Liu, Q., van den Berg, VincentA.C., Verhoef, ErikT. and Jiang, R. (2025). Pricing in the stochastic bottleneck model with price-sensitive demand Transportation Research. Part B, Methodological, 194:1--25.

We analyse time-varying tolling in the stochastic bottleneck model with price-sensitive demand and uncertain capacity. We find that price sensitivity and its interplay with uncertainty have important implications for the effects of tolling on travel costs, welfare and consumers. We evaluate three fully time-variant tolls and a step toll proposed in previous studies. We also consider a uniform toll, which affects overall demand but not trip timing decisions. The first fully time-variant toll is the {\textquoteleft}first-best{\textquoteright} toll, which varies non-linearly over time and results in a departure rate that also varies over time. It raises the generalised price (i.e. the sum of travel cost and toll), thus lowering demand. These outcomes differ fundamentally from those found for first-best pricing in the deterministic bottleneck model. We call the second toll {\textquoteleft}second-best{\textquoteright}: it is simpler to design and implement as it maximises welfare under the constraint that the departure rate is constant over time. While a constant rate is optimal without uncertainty, it is not under uncertain capacity. Next, {\textquoteleft}third-best{\textquoteright} tolling adds the further constraint to the second-best that the generalised price should stay the same as without tolling. It attains a lower welfare and higher expected travel cost than the second-best scheme, but a lower generalised price. All our other tolls raise the price compared to the no-toll case. In our numerical study, when there is less uncertainty: the second-best and third-best tolls achieve welfares closer to that of the first-best toll, and the three schemes become identical without uncertainty. As the degree of uncertainty falls, the uniform and single-step tolls attain higher welfare gains. Also, when demand becomes more price-sensitive, the uniform and single-step tolls attain relatively higher welfare gains. Our step toll would lower the generalised price without uncertainty but raises it in our stochastic setting.