Answer to Question #169000 in Linear Algebra for Bernice maina

Question

In a study of the domestic market share of three major automobile manufacturers A, B and C in a

certain country, it was found out 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 the rest

wouldbuyacarmanufacturedbyC.OfthecustomerswhoboughtacarmanufacturedbyB,90%

would again buy a car manufactured by B, 5% would buy a car manufactured by A and the rest

would buy a car manufactured by C. 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 the rest

would buy a car manufactured by B.

Required

The long run market share of the manufacturers (14 Marks)

Answer 1

We assume that the periods of use of cars of all manufacturers are the same. Then we can divide the time scale into segments of such a period. We also assume that all customers are the same in each period.

Let $a_n, b_n, c_n$ be the number of customers who, during the n-th period, buy/use automomobiles from the manufacturers A, B and C respectively. The conditions give us the relation between $a_n, b_n, c_n$ and $a_{n+1}, b_{n+1}, c_{n+1}$:

$a_{n+1}=0.75a_n+0.05b_n+0.05c_n$

$b_{n+1}=0.15a_n+0.90b_n+0.10c_n$

$c_{n+1}=0.10a_n+0.05b_n+0.85c_n$

In the long run, the market share of the manufacturers will stabilize. Hence we should suppose that $a_n\to a, b_n\to b, c_n\to c$. This gives us the system of linear equations.

$a=0.75a+0.05b+0.05c$

$b=0.15a+0.90b+0.10c$

$c=0.10a+0.05b+0.85c$

This system is equivalent to the follows

$-5a+b+c=0$

$3a-2b+2c=0$

$2a+b-3c=0$

The general solution of this system is

a=4t, b=13t, c=7t

Therefore, the shares of manufacturers in long term will be as follows:

if A then 4/24=16.67% ;

if B then 13/24=54.17%;

if C then 7/24=29.17% .