Sandy Sprunger is part owner in one of the largest quick-oil-change operations for a medium-sized city in the Midwest. Currently, the firm has 60% of the market. There are a total of 10 quick lubrication shops in the area. After performing some basic marketing research, Sandy has been able to capture the initial probabilities, or market shares, along with the matrix of transition, which represents probabilities that customers will switch from one quick lubrication shop to another. These values are shown in the table on the shown below: Initial probabilities, or market share, for shops 1 through 10 are 0.6, 0.1, 0.1, 0.1, 0.05, 0.01, 0.01, 0.01, 0.01, and 0.01.

(a) Given these data, determine market shares for the next period for each of the 10 shops.

(b) What are the equilibrium market shares?

(c) Sandy believes that the original estimates for market shares were wrong. She believes that shop 1 has 40% of the market, and shop 2 has 30%. All other values are the same. If this is the case, what is the impact on market shares for next-period and equilibrium shares?

(d) A marketing consultant believes that shop 1 has tremendous appeal. She believes that this shop will retain 99% of its current market share; 1% may switch to shop 2. If the consultant is correct, will shop 1 have 90% of the market in the long run?

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To 10 0.0 0.0 0.0 0.0 0.01 0.10 0.0 0.0 0.01 0.03 0.10 0.0 0.05 0.05 0.05 0.0 0.0 0.0 0.01 0.02 0.0 0.030.0 0.0 0.01 0.91 0.0 0.0 0.01 0.01 0.10 0.70 0.0 0.10 0.04 0.10 0.03 0.80 0.01 0.01 0.100.00.10 0.70 0.04 0.10 0.10 0.05 0.00 0.70 FROM 4 9 0.60 0.10 0.10 0.10 0.05 0.01 0.80 0.0 0.0 0.01 0.0 0.0 0.70 0.0 0.01 0.01 0.0 0.0 0.90 0.01 0.0 0.0 0.0 0.10 080 0.0 0.0 0.0 0.0 0.01 0.01 0.01 0.0 00 0.01 0.01 0.0 0.0 0.00.01 0.01 0.01 0.0 0.0 0.01 0.0 0.0 0.0 0.0 0.01 4 6 10