Yu, X., van den Berg, VincentA.C. and Verhoef, ErikT. (2025). Preference heterogeneity in a dynamic flow congestion model Transportation Research Part B: Methodological, 195:1--23.

The bottleneck model has dominated the field on dynamic congestion, as it is tractable and has closed-form solutions. But it has rather specific outcomes that do not occur in most other dynamic models or under static congestion. In the bottleneck model, pricing can remove all travel delays while keeping the generalized price unchanged, making pricing much more beneficial than in other models. Thus, it is important to see how pricing performs in other models. We do so by adding discrete preference heterogeneity to a dynamic flow congestion model, and we study the efficiency and distributional effects of pricing. By formulating a multi-point optimal-control problem using Hamiltonians, we derive the social optimum and analyze the properties of travel equilibrium. Solving our model is more complex than solving the bottleneck model, but still results in closed-from solutions, whereas most other dynamic congestion models have no such solutions. Without 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 tolling cannot eliminate all travel time delays). In our model, users with a lower VOT always lose from tolling, while those with a higher VOT may gain or lose, depending on the parameters. Compared to the bottleneck model, tolling is less beneficial for society, lengthens the peak, and hurts users more. Our findings reveal the significance of considering congestion type and preference heterogeneity in assessing the implementation of congestion tolling.