\ - ez. 



One advantage to the smoothing cost approach 

 over other approaches is that p, e, and c can be 

 normalized so as to be interpreted as relative 

 prices. That is, the normalized values p - 1, elp, 

 and dp can be interpreted as the value of having 

 the between period harvest "smoothed" by one unit 

 relative to the value of one unit of additional har- 

 vest. Actual relative values are often difficult to 

 determine. But by parameterizing on e and c, it is 

 possible to present a decision maker not only a 

 range of possible "optimal" policies and their con- 

 sequences, but also some feeling for the relative 



■•Mendelssohn, R. 1976. Harvesting with smoothing 

 costs. SWFC Admin. Rep. 9H, 26 p. Southwest Fish. Cent. 

 Honolulu Lab.. Natl. Mar. Fish, Serv., NOAA, Honolulu, HI 

 96812. 



trade off between total income and the smoothness 

 of the received income stream. 



For the Wood and Branch Rivers, two sets of 

 computations were performed. The first set as- 

 sumes that y = e, i.e., there is an equal concern for 

 increases in allowable harvest as well as for de- 

 creases. This is equivalent to e = 0.0 and c = y (or 

 equivalently e). The motivation for this cost struc- 

 ture is that fishermen typically resist any decrease 

 in the allowed harvest, hence y>0. However, al- 

 lowing increases in the harvest size often signals 

 fishermen to gear up and invest in equipment, 

 thereby making it even more difficult to decrease 

 the allowable harvest later on. Therefore this cost 

 should be equal to a cost due to a decrease in the 

 harvest. 



As a counterbalance to this, a second set of com- 

 putations were performed with y>0 but e = 0, i.e., 

 a cost only if the harvest is decreased. This is 

 equivalent toe = e = y/2. 



For the first set of computations, with e = 0.0 

 andp = 1.0, values of c of 0.25, 0.50, 0.75, 1.00, 

 1.25, 1.50, 1.75, and 2.00 were used. These are 

 equivalent to relative values of Vs, V4, %, ¥2, %, %, 

 ■%, and 1. For the second set of runs, withe = e, and 

 p = 1.0, values of 0.25, 0.50, 0.75, 1.00 and 1.25 

 were used. These are equivalent to a ratio of yip 

 equal to Va, V2, %, 1, VA. The results are sum- 

 marized in Figure 2(a)-(m) and Figure 3(a)-(m), 

 which show an optimal policy for each river for 

 each of these cases. All computations were per- 

 formed on 26-point grids. 



The figures are read as follows. Suppose 2 was 

 harvested last year and x is the observed popula- 

 tion size this period. Find the point {x, z) on the 

 graph and follow the arrow in that zone to the 

 appropriate boundary as indicated. Then read off 

 the 2 value of this point; this is the optimal amount 

 to harvest during this period. 



For example, if e = 0.50, e = 0.00, x^ = 0.84, and 

 the harvest last period was 0.28, Figure 2(b) shows 

 that the optimal policy for the Wood River is to 

 harvest 0.28 this period. Note that the dashed line 

 is the equivalent base stock harvest with no 

 smoothing costs. 



While the policies in Figures 2 and 3 are optimal 

 for the given relative values of p, e, and c, they are 

 complex in nature and would be difficult for a 

 layperson to understand. Practical management 

 often implies determining simpler, good but sub- 

 optimal policies that achieve the same objectives. 

 These policies are often more desirable since they 



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