Question

Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

Answer 1

A and B are the two types of soaps.

The current market shares are 15% and 85%.

This is represented as

(A B)

(0.15 0.85)

(A A) ⇒ those who bought A previous year will again buy A = 65 % = 0.65

(A B) ⇒ those who bought A previous year will buy soap B now = 35 % = 0.35

(B A) ⇒ those who bought B previous year will buy A now = 45 % = 0.45

(B B) ⇒ those who bought B previous year will buy it again = 55 % = 0.55

The transition probability matrix is

\[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T=\begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.65&0.35\\0.45&0.55\end{pmatrix}\\ \end{matrix}}\]

(i) Their market shares after one year is given by

`((0.15, 0.85)) ((0.65, 0.35),(0.45, 0.55)) = ((0.0975 + 0.3825, 0.0525 + 0.4675))`

= `(("A", B"),(0.48, 0.52))`

i.e. A = 0.48 = 48%

B = 0.52 = 52%

So after one-year market shares of soap A will be 48% and soap B will be 52%

(ii) At equilibrium, A + B = 1

We have `(("A", "B")) ((0.65, 0.35),(0.45, 0.55)) = (("A", "B"))`

By matrix multiplication,

(0.65A + 0.45B 0.35A + 0.55B) = (A B)

Equating the corresponding elements,

0.65A + 0.45B = A

0.65A + 0.45A(1 – A) = A  ......(Using A + B = 1)

0.65A + 0.45 – 0.45A = A

0.45 = 0.8A

A = 0.5625 or A = 56.25 %

B = 100 – 56.25 = 43.75%

The equilibrium is reached when the market share of soap A is 56.25% and the market share of soap B is 43.75%