Alternative marginal-cost pricing for road networks

In the literature, several studies have algebraically characterized the set of toll vectors or patterns that, when added to a user equilibrium problem, its solution would be system optimal. Toll vectors in this set are termed “valid.” While the toll vector commonly advocated in the literature, i.e., one that equates the toll on each link to its marginal external cost, is always valid, other valid toll vectors generally exist and many leave some utilized links in the network untolled. On the surface, this may appear unreasonable and seems to violate the principle of marginal-cost pricing. This note shows that, when travel demands are elastic, all valid toll vectors satisfy this principle, in that the total tolls for each path equals the congestion externality an additional traveler on the path imposes on others.