Question

# The purchase patterns for two brands of toothpaste can be expressed as a Markov process with...

The purchase patterns for two brands of toothpaste can be expressed as a Markov process with the following transition probabilities:

 To From Special B MDA Special B 0.80 0.20 MDA 0.25 0.75
1. Which brand appears to have the most loyal customers? (MDA or special B)

Explain.
2. What are the projected market shares for the two brands? If required, round your answer to three decimal places. π1 =_ , π2 =_

a). Special B appears to have more loyal customers because it retains 80% of its customers and more customers are shifting from MDA to Special B (0.25) than from Special B to MDA (0.20).

b). Let q1 and q2 be the projected market shares of the Special B and MDA. Then,

q1 = 0.80q1 + 0.20q2 ----- (a)

q2 = 0.25q1 + 0.75q2 ----- (b)

q1 + q2 = 1 ---- (c)

Then, q1 = 1-q2

Substituting in (b), we get

q2 = 0.25(1-q2) + 0.75q2

q2 = 0.5

q1 = 1-q2 = 0.5

Projected market share for both brands is 50%, respectively.