A consumer is trying to decide between two long-distance calling plans.

The first one charges a flat rate of 10¢ per minute, whereas the second charges a flat rate of 99¢ for calls up to 20 minutes in duration and then 10¢ for each additional minute exceeding 20 (assume that the calls lasting a noninteger number of minutes are charged proportionately to a whole-minute’s charge). Suppose the consumer’s distribution of call duration is exponential with parameter λ.
(a) Explain intuitively how the choice of calling plan should depend on what the expected call duration is.
If a consumer’s calls are typically short, the [(a)Select— first or second plan] makes more sense. If a consumer’s calls are somewhat longer, then the [(b)Select— first or second plan] makes more sense.

(b) Which plan is better if expected call duration is 9 minutes? 19 minutes? [Hint: Let h1(x) denote the cost for the first plan when call duration is x minutes and let h2(x) be the cost function for the second plan. Give expressions for these two cost functions, and then determine the expected cost for each plan.]
If the expected call duration is 9 minutes, then the [(c)Select— first or second plan] is better. If the expected call duration is 19 minutes, then the [(d)Select— first or second plan] is better.
a ?
b ?
c ?
d ?

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