QUESTION $7$:

Suppose there are three travel modes $-$ automobile, bus and light rail. A calibrated utility function for each mode is as follows:

$V_a=-0\mathrm{.}30-0\mathrm{.}002**T{C}_a-0\mathrm{.}05**T{T}_a$

$V_b=-0\mathrm{.}35-0\mathrm{.}002**T{C}_b-0\mathrm{.}05**T{T}_b$

$V_r=-0\mathrm{.}40-0\mathrm{.}002**T{C}_r-0\mathrm{.}05**T{T}_r$

where $V_a,V_b$ and $V_r=$ observable utilities for auto, bus and light rail, respectively; TC $,T{C}_b$ and $T{C}_r=$ cost of travel $($cents$)$ for auto, bus and light rail, respectively; $T{T}_a,T{T}_b$ and $T{T}_r=$ travel time $($minutes$)$ for auto, bus and light rail, respectively. The travel cost and travel time for each mode is shown below.

$\backslash$table$[[$Mode$,$Travel cost $($cents$),$Travel time $($minutes$)],[$Automobile$,200,30],[$Bus$,100,40],[$Light rail,$150,45]]$

$($a$)$ Calculate the share of each mode $($i$.$e$.$ modal split$)$ using the multinomial logit model.

$($b$)$ In the part $($a$),$ if the fare of light rail is reduced to $75$ cents, what would be the share of each mode?

$($c$)$ Since both bus and light rail are public transportation modes, the choice of these two modes is likely to be correlated. Explain how the multinomial logit model will yield unrealistic results of modal split due to this correlation in part $($b$).$