Solving for equilibrium in the basic bathtub model

Abstract

The basic (identical individuals) bathtub model has an unfamiliar mathematical structure, with a delay differential equation with an endogenous delay at its core. The early papers on the model circumvented this complication by making approximating assumptions, but without solution of the proper model it is unclear how accurate the results are. More recent work has either considered special cases that can be solved analytically using familiar methods, or has turned to generic computational solution. This paper develops a customized method for computational solution of equilibrium in the basic bathtub model with smooth preferences that exploits the mathematical structure of the problem. An inner loop solves numerically for the entry rate, conditional on the equilibrium utility level, by verifying a trip distance condition. An outer loop uses the computed start time from the inner loop to solve for the population that commutes over the rush hour, then lowers the equilibrium utility level to repeat the inner loop for a new level of utility. One result in that, even though tastes and the congestion technology are smooth, the entry rate and exit rate functions exhibit discontinuities at breakpoints. Another result is that, depending on the form of tastes and the congestion technology, the user cost curve as a function of population and may be backward bending.