Pricing in the stochastic bottleneck model with price-sensitive demand

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 that have been proposed in previous literature. We also consider a uniform toll, which affects overall demand but not trip timing decisions. The first fully time-variant toll is the "first-best" 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. The second we call a "second-best" toll, as it aims to keep the departure rate constant over time. This is optimal without uncertainty, but it lowers welfare with uncertain capacity. Next, "third-best" tolling adds the further constraint that, besides a constant departure rate, 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 tolling schemes-except for the third-best one-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 are identical without uncertainty. As the degree of uncertainty falls, the uniform and single-step tolls attain welfare gains closer to that from first-best pricing. Also, when demand becomes more price-sensitive, the uniform and single-step tolls attain outcomes closer to those in the first-best outcome. Our ADL equilibrium step toll would lower the generalised price without uncertainty but raises it in our stochastic setting.