Regulating dynamic congestion externalities with tradable credit schemes: Does a unique equilibrium exist?

Yue Bao, Erik T. Verhoef*, Paul Koster

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


Tradable credit schemes offer a potentially efficient, revenue-neutral policy alternative to classical dynamic pricing of congestion externalities. We show in this paper that the resulting equilibrium may not be unique for particular models of congestion, including the first-best solution for the conventional Vickrey's bottleneck model. This can have substantial detrimental impacts on social welfare and social acceptance of tradable credit schemes. The reason underlying this result is that the credit supply-demand condition can be satisfied for a continuum of credit prices. This is because any marginal change in the credit price will be matched by a compensating change in queuing times, keeping user price fixed but deviating from the first-best optimum in which no queueing should occur. We find that the problem of non-uniqueness does not occur for the dynamic flow congestion model proposed by Chu. A unique equilibrium can be obtained in the bottleneck model if the buying and selling of credits with a bank is allowed, against a pre-determined price. Credits are then still tradable so that the use can deviate from the initial distribution, but the credit price is determined by the perfectly elastic demand and supply from the bank.

Original languageEnglish
Pages (from-to)225-236
Number of pages12
JournalTransportation Research Part B: Methodological
Early online date29 Jul 2019
Publication statusPublished - Sept 2019


  • Non-uniqueness
  • Road pricing
  • Tradable credits
  • Tradable permits
  • Traffic congestion

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