Verhoef, E. (2002). Second-best congestion pricing in general static transportation networks with elastic demands Regional Science and Urban Economics, 32:281--310.

This paper studies the second-best problem where not all links of a congested transportation network can be tolled. The second-best tax rule for this problem is derived for general static networks, so that the solution presented is 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 the general solution presented. It is further demonstrated that, for instance by using the concept of 'virtual links', the same method can be applied to a broader class of second-best problems in static networks. Finally, a small network is used to demonstrate numerically that an interior second-best optimum need not always be unique, and need not always exist. However, both examples require extreme differences in marginal external costs across links, and a non-optimal toll-point to be used, which casts doubt on the practical relevance of this complication. {\textcopyright} 2002 Elsevier Science B.V. All rights reserved.