Consider an open-access highway of fixed length and capacity, which is nonrival only for sufficiently low traffic volumes. Assume the highway is used exclusively by identical commuters who make either one or zero daily round trips on the highway, depending on whether they obtain a net benefit from taking a round trip. Assuming there is one commuter per vehicle, the daily number of round trips, vehicles and commuters on the highway are equal. A commuter's travel time per round trip is given by

T(v) = 1 if v vc

(v/vc)^? if v > vc

in hours, where 0 is traffic volume on the highway in the number of vehicles per day, > 0 is the critical traffic volume at which congestion sets in and > 0 is a congestion parameter. Due to foregone opportunities, fuel consumption, vehicular wear and tear etc., using the highway costs a commuter > 0 dollars per hour of travel. Therefore, a commuter's total cost of using the highway is () = () dollars per round trip when the traffic volume is . With the highway providing access to employment, the benefit to a commuter of using the highway is dollars per round trip, where it is assumed that > c.

(i.e. that using the highway is worthwhile to a commuter if congestion is absent or insufficiently high). Let e and * denote the equilibrium and efficient traffic volumes, respectively.

How to derive the solution for in terms of the model's parameters, and verify that < ?

How to derive the solution for * in terms of the model's parameters, and verify that * < ?

How to very briefly explain why the highway is overused (i.e. excessively congested) in equilibrium?

Suppose the highway authority levies a toll of dollars per vehicle per round trip. How to derive the solution for the optimal toll, * , in terms of the model's parameters?