An ordinary differential equation formulation of the bottleneck model with user heterogeneity

•Study user heterogeneity in terms of values of time and schedule delay early.•Develop an ordinary differential equation formulation of the bottleneck model.•Derive the explicit condition for Pareto-improving by pricing scheme. Considering heterogeneous values of time and schedule delay early, in this paper we develop an ordinary differential equation formulation of the bottleneck model without allowance of arriving late. We show that in no-toll equilibrium, the generalized travel cost increases with departure time as the ratio of value of schedule delay to value of time increases with these two values, and commuters with higher values would experience higher generalized costs. We then derive the first-best toll and analyses its efficiency and distributional impacts. We obtain the sufficient Pareto-improving condition for the first-best scheme without imposing specific functional forms on heterogeneity. It is a Pareto improvement when the gap between values of time and schedule delay early increases as the ratio of value of schedule delay early to value of time increases. The proposed approach is applied to deal with such user heterogeneity that both the values of time and schedule delay early follow uniform distributions. In this case, the Pareto-improving condition is not only sufficient but also necessary. The individual gain from tolling is strictly monotonically increasing with respect to values of time and schedule delay early. Outside this condition, the first-best toll scheme makes users with relatively higher values of time and schedule delay early better off and those with relatively lower values worse off. And, users with higher values of time and schedule delay early could gain more or lose less from tolling. In contrasts with previous literature, our approach is robust in the sense that it can deal with not only discrete but also continuous user heterogeneity.