Question

2. Set up both the vector of state probabilities and the matrix of transition probabilities given the following information: Store 1 currently has 40% of the market; store 2 currently has 60% of the market. In each period, store 1 customers have an 80% chance of returning; 20% of switching to store 2.

In each period, store 2 customers have a 90% chance of returning; 10% of switching to store 1. a.

Find the percentage of market for each store after 2 periods. b. Find the equilibrium conditions of 2 stores (limiting probabilities). What’s the meaning of these probabilities?

Answer 1

(a) The R code is:

mat1 <- matrix(c(0.4,0.6), ncol = 2, nrow = 1)
mat1 <- as.matrix(mat1)
mat2 <- as.matrix(matrix(c(0.8, 0.9, 0.2, 0.1), ncol = 2, nrow = 2))
mat2 <- as.matrix(mat2)
i = 0
while(i < 2)
{
mat3 = mat1%*%mat2
mat1 = mat3
i = i + 1
}
mat3

The percentage of market for store 1 after 2 periods = 81.4%

The percentage of market for store 2 after 2 periods = 18.6%

(b) The R code is:

mat1 <- matrix(c(0.4,0.6), ncol = 2, nrow = 1)
mat1 <- as.matrix(mat1)
mat2 <- as.matrix(matrix(c(0.8, 0.9, 0.2, 0.1), ncol = 2, nrow = 2))
mat2 <- as.matrix(mat2)
mat3 = mat1%*%mat2
while(abs(mat1 - mat3) > 0.0001)
{
mat3 = mat1%*%mat2
mat3=mat1+mat3
mat1=mat3-mat1
mat3=mat3-mat1
}
mat3

The limiting probability of market for store 1 = 81.814%

The limiting probability of market for store 2 = 18.186%

It means that market for store 1 and store 2 will be respectively 81.814% and 18.816% after long period of time