In a study of the domestic market share of the three major automobile manufacturers A, B, and C in a certain country, it was found that their current market shares were 60%, 30%, and 10%, respectively. Furthermore, it was found that of the customers who bought a car manufactured by A, 75% would again buy a car manufactured by A, 15% would buy a car manufactured by B, and 10% would buy a car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manufactured by B, whereas 5% each would buy cars manufactured by A and C. Finally, of the customers who bought a car manufactured by C, 85% would again buy a car manufactured by C, 5% would buy a car manufactured by A, and 10% would buy a car manufactured by B. Assuming that these sentiments reflect the buying habits of customers in the future, determine the market share that will be held by each manufacturer after the next two model years
Present the state vector and the transition matrix for the above case.
To determine the market share held by each manufacturer after the next two model years, we can use the given information to calculate the future market shares iteratively.
Let's start with the current market shares:
Manufacturer A: 60%
Manufacturer B: 30%
Manufacturer C: 10%
After the first model year:
Manufacturer A:
- 75% of customers who bought a car from A will buy from A again.
- 15% will switch to B.
- 10% will switch to C.
Therefore, the market share for A after the first year will be:
0.75 * 60% + 0.05 * 30% + 0.10 * 10% = 48% + 1.5% + 1% = 50.5%
Manufacturer B:
- 90% of customers who bought a car from B will buy from B again.
- 5% will switch to A.
- 5% will switch to C.
Therefore, the market share for B after the first year will be:
0.90 * 30% + 0.05 * 60% + 0.05 * 10% = 27% + 3% + 0.5% = 30.5%
Manufacturer C:
- 85% of customers who bought a car from C will buy from C again.
- 5% will switch to A.
- 10% will switch to B.
Therefore, the market share for C after the first year will be:
0.85 * 10% + 0.05 * 60% + 0.10 * 30% = 8.5% + 3% + 3% = 14.5%
After the second model year, we repeat the calculations using the updated market shares:
Manufacturer A:
0.75 * 50.5% + 0.05 * 30.5% + 0.10 * 14.5% = 37.875% + 1.525% + 1.45% = 40.85%
Manufacturer B:
0.90 * 30.5% + 0.05 * 50.5% + 0.05 * 14.5% = 27.45% + 2.525% + 0.725% = 30.7%
Manufacturer C:
0.85 * 14.5% + 0.05 * 50.5% + 0.10 * 30.5% = 12.325% + 2.525% + 3.05% = 17.9%
After the second model year, the estimated market shares will be:
Manufacturer A: 40.85%
Manufacturer B: 30.7%
Manufacturer C: 17.9%
Please note that these calculations are based on the given assumptions and may not account for other factors that can influence market share changes, such as overall market growth or competition from other manufacturers.
Now, let's present the state vector and transition matrix for this scenario:
State vector:
S = [60% 30% 10%]
This represents the initial market share of each manufacturer.
Transition matrix:
T =
[0.75 0.05 0.10]
[0.05 0.90 0.05]
[0.05 0.05 0.85]
This matrix represents the probabilities of customers switching between manufacturers. For example, the element T[1][2] = 0.05 means that 5% of customers who bought a car from manufacturer A will switch to manufacturer B in the next model year. The diagonal elements of the matrix represent the probability of customers sticking with the same manufacturer.
To determine the market share that will be held by each manufacturer after the next two model years, we can use a matrix to represent the current market shares and the transition probabilities for customers switching between different manufacturers.
First, let's represent the current market shares:
- Manufacturer A: 60%
- Manufacturer B: 30%
- Manufacturer C: 10%
Next, let's create a transition matrix that represents the probabilities of customers switching between manufacturers based on the given information:
A B C
A 0.75 0.15 0.10
B 0.05 0.90 0.05
C 0.05 0.10 0.85
In this matrix, the row represents the current manufacturer and the column represents the manufacturer customers switch to.
Now, to determine the market shares after the next two model years, we need to apply this transition matrix twice.
For the first year:
1. Multiply the current market shares vector [60%, 30%, 10%] by the transition matrix.
2. The resulting vector will represent the new market shares after the first year.
For the second year:
3. Multiply the resulting vector from step 2 by the same transition matrix again.
4. The resulting vector will represent the final market shares after the second year.
Let's perform these calculations:
First-year market shares:
[60%, 30%, 10%] * [[0.75, 0.15, 0.10], [0.05, 0.90, 0.05], [0.05, 0.10, 0.85]] = [0.525, 0.365, 0.11]
Second-year market shares:
[0.525, 0.365, 0.11] * [[0.75, 0.15, 0.10], [0.05, 0.90, 0.05], [0.05, 0.10, 0.85]] = [0.44475, 0.40125, 0.154]
Therefore, after the next two model years, the estimated market shares will be approximately:
- Manufacturer A: 44.475%
- Manufacturer B: 40.125%
- Manufacturer C: 15.4%
Please note that these results are approximate and assume that the customer sentiments and buying habits remain constant over the two-year period. Additionally, keep in mind that market shares can also be influenced by other factors such as competition and market conditions.