First-best marginal cost toll for a traffic network with stochastic demand

Agachai Sumalee, Wei Xu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

49 Citations (Scopus)


First-best marginal cost pricing (MCP) in traffic networks has been extensively studied with the assumption of deterministic travel demand. However, this assumption may not be realistic as a transportation network is exposed to various uncertainties. This paper investigates MCP in a traffic network under stochastic travel demand. Cases of both fixed and elastic demand are considered. In the fixed demand case, travel demand is represented as a random variable, whereas in the elastic demand case, a pre-specified random variable is introduced into the demand function. The paper also considers a set of assumptions of traveler behavior. In the first case, it is assumed that the traveler considers only the mean travel time in the route choice decision (risk-neutral behavior), and in the second, both the mean and the variance of travel time are introduced into the route choice model (risk-averse behavior). A closed-form formulation of the true marginal cost toll for the stochastic network (SN-MCP) is derived from the variational inequality conditions of the system optimum and user equilibrium assignments. The key finding is that the calculation of the SN-MCP model cannot be made by simply substituting related terms in the original MCP model by their expected values. The paper provides a general function of SN-MCP and derives the closed-form SN-MCP formulation for specific cases with lognormal and normal stochastic travel demand. Four numerical examples are explored to compare network performance under the SN-MCP and other toll regimes.
Original languageEnglish
Pages (from-to)41-59
Number of pages19
JournalTransportation Research Part B: Methodological
Issue number1
Publication statusPublished - 1 Jan 2011


  • Demand uncertainty
  • First-best toll
  • Marginal cost pricing
  • Stochastic network

ASJC Scopus subject areas

  • Transportation


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