A certain grocery store sells two types of apples Ambrosia a

A certain grocery store sells two types of apples: Ambrosia and Braeburn. Every week, 20% of the people who bought Ambrosia last week switch to Brae-burn and 80% buy Ambrosia again. Similarly, 30% of customers who bought Braeburn last week switch to Ambrosia and the remaining 70% buy Braeburn again. In the long run, what will the share of each type of apple be? (find the steady state vector).

Solution

The stochastic matrix for the described matter is A =

0.8

0.3

0.2

0.7

The steady state vector X = (x,y)^T is given by AX = X or, (A-I₂)X = 0. The RREF of A-I₂ is

1

-1.5

0

0

Thus, the equation (A-I₂)X = 0 is equivalent to x-1.5 = 0 or, x = 1.5y. Then = (1.5y, y)^T = y(1.5,1)^T. Hence, the steady state vector is (1.5,1)^T.

0.8	0.3
0.2	0.7