26. MARKET SHARt OF AUTo MANUFACTURERS In a study of the domestic market share o

Question

26. MARKET SHARt OF AUTo MANUFACTURERS In a study of the domestic market share of the three major automobile man- ufacturers A. B, and C in a certain country, it was found that of the customers who bought a car manufactured by A, 75% would again buy a car manufactured by A, 15% would buy a car manufactured by B. and 10% would buy a car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manu factured by B, whereas 5% each would buy cars manufac. tured by A and C. Finally, of the customers who bought a car manufactured by C. 85% would again buy a car manu- factured by C. 5%, would buy a car manufactured by A and 10% would buy a car manufactured by B. Assuming that these sentiments reflect the buying habits of custom- ers in the future model years, determine the market share that will he held by each manufacturer in the long run.

Explanation / Answer

I think the current market share in this problem is missing.

I have the similar type of question with the current market share:

Create a 3-by-3 (single-step) transition matrix, where the columns have the conditional probabilities of a customer buying a car manufactured by A, B, and C, respectively, and the rows correspond to A, B, and C, the respective manufacturer of the car currently owned by a customer. Call this matrix T.
T =
[0.75, 0.15, 0.10]
[0.05, 0.90, 0.05]
[0.05, 0.10, 0.85].

Now define Tn = T^n as the n-step transition matrix. Clearly, T1 = T^1 is identical to T. In the context of this problem, n is the number of model years.

Since there are two model years involved, compute T2 = T^2.
T2 =
[0.575, 0.2575, 0.1675]
[0.085, 0.8225, 0.0925]
[0.085, 0.1825, 0.7325].

Now let s0 = [0.57, 0.24, 0.19] be the current market shares of A, B, and C respectively. Then the corresponding market shares after two years, s2, can be determined by
s2 = s0 x T2
= [0.3643, 0.37885, 0.25685].

In other words, the respective market shares after two years will be about 36%, 38%, and 26%. (Of course, this problem doesn't present a very realistic model unless you believe people replace their cars every year.)

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