Three FM radio stations (A, B, and C) are competing for customers in the same ma

Question

Three FM radio stations (A, B, and C) are competing for customers in the same market area. Through an aggressive advertising campaign, each month station A s capturing 896 of station B's customers and 5% of station C's customers. Hint: The phrase station As capturing 8 of station B's customers means that 89 of the listeners are moving from B to A, not the other way around.) At the same time, station A is losing only 2% of its customers to B and 2% of its customers to C. Additionally, each month 1% of B's customers switch to C, and 1% of C's customers switch to B. Initially, A and B each have 30% of the listeners and C has 40%. Use this information to answer the following questions. (Give your answers correct to three decimal places. Note that you are asked to give decimal values and not percentages. For example, you would need to enter 0.628 as opposed to 62.800%.) (a) What fraction of the listeners will A have after 3 months? 0.06 What fraction of the listeners will B have after 3 months? 0.06 What fraction of the listeners will C have after 3 months? 0.06 (b) In the long run, what fraction of the listeners will A have? 0.06 In the long run, what fraction of the listeners will B have? 0.06 In the long run, what fraction of the listeners will C have? 0.06

Explanation / Answer

Initially,A=B=30% , C=40%

A= A+8% B + 5%C - 2%A - 2%A= (30 + 2.4 + 2- 0.6 - 0.6)% = 33.2%

B =B +2%A + 1% C-1%oB- 8% B = (30 + 0.6 + 0.4 - 0.3 - 2.4)% = 28.3%

C=C+2%A + 1%oB - 1% C - 5%C = (40 + 0.6 + 0.3 - 0.4 - 2)% = 38.5%

a) In one month A's listeners increase by 6.6% from the initial value

Number of listeners A will have in 3 month = 106.6 * 106.6 * 106.6 * 30% / 100 * 100 *100 = 36.34% = 0.363

In one month B's listeners decrease by 5.6% from the initial value

Number of listeners A will have in 3 month = 94.4 * 94.4 * 94.4 * 30 % / 100 * 100 *100 = 25.23 % = 0.252

In one month C's listeners decrease by 3.75% from the initial value

Number of listeners A will have in 3 month = 96.25 * 96.25 * 96.25 * 40 % / 100 * 100 *100 = 35.66 % = 0.356

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