drawal in year i for sundry expenses; 7 is tho 

 interest rate paid (or received) on the cash ac- 

 count 2; o), is the unl<nown revenue per unit of 

 capacity in the i"' year; N is the number of years 

 in the planning period; /3 is the fraction of the 

 value of the capacity recoverable at the end of 

 the planning period; 5 is the income tax rate; 

 and e is the straightline depreciation fraction. 

 Also E will be used to denote the mean of the 

 random variable cj,; and L will be used to denote 

 the smallest possible annual net revenue having 

 a positive probability of occuring. The symbol 

 a, is used to denote the output price where only 

 the yield is a random variable in the application 

 below. 



Using the above development, the survival 

 model may be stated as follows: 



Maximize £(2/v + |3a/v + iyw) over all n-tuples 

 of functions s,(coi , GJ2 , • • ., <^i-i ), ( = 1,2,.. ., 

 N, satisfying the difference equations 



(1) Xi—Xi-[ =OiSi,Xc, =00^0, 



where 



y, — y,_i = s,, yo given and non-negative, 



(2) 2, — 2,- 1 = 72/- 1 + y, (coi—Ti) — a„s, 



— A,. — 6 |y, (cj,— r,) + 72,-1 — A,- 

 -exi\,e = 0.091, 



THE DECISION RULE FOR INVESTMENT 



By the use of dynamic programming 

 methods, the method developed by Thompson 

 and George was extended, as mentioned above, 

 to allow for depreciation and income taxes. The 

 extended mle for optimal investments is sum- 

 marized in the following theorem. 



Theorem: Suppose H|(zo, yo, Xo )>0, i.e. the 

 upper bound for investments in the first year 

 is non-negative. Let R/^ be the expected mar- 

 ginal value of capacity for survival investment 

 decisions— the marginal value of capacity vis- 

 uahzing the worst. Then the decision rule for 

 optimal survivable investment is as follows: 



(3) Sk' = Hk izk'- 1 , yk'- 1 , Xk^ , ) if Rk>0, 

 and Sk = o if Rk<0 



\ with the feasible value of Sk being immaterial 

 \tRk=0. 



In other words, the decisionmaker buys the 

 survivable limit of capacity in year k if the 

 marginal value of capacity visualizing the worst 

 is positive in that year, and he makes no capacity 

 purchases if this marginal value is negative. It 

 also follows that the optimal purchase is im- 

 material in any year (because of the linearity 

 of the problem) whenever the decision rule is 

 zero. The upper bound for investments in the 

 first year insures the existence of a feasible 

 investment solution in each year of the planning 

 horizon. 



where Zo given, and ; = 1,2, 

 ing the inequalities 



N, and satisfy- 



OCs,<H, (2,-, , y,-i , Xi-i), / = 1, 2, . . ., N. 



In words, the decisionmaker desires to maxi- 

 mize expected net worth at the end of the 

 decision period where the purchases of capacity 

 are selected from the survivable set in each 

 year (delineated by the inequality restrictions). 

 Thus, in the maximization process, the decision- 

 maker, who takes into account all of the informa- 

 tion known at the time of the decision, selects 

 the investment from the survivable set of capa- 

 city purchases that maximizes expected net 

 worth at the end of the planning horizon. 



An Application to Shrimp Fishing 



To indicate how the model may be applied to 

 a shrimp fishing firm, parameters were specified 

 for a relatively small fishing firm operating 

 73-foot steel hull trawlers (see Table 1). In the 

 specifications, the values of the parameters were 

 specified to reflect prices, costs, and landings per 

 vessel as reported by the firms cooperating in 

 the study. There is an exception with regard to 

 Problem 3. Average landings per vessel which 

 were found to be 57,560 pounds of heads-off 

 shrimp per year in the years 1958 through 1969 

 were specified to be one standard deviation 

 above the mean to evaluate the effect of better 

 than average management. That is, in Problem 



114 



