Verhoef, E., Koh, A. and Shepherd, S. (2010). Pricing, capacity and long-run cost functions for first-best and second-best network problems Transportation Research. Part B, Methodological, 44(7):870--885.

This paper considers the use of 'long-run cost functions' for congested networks in solving second-best network problems, in which capacity and tolls are instruments. We derive analytical results both for general cost and demand functions and for specific functional forms, namely Bureau of Public Roads cost functions and constant-elasticity demand functions. The latter are also used in a numerical simulation model. We consider second-best cases where only a sub-set of links in a network is subject to tolling and/or capacity choice, and cases with and without a self-financing constraint imposed. We will demonstrate that, under certain assumptions, second-best long-run cost (or actually: generalized price) functions can be derived for most of the cases of interest, which can be used in an applied network model as a substitute for the conventional short-run user cost functions. Doing so reduces the dimensionality of the problem and should therefore be helpful in speeding up procedures for finding second-best optima. {\textcopyright} 2009 Elsevier Ltd.