Road Toll Optimisation and the Rule of a Half

Road Toll Optimisation and the Rule of a Half


G Hyman, Department for Transport, UK



All methods for computing optimum road user charges seek to maximise some measure of social welfare. Some methods explicitly compute the change in welfare, others rely entirely on first order conditions (e.g. marginal cost pricing). Occasionally, ?optimum? road user charges are reported without a full statement of the welfare measure, or an appraisal of the welfare gains. The motivation for this paper was to help to overcome this apparent ?blind spot? in such studies. While the focus will initially be on outlining how to implement algorithms and illustrating typical results, the need for explicit, and detailed, welfare calculations will become apparent when more difficult problems are addressed.

Algorithms that are based purely on first order conditions are potentially flawed, because such conditions are not sufficient to ensure a welfare maximum. Some first order conditions may also be violated when a constrained optimum lies on the boundary of the feasible region. However, under suitable conditions, both approaches should yield the same results. Both methods require iterative solution, so there may be little to choose between them on computational grounds. Apart from these considerations, explicit welfare optimisation offers practical advantages: the adaptability of the resulting algorithms, a strong link with economic appraisal and a basis for conducting sensitivity analysis. The main practical problem is to compute the welfare measure itself, but this is not as difficult as it might seem initially.

A suitable measure of welfare is the difference between the total willingness to pay for travel and the social cost of the trips that are made. The social cost of travel includes the cost of road congestion and the variable costs of operating vehicles. In a European context, an appropriate means of treating fuel duties is also required. The willingness to pay for a single unit of travel is the traveller?s maximum bid price for that unit, often expressed as the inverse of a travel demand function. The total willingness to pay for all of the units of travel is an integral of the inverse of the travel demand function. A fully analytic solution for the optimum road user charges would require a solution for this integral. However, the travel demand function cannot always be inverted. Even when it can be inverted, the willingness to pay integral is not always expressible in closed form. Hence numerical optimisation methods are needed in most practical applications.

A simple algorithm is described, based on a well-established approximation, known as the rule of a half. This approximation is accurate to second order and provides an objective function that is suitable for local optimisation (i.e once the solution is approached sufficiently closely). It will be demonstrated that the rule of a half can be applied in a numerically accurate algorithm for calculating optimum charge levels. These methods apply to a wide class of optimisation problem e.g. to problems for which the marginal social cost is not known analytically and to overcome problems where simple marginal cost pricing methods are inconsistent with welfare optimisation.

The project has been conducted in conjunction with a more detailed study, using outputs from the National Transport Model for Great Britain, which is used for national road traffic (and multimodal) forecasting. The illustrations quoted here are of a generic nature and are intended to assess the overall plausibility of more specific results.

Quantitative illustrations are provided, for a broad range of road types and conditions, of the relationships between optimal charges, traffic intensities and welfare benefit levels per km. Suboptimal charging policies are investigated, in terms of the welfare gains that can obtained from alternative charging strategies. Situations under which simple, but frequently used, marginal cost pricing criteria do not result in welfare maximising road user charges are identified. The resulting estimates of road user charges and welfare gains are contrasted with equivalent optimal estimates. Full analytical conditions for the maximisation of welfare are provided. Operational tests are described for the violation of these conditions which can provide the basis for improved charging strategies.


Association for European Transport