Fundamental properties of volume-capacity ratio of a private toll road in general networks

It has been addressed in the existing literature that the volume-capacity (v/c) ratio on a private toll road is constant, regardless of the toll value and capacity set by the private companies for each toll road in the network. The previous derivation depends on the assumption that the user equilibrium flow on the private toll road is continuously differentiable with respect to its capacity and toll charge. In this paper, it is shown that the constant v/c ratio phenomenon still holds via relaxing this assumption. Such phenomenon is further examined in different scenarios: the v/c ratio remains constant under the demand regulation, markup charge regulation, and social cost minimization scheme; it may change under the rate-of-return regulation; it may decrease under capacity regulation and increase under the price-cap regulation. Moreover, this paper demonstrates that the v/c ratio may change if there are decreasing or increasing returns to scale in road construction, or the road capacity can only take discrete values, or link flow interactions are involved, whereas the v/c ratio remains constant in case of elastic demand.