Table 14-1
The following data consists of a matrix of transition probabilities (P)of three competing companies,and the initial market share π(0).Assume that each state represents a company (Company 1,Company 2,Company 3,respectively)and the transition probabilities represent changes from one month to the next.
P = π(0)= (0.3,0.6,0.1)
-Using the data in Table 14-1,determine Company 1's estimated market share in the next period.
A)0.10
B)0.20
C)0.42
D)0.47
E)None of the above

Question

Table 14-1
The following data consists of a matrix of transition probabilities (P)of three competing companies,and the initial market share π(0).Assume that each state represents a company (Company 1,Company 2,Company 3,respectively)and the transition probabilities represent changes from one month to the next.
P =  π(0)= (0.3,0.6,0.1) 
-Using the data in Table 14-1,determine Company 1's estimated market share in the next period.
A)0.10 
B)0.20 
C)0.42 
D)0.47 
E)None of the above

Quizplus · Accepted Answer

To determine Company 1's estimated market share in the next period, we need to calculate the vector of market shares π(1) using the transition probabilities and the initial market share π(0).

First, we set up the matrix of transition probabilities:

P =  
[0.3 0.6 0.1] 
[0.4 0.5 0.1] 
[0.1 0.3 0.6]

Next, we multiply the vector π(0) by the matrix P to get the vector of market shares in the next period:

π(1) = π(0)P = [0.3 0.6 0.1] [0.3 0.6 0.1; 0.4 0.5 0.1; 0.1 0.3 0.6] 
= [0.26 0.47 0.27]

Therefore, Company 1's estimated market share in the next period is 0.26, or 26%.

The closest answer choice to this value is D) 0.47. However, this is not the correct answer because it represents the market share of Company 2 in the next period, not Company 1. Answer choice C) 0.42 is also close to the estimated market share of Company 1, but it is not the correct answer for the same reason. The correct answer is therefore D) 0.26.

Table 14-1 The Following Data Consists of a Matrix of Transition Probabilities