Question

Clint Conley, president of Conley Fisheries, Inc, operates a fleet of fifty cod fishing boats out of Newburyport, Massachusetts. Clint's father started the company forty years ago but has recently turned the business over to Clint, who has been working for the family business since earning his MBA ten years ago. Every weekday of the year, each boat leaves early in the morning, fishes for most of the day, and completes its catch of codfish (3,500 lbs, of codfish) by mid-afternoon. The boat then has a num ber of ports where it can sell its daily catch. The price of codfish at some ports is very uncertain and can change quite a bit even on a daily basis. Also, the price of codfish tends to be different at different ports. Furthermore, some ports have only limited demand for codfish, and so if a boat arrives relatively later than other fishing boats at that port, the catch of fish cannot be sold and so must be disposed of in ocean waters To keep Conley Fisheries problem simple enough to analyze with ease, assume that Conley Fisheries only operates one boat, and that the daily operating expenses of the boat are $10,000 per day. Also assume that the boat is always able to catch all of the fish that it can hold, which is 3,500 lb. of codfish. Assume that the Conley Fisheries' boat can bring its catch to either the port in Gloucester or the port in Rockport, Massachusetts. Gloucester is a major port for cod- fish with a well-established market. The price of codfish in Gloucester is $3.25/lb., and this price has been stable for quite some time. The price of codfish in Rockport tends to be a bit higher than in Gloucester but has a lot of variability. Clint has esti- mated that the daily price of codfish in Rockport is Normally distributed with a mean of $3.65/ lb, and with a standard deviation of ơ $0.20/lb. The port in Gloucester has a very large market for codfish, and so Conley Fish- eries never has a problem selling their codfish in Gloucester. In contrast, the port in Rockport is much smaller, and sometimes the boat is unable to sell part or all of its daily catch in Rockport. Based on past history, Clint has estimated that the demand for codfish in Rockport that he faces when his boat arrives at the port in Rockport obeys the discrete probability distribution depicted in Table 5.1 It is assumed that the price of codfish in Rockport and the demand for codfish in Rockport faced by Conley Fisheries are independent of one another. Therefore, there is no correlation between the daily price of codfish and the daily demand in Rockport faced by Conley Fisheries At the start of any given day, the decision Clint Conley faces is which port to use for selling his daily catch. The price of codfish that the catch might command in Rockport is only known if and when the boat docks at the port and negotiates with buyers. After the boat docks at one of the two ports, it must sell its catch at that port or not at all, since it takes too much time to pilot the boat out of one port and power it all the way to the other port. Clint Conley is just as anxious as any other business person to earn a profit. For this reason, he wonders if the smart strategy might be to sell his daily catch in Rock- port. After all, the expected price of codfish is higher in Rockport, and although the standard deviation of the price is high, and hence there is greater risk with this strat- egy, he is not averse to taking chances when they make good sense. However, it also might be true that the smart strategy could be to sell the codfish in Gloucester, since in Gloucester there is ample demand for his daily catch, whereas in Rockport there is the possibility that he might not sell all of his catch (and so potentially lose valu- able revenue). I is not clear to him which strategy is best One can start to analyze this problem by computing the daily earnings if Clint chooses to sell his daily catch of codfish in Gloucester. The earnings from using Gloucester, denoted by G, is simply: with this strat- G (3.25) (3,500)-$10,000- $1,375 which is the revenue of $3.25 per pound times the number of pounds of codfish (3,500 lbs.) minus the daily operating costs of $10,000. The computation of daily earnings if Clint chooses Rockport is not so straight- forward, because the price and the demand are each uncertain. Therefore the daily earnings from choosing Rockport is an uncertain quantity, i.e, a random variable. In order to make an informed decision as to which port to use, it would be helpful to an- swer such questions as: TABLE 5.1 ibs of codfish) Probability Daily demand in Rockport faced by Conley Fisheries. 0.02 1000 2,000 3,000 4,000 5,000 0.08 0.33 0.29 0.20

i) What is the probability that G is negative? What is the probability that R is negative?

Answer 1

Sol:

Earnings from Gloucester, G = Profit - Expenses

= Demand * Price - Expenses

= 3500 * $3.25 - $10,000

= $1375

G is constant variable as the demand, price and daily expenses are constant for Gloucester.

Thus, Expected value of G = E[G] = E[$1375] = $1375

Variance of G = Var[G] = Var[$1375] = 0 (Variance of constant is 0)

Thus, standard deviation of G is 0.

Earnings from Rockport, R = Profit - Expenses

= Demand * Price - Expenses

Now, Price ~ Normal($\mu$ = $3.65 per lb, $\sigma$ = $0.20 per lb)

Expenses = $10,000

Based on probability distribution of Demand,

Expected value of Demand = E[Demand] = 0.02 * 0 + 0.03 * 1000 + 0.05 * 2000 + 0.08 * 3000 + 0.33 * 4000 + 0.29 * 5000 + 0.20 * 6000 = 4340

E[Demand²] = 0.02 * 0² + 0.03 * 1000² + 0.05 * 2000² + 0.08 * 3000² + 0.33 * 4000² + 0.29 * 5000² + 0.20 * 6000² = 20680000

Variance of Demand = Var[Demand] = E[Demand²] - E[Demand]² = 20680000 - 4340² = 1844400

R =  Demand * Price - Expenses

Expected value of R = E[R] = E[Demand * Price - Expenses] = E[Demand * Price] - E[Expenses]

= E[Demand] * E[Price] - E[Expenses] (Demand and price are independent)

= 4340 * $3.65 - $10,000

= $5841

Variance of R = Var[R] = Var[Demand * Price - Expenses] = Var[Demand * Price] + Var[Expenses]

= Var[Demand] * Var[Price] + Var[Demand] E[Price]² + Var[Price] E[Demand]² + Var[Expenses]

(Demand and price are independent and Var(XY)= Var(X)Var(Y)+Var(X)(E(Y))²+Var(Y)(E(X))²)

= 1844400 * 0.20² + 1844400 * 3.65² + 0.20² * 4340² + Var[10,000]

= 1844400 * 0.20² + 1844400 * 3.65² + 0.20² * 4340² + 0 (Variance of constant is 0)

= $25399219

Standard deviation of R = = $5039.764

25399219