Data from customer reward cards at a major grocery store chain show that of all customers who purchase Coke in their weekly grocery shopping, 90% will purchase it again the next week while 10% will purchase Pepsi instead. Of all customers who purchase Pepsi one week, 80% will purchase it again the next week and 20% will purchase Coke.
Assuming that the probability that any given customer purchases Coke (or Pepsi) in any week depends only on their purchase in the previous week, model a customer’s weekly Coke and Pepsi purchasing as a Markov chain and answer the following questions.
a) What are the states of Markov chain?
b) Determine the one-step transition matrix, P, for the Markov chain.
c) If a customer purchases Pepsi this week, what is the probability that she will purchase Coke two weeks from now?
d) Write out and solve the equations for the steady-state probabilities:
and .
e) Based on your answer to part d), what will Coke’s market share be one year from now?
f) Write out and solve the equations for the expected first passage times:
g) For the customers who purchased Pepsi last week, what is the average number of weeks in a row that they will purchase Pepsi before switching to Coke? Explain your answer.