Optimal Toll Rate and Expansion of Urban Expressway

A welfare economic approach is tried to an optimal decision of toll rate and expansion of urban expressway network in an equilibrium of toll revenues and cost of service supplied. The model, originated with Yamada, is such that the decision comes into optimality when the maximum consumers' surplus is reached in the equilibrium condition. The paper is concerned with some general aspects of the optimal solution and reexamination of the solution obtained in the past when used a specific demand curve. General aspects obtained are as follows; The extremum condition to consumers' surplus is equivalent to that to diverted traffic (the realized number of expressway users) only when demand curve has such a property that the marginal consumers' surplus to network expansion vanishes. In case that the marginal consumers' surplus does not vanish, the extrema of consumers' surplus is found in the regions of negative marginal diverted traffic if demand curve yields positive marginal surplus, and in the regions of the positive if it gives negative marginal surplus. The contact points of demand and average cost curves give extrema of neither consumers' surplus nor diverted traffic. An implicative finding, made out by Yamada by using a specific demand curve, that optimal network expansion is reached when the marginal service cost to expansion averaged to the marginal diverted traffic to expansion is equal to the value of time saved by using expressway just by mean trip length holds good at the points of the maximum diverted traffic, but not at the contacts of demand and average cost curves. In case of the demand curve, the condition for an equilibrium of revenues and cost to come into existence is that the minimum of the ratio of service cost averaged to the whole population of expressway users to the value of time mentioned above is less than or equal to e(-1)