Social Optimum in the Basic Bathtub Model

The basic bathtub model extends Vickrey’s bottleneck model to admit hypercongestion (traffic jam situations). A fixed number of identical commuters travel a fixed distance over a dense network of identical city streets between home and work in the early morning rush hour under dynamic macroscopic fundamental diagram congestion. This paper investigates social optima in the basic bathtub model and contrasts them with the corresponding competitive equilibria. The model gives rise to delay-differential equations, which considerably complicate analysis of the solution properties and design of computational solution algorithms. This paper considers the cases of smooth and strictly concave travel utility functions and of α–β–γ tastes. For each it develops a customized solution algorithm, which it applies to several examples, and for α–β–γ tastes, it derives analytical properties as well. Departures may occur continuously, in departure masses, or a mix of the two. Additionally, hypercongestion may occur in the social optimum. This paper explores how these qualitative solution properties are related to tastes.