van den Berg, V.A.C. and Verhoef, E. (2012). Is the travel time of private roads too short, too long, or just right? Transportation Research. Part B, Methodological, 46(8):971--983.

We consider price and service-quality setting in oligopolistic markets for congestible services, applied to the case of private roads. Previous studies show that parallel competitors set a volume/capacity ratio (and thereby a travel time or service quality) that is socially optimal if they take the actions of the others as given. We find that this result does not hold when capacity and toll setting are separate stages-as then firms aim to limit toll competition by setting lower capacities, and thus higher travel times-or when firms set capacities sequentially, as then firms aim to limit the capacities of later entrants by setting higher capacities. In our Stackelberg competition, the last firm to act has no capacity decisions to influence. Hence, it is only concerned with the toll-competition substage, and sets a travel time that is longer than socially optimal. The first firm cares mostly about the competitors' capacities that it can influence: it sets a travel time that is shorter than socially optimal. The average travel time will be too short from a societal point of view. {\textcopyright} 2012 Elsevier Ltd.