math help!!

Question

If the current market share for each company is Chevy 45%, Ford 35% and Honda 20%, what is the market 
share for each company after the purchase of the next car?   

 

(i.e. find the state vector after one transition) 

Answer 1

To find the market share for each company after the purchase of the next car, we need to apply a transition matrix based on the current market shares. Let's represent the market shares as a vector:

S = [Chevy, Ford, Honda]

The initial market shares are given as:

S_initial = [45%, 35%, 20%] = [0.45, 0.35, 0.20]

Now, we need a transition matrix that represents the changes in market shares due to the purchase of the next car. Since we don't have specific information about the transition, I'll make a simple assumption. Let's say that each company gains or loses 1% market share with each car purchase, with the gains distributed proportionally based on their initial market shares. This would result in the following transition matrix:

T = | 0.99 0.01 0 |

| 0.01 0.99 0 |

| 0 0 1.00 |

Now, we can calculate the new market shares (S_new) by multiplying the initial market shares (S_initial) by the transition matrix (T):

S_new = S_initial * T

S_new = [0.45, 0.35, 0.20] * | 0.99 0.01 0 |

| 0.01 0.99 0 |

| 0 0 1.00 |

To perform the matrix multiplication:

S_new[Chevy] = (0.45 * 0.99) + (0.35 * 0.01) + (0.20 * 0) = 0.4455 + 0.0035 + 0 = 0.4490 (rounded to 4 decimal places)

S_new[Ford] = (0.45 * 0.01) + (0.35 * 0.99) + (0.20 * 0) = 0.0045 + 0.3465 + 0 = 0.3510 (rounded to 4 decimal places)

S_new[Honda] = (0.45 * 0) + (0.35 * 0) + (0.20 * 1.00) = 0 + 0 + 0.20 = 0.20

So, after the purchase of the next car, the market shares for each company would be approximately:

- Chevy: 44.90%

- Ford: 35.10%

- Honda: 20.00%