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
4 answers
Yeah sorry im not going to read all of that
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.
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.
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.