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 commuters travel time per round trip is given by

() = { 1 _c

( / ) > v_c

in hours, where 0 is traffic volume on the highway in number of vehicles per day, _c > 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 commuters 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.

a. Derive the solution for in terms of the models parameters, and verify that _c < _e .

b. Derive the solution for * in terms of the models parameters, and verify that < ^e .

c. Very briefly explain why the highway is overused (i.e. excessively congested) in equilibrium.