Question 4 There is a tough competition between three stores, Dree Rite, Fashion Inc and Luxury Living when it comes to luxury goods. Not all customers are loyal to the respective stores. Over any given month, Dree Rite can retain only 70% of its customers. Dree Rite loses 10% of its customers to Fashion Inc, and 20% of its market to Luxury Living. Fashion Inc is able to retain 85% of its customers. Fashion Inc loses 5% of its market to Dree Rite and 10% of its market to Luxury Living each month. Luxury Living is able to maintain 90% of its customers. Luxury Living loses 5% of its market to Fashion Inc and 5% of its market to Dree Rite. In the month of August, each of these clothing stores has an equal share of the market.

(a) Determine the transition probability matrix. (5 marks)

(b) Write the current market share vector. (2 marks)

(c) What will be the market shares of the three stores in September? (10 marks)

(d) What will be the market shares of the three stores in October? (8 marks)