Question

41. Iwo competing soft-drink companies, lubCola and MajesticKola, presently have one third and two thirds of the market, respectively. Market research indicates that each week 92% of KlubCola drinkers buy that brand again and the remainder switch to MajesticKola. Also, 85% of MajesticKola drinkers buy Majestic again while the rest switch to KlubCola. a) What are the initial and transition probability matrices for this situation? b) What is the market share for each company after i) 1 week? Express your answers as percents accurate to one decimal place. ii) 2 weeks? c) Will the market shares reach a steady state? If so, what is it, and what does this result represent?

Answer for b is I) Klub: 50.7%

Majestic:59.3%

ii) klub : 46.3%

Majestic: 53.7%

C) klub = 15/23

Majestic = 8/23

Show steps for part b and c

Answer 1

a)

transition matrix,T is

	Klub cola	Majestic Kola
Klub cola	0.92	0.08
Majestic Kola	0.15	0.85

intital matrix [ 1/3 , 2/3 ]

b)

1) after 1 week=initial matrix*transition matrix =

1/3	*	0.92	0.08
2/3		0.15	0.85

= [ 1/3*0.92 + 2/3*0.15 , 1/3*0.08+2/3*0.85] = [ 0.40667 , 0.59333]

=[ 40.7% , 59.3%]

so,

Klub cola = 40.7%
Majestic Kola=59.3%

ii)

after 2 week=initial matrix*(transition matrix )²

=square of transition matrix is

	A1	A2
1	0.8584	0.1416
2	0.2655	0.7345

so, after 2 week =

1/3	*	0.8584	0.1416
2/3		0.2655	0.7345

=[0.4631,0.5367]

so,

Klub cola = 46.3%
Majestic Kola=53.7%

c)

let π1,π2 be steady state of Klub cola and Majestic

then π=π'T

π= [π1,π2] *[transition matrix]

π1=0.92π1+0.15π2

π1=1.875π2........1)

π2=0.08π1+0.85π2............2)

also,

π1+π2=1...........3)

solvine 1 , 2) and 3)

we, get π1=0.652174 or 15/23

π2=0.3478 or 8/23

so, in the long run,Probbaility that drinker will buy Klub cola again is 15/23

and majestic is 8/23