A bathtub model of downtown traffic congestion

In standard economic models of traffic congestion, traffic flow does not fall under heavily congested conditions. But this is counter to experience, especially in the downtown areas of major cities during rush hour. This paper analyzes a bathtub model of downtown rush-hour traffic congestion that builds on ideas put forward by William Vickrey. Water flowing into the bathtub corresponds to cars entering the traffic stream, water flowing out of the bathtub to cars exiting from it, and the height of water in the bathtub to traffic density. Velocity is negatively related to density, and outflow is proportional to the product of density and velocity. Above a critical density, outflow falls as density increases (traffic jam situations). When demand is high relative to capacity, applying an optimal time-varying toll generates benefits that may be considerably larger than those obtained from standard models and that exceed the toll revenue collected.