please explain each step of this problem 8. In a particular city, cellphones are

Question

please explain each step of this problem

8. In a particular city, cellphones are sold only three companies A, and C. Every year 15% of A's customers go to B, 10% of A's customes go to 20% of B's customers go to A, 30% of B's customers go to C, 10% of C's customers go to A, and 22% of C's customers go to B. At the end of 2014, A had three times the market share of C and B had six times the market share of C. Assuming this trend persists, find: 7S 2o 12 IS SO 0 a) The market distribution by the end of 2015. 3S 7 27p b) The final stable market distribution. 1 31 24 349 3

Explanation / Answer

Market share ratio is 3:6:1

3C+6C+1=1

10C=1

C=0.10

A= [.75    0.20     0.12]                        X= [0.30]

     [.15      0.50     0.22]                            [ 0.60]

     [.10      0.30      0.66]                           [0.10]

We are given that by the end of 2014 the market share is of ratio 3:6:1

hence we calculated the maket share of C= 0.10 , B=0.60 and A is 0.30 by which we calculate the matrix X of market share.

Now we are given information about the % of customers which goes from one company to another

we prepare the matrix A such that the 1st column gives the % of customers remining with A,B and C

so after 15% customer who goes to B and 10% customers who goes to C, A is left with 75% customers.

Similarly for column second and third.

a) Market distribution by end of 2015 is

% of customers * market share of 2014

A*X

[.75    0.20     0.12]              *          [0.30]

[.15      0.50     0.22]                         [ 0.60]

[.10      0.30      0.66]                         [0.10]

X'=[0.357]

[0.367]

[0.276]

b) Final stable market distribution

=AX'

=[0.38765]

[0.29777]

[0.32796]

we get this value from A Matrix being the same but with new market distribution matrix.

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