An Analytic Assessment of Subsidies to Bus Services

A simple mathematical model has been constructed of a bus service, in a way which allows the requirements for maximum benefit, and the rate of return on subsidy, to be solved analytically. Although the model is very aggregate, it permits categorization of different passenger types and the inclusion of external (i.e., nonpassenger) benefits, and incorporates the effects of changing bus occupancy on speed, and therefore on operating cost, and on average passenger waiting time (because some buses may be full). The physical meaning of the rather complicated equations which result is explained and, where necessary, the assumptions are simplified to illustrate the often very simple trade-off between service cost and passenger benefit which occurs at maximum benefit. Equations are derived which describe how patronage, fares and the number of bus-kilometers operated change with the amount of subsidy provided, and how average bus occupancy, the elasticities of demand with respect to fares and service, and the amount of subsidy per additional passenger depend on the level of subsidy. A numerical example, typical of urban bus services in the United Kingdom, illustrates the model and shows the relative sizes of the contributions to cost or benefit made by the different components. The results of the model agree well with a cost/benefit assessment from a more sophisticated computer model, and the implications of using subsidy to reduce fares or improve service in a suboptimal way are investigated.