Transportation network with externalities

In a transportation network comprised of parallel routes with linear latency functions, we study how externalities among different routes affect the socially optimal allocation and the equilibrium allocation of traffic flows. Assuming that the externalities are not too severe, we analytically derive a system of equations that define the optimal distribution of the traffic flow with minimum social cost. We also solve for the Wardrop equilibrium without route charges in which every commuter minimizes the travel time. Furthermore, we characterize the Wardrop equilibrium with route charges in which two carriers set prices to maximize their own profits and commuters choose which carrier's service to use in order to minimize the sum of monetary cost (service price paid) and waiting cost (travel time spent). Our results show that Wardrop's first principle (user equilibrium) remains valid with mild externality level, and his second principle (social optimum), which usually fails to hold in equilibrium in a standard setup without externalities, can be achieved together with the first principle under some externality conditions, leading to the price of anarchy equal to 1. We also show that when the traffic flow goes to infinity, the price of anarchy can still be greater than 1 with the existence of externalities. © 2020 Elsevier B.V.