Externalities, average and marginal costs, and tolls on congested networks with time-varying flows

For congested networks on which flows vary over time, the authors derive system marginal costs, user perceived costs and user externality costs, for each arc and path. They also obtain a set of optimal congestion tolls and flow controls which may be used to shift the user determined flows toward a socially preferred pattern. An important way in which the present results differ from the usual static analysis is that the social cost externality depends not only on the level of congestion, but also on the rate of increase or decrease of congestion. This is intuitively explicable as follows. Consider users delayed on an arc. Their delays will be further compounded or multiplied if congestion has increased during the time they are delayed. On the other hand, their delays will be reduced if congestion has declined during the time they are delayed. This multiplier effect is such that the resultant dynamic externalities can easily be a few times larger, or smaller, than the externalities derived in the usual static analysis. As a result, the congestion tolls or tariffs which are usually proposed or advocated, based on static analysis, may be inappropriate. The results are illustrated with a numerical application to a small network example.