Page 118 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

Thus x = 1,10,000 and y = 60,000. Notes
Since 90% shares of A and 80% shares of B are allocated to outside shareholders, we can write the
vector S = [0.9 0.8].

Let P = [x y] = [1,10,000 60,000] be vector of profits of the two companies A and B. Thus, the total
profits allocated to the outside shareholders is the scalar product of S and P.
S.P = 0.9 × 1,10,000 + 0.8 × 60,000 = 1,47,000.

Note that this is equal to 98,000 + 49,000 which is the total of separately earned profits.
Markov Brand-Switching Model

Let there be only two brands, A and B, of a toilet soap available in the market. Let the current
market share of brand A be 60% and that of B be 40%. We assume that brand-switching takes
place every month such that 70% of the consumers of brand A continue to use it while remaining
30% switch to brand B. Similarly, 80% of the consumers of brand B continue to use it while
remaining 20% switch to brand A.
The market shares of the two brands can be written as a row vector, S = [0.6 0.4] and the given
brand switching information can be written as a matrix P of transition probabilities,

A B
A 0.7 0.3
P = .
B 0.2 0.8
Given the current information, we can calculate the shares of the two brands after, say, one
month or two months or ...... n months. For example, the shares of the two brands after one
month is

0.7 0.3
S(1) = 0.6 0.4 0.5 0.5
0.2 0.8
Similarly, the shares after the expiry of two months are given by

S(2) = S(1)×P
Proceeding in a similar manner, we can write the shares of the two brands after the expiry of
n months as

S(n) = S(n – 1)×P
= S(n – 2)×P×P = S(n – 2)×P 2
....................................
n–1
= S(1)×P = S(0)×P n
where S(0) = [0.6 0.4] denotes the current market share vector of the two brands.
We note that as n ® ¥, the market shares of the two brands will tend to stabilize to an equilibrium
position. Once this state is reached, the shares of the two brands become constant. Eventually,
we have S(n) = S(n – 1). Thus, we can write S = S.P, where S = [s s ] is the vector giving the
A B
equilibrium shares of the two brands.
The above equation can also be written as
S[I – P] = 0

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