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Sharp EL-W531TL Manual

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Matrix Calculation
In a certain year (year 0), the share of manufacturer A is 10% and the
share of manufacturer B is 90%. Manufacturer A then releases a new
product, and each following year it maintains 90% of the share ak it had
the previous year (year k), and usurps 20% of the share bk of
manufacturer B.
Find the transition matrix for this process and the shares of
manufacturers A and B after 2 years.
The share of each company after one year is expressed as follows using
a0 and b0.
Thus, a1 and b1 are
The transition matrix is
: This is equal to matA2.
Answer
0.9 0.2
0.1 0.8
A =
0.83 0.34
0.17 0.66
A
2
=
a1 = 0.9a0 + 0.2b0
b1 = (1-0.9)a0 + (1-0.2)b0
Expressing a2 and b2 using a0 and b0 gives
a2 = 0.9(0.9a0 + 0.2b0) + 0.2(0.1a0 + 0.8b0)
= (0.9 x 0.9 + 0.2 x 0.1)a0 + (0.9 x 0.2 + 0.2 x 0.8)b0
= 0.83a0 + 0.34b0
b2 = 0.1(0.9a0 + 0.2b0) + 0.8(0.1a0 + 0.8b0)
= (0.1 x 0.9 + 0.8 x 0.1)a0 + (0.1 x 0.2 + 0.8 x 0.8)b0
= 0.17a0 + 0.66b0
In the same way, after two years
a2 = 0.9a1 + 0.2b1
b2 = 0.1a1 + 0.8b1
a1 = 0.9a0 + 0.2b0
b1 = 0.1a0 + 0.8b0
In summary,
a2 = 0.83a0 + 0.34b0
b2 = 0.17a0 + 0.66b0
Manufacturer A
Share 10%
20%
10%
Manufacturer B
Share 90%
<Example>
(This example is for EL-W506T only.)
… 67


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