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Question

ome File view History Bookmarks People Window He Edit 86% https://invent ?C. Find X2 (the l ? Welcome, Jal × ..-, MindTap . Ce x v G A Study Conc x × % Assignment i × X H could you te: × Jake . c ? Secure https://www.webassign.net/web/Student/Assignment-Responses/submit?dep-18382341 JT3601 Free Listen." Compass | Moodie Gmail MU of IMail r Linkedin Handshake D UIUC Enterprise 11 Netflix ? Thesaurus Need Help?Warich Mastor ?1.11 points TanFin12 9.2.026 CMI. My Notes Ask Your In a study of the domestic market share of the three major automobile manufacturers 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 and car manufactured by C. Of the customers who bought a car manufactured by B, 50% would again buy a car manufactured by 8, whereas 25% each would buy cars manufactured by A and C. Finally, of the customers who bought a car manufactured by C, 85% would again buy a car manufactured 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 customers in the future model years, determine the market share that will be held by each manufacturer in the long run I -41.11 points TanFin12 9.3.022 CMI Notes Ask Your T Compute the steady-state matrix of the stochastic matrix. 10.2.1 0 1 4 .2 0 0 0.4 0 0 4 .3

Explanation / Answer

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]
[0.25, 0.50, 0.25]
[0.05, 0.10, 0.85].

Since there are two model years involved, compute T2 = T^2.
T2 =
[0.605, 0.1975, 0.1975]
[0.325, 0.3125, 0.3625]
[0.1050, 0.1425, 0.7525].

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
= [0.4428, 0.21465, 0.34255].

In other words, the respective market shares after two years will be about 44%, 22%, and 34%.

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