Road toll optimisation and the rule of a half.

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. 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 be 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. For the covering abstract see ITRD E126595.