Within a certain market area there are two brands of soap th

Within a certain market area, there are two brands of soap that most people use: \"super soap\" and \"cheap soap\", with the current market split evenly between two brands. A company is considering introducing a third brand called \"extra clean soap\" and they have done some initial studies of market conditions. The estimates of weekly shopping patterns are as follows: If a customer buys super soap this week, there is a 75% chance that next week the super soap will be used again, a 10% chance that extra clean will be used, and a 15% chance that the cheap soap will be used. If a customer buys the extra clean this week, there is a 50-50 chance the customer will switch, and if a switch is made it will only be to super soap. If a customer buys cheap soap this week, it is equally likely that the next week the customer will buy any of the three brands. (a) Give the Markov matrix for the above data, where Xn is the type of soap used in week n. (b) What is the long-run market share for the new soap? (c) What will be the market share for the new soap two weeks after it is introduced? (d) The market consists of approximately 1 million customers each week. Each purchase of super, cheap and extra clean soap yield a profit of 15, 10, and 25 cents respectively. Assume the market was at steady state with an even split between the 2 products. The initial advertising to introduce the new product was $100,000. How many weeks will it be until the $100,000 is recovered from the added revenue of the new product? (explain your answer)

Solution

Expected purchase of new soap in 2 weeks = 0.10+0.333 = 0.433

a) Long run market share for new soap = 0.2165 per week

Extra cost incurred = 100000

Profit increase due to new soap in 2 weeks =

1.583(15)+0.833(10)+0.433(25)

= 23.745+8.33+10.825

= 42.9

For 1000 to be adjusted no of weeks = 1000x2/42.9 = 46.62

or 47 weeks.