I have to develop a model to estimate the travel mode choice (car, public transport or bicyle) between two zones, based on variables like for example the car ownership, the travel time of the different modes, the percentage elderly people, the percentage of students, and the parking fee.

I have aggregated data about all these variables for all the zones. Also I have data about the observed %car, %Public Transport (PT) and %bicycle between all pairs of zones.

The question is now how I can estimate a model based on this data. I am not sure but I think the model has to look like as following:
$\displaystyle P_{car} = \frac{e^{V_{car}}}{e^{V_{car}}+e^{V_{bicyle}}+e^{V _{PT}} }$
$\displaystyle V_{car} = \alpha +\beta_1 * CarOwnership + \beta2*PercetageStudents +...$

(To estimate the % bicycle use and the % public transport the same type of model will be used)

My question is how I can estimate the values for alpha, beta1, beta2, beta_n?

Maybe I am using the wrong type of model, in that case please let me know! But I am not an expert in this field.