A large number of commuters decide to use either their cars or the bus. Commuting time by bus is increasing with the proportion of commuters using

A large number of commuters decide to use either their cars or the bus. Commuting time by bus is increasing with the proportion of commuters using car (traffic congestion). Commuting by car takes ????(????) = 20 + 60???? minutes while commuting by bus takes ????(????) = 45 + 15???? minutes, where N is the proportion of commuters by car, 0 ≤ ???? ≤ 1.

i) Plot the curves of the commuting time by car and the commuting time by bus as a function of the proportion of car users.

ii) What is the proportion of commuters who will take their car if everyone is taking his/her decision freely and independently to minimize his/her own commuting time?

iii) What is the proportion of car users that minimizes the total commuting time? (Hint: total commuting time is (20 + 60????)???? + (45 + 15????)(1 -????) and is minimized when 20 + 120???? - 30 - 30???? = 0).

iv) Compare your answer in (iii) to that in (ii) and interpret the difference. How large is the deadweight loss from externality?

v) Explain how a toll could achieve the efficient of commuters between train and car which will be beneficial to everyone.