Second-Best Congestion Pricing in General Static Transportation Networks with Elastic Demands

This paper studies the second-best problem where not all links of a congestedtransportation network can be tolled. The second-best tax rule for thisproblem is derived for general static networks, so that the solution presentedis valid for any graph of the network and any set of tolling points available.A number of known second-best tax rules are shown to be special cases of thegeneral solution presented. It is further demonstrated that, for instance byusing the concept of 'virtual links', the same method can be applied to abroader class of second-best problems in static networks. Finally, a smallnetwork is used to demonstrate numerically that an interior second-bestoptimum need not always be unique, and need not always exist. However, bothexamples require extreme differences in marginal external costs across links,and a non-optimal toll-point to be used, which casts doubt on the practicalrelevance of this complication.