Preference heterogeneity in a dynamic flow congestion model

We study how preference heterogeneity affects travel behavior and congestion pricing in a dynamic flow congestion model. We formulate and solve a multi-point optimal control problem using a Hamiltonian-based method to derive the social optimum. The properties of the travel equilibrium are explored analytically, particularly for travelers' arrival rates, arrival intervals, congestion externalities, and tolls. In the absence of tolling, the arrival order is determined by the ratio of the value of time (VOT) to the value of schedule delay, as in the bottleneck model. However, unlike the bottleneck model, the same holds for the social optimum when only the VOT differs across users, as travel delays will not be fully eliminated. In social optimum, the arrival rate, travel delay, and toll jump discontinuously at the boundary time between user types, but these discontinuities do not undermine the stability of the socially optimal equilibrium. Assessment of the distributional effects indicates that users with a lower VOT always lose from tolling, whereas users with a higher VOT may gain or lose from tolling. The latter depends on the type and degree of heterogeneity, the elasticity of travel delay with respect to arrival rate, and the number of users for both types. Compared to the bottleneck model, tolling is less beneficial for society and hurts users more. Our findings reveal the significance of the type of congestion and preference heterogeneity when assessing the implementation of congestion tolling.