The effect of private road supply on the volume/capacity ratio when firms compete Stackelberg in Road Capacity

We study road supply by competing firms between a single origin and destination. In previous studies, firms simultaneously set their tolls and capacities while taking the actions of the others as given in a Nash fashion. Then, under some widely used technical assumptions, firms set the same volume/capacity ratio as a public operator would and thus have the same amount of congestion and travel time. We find that this result does not hold if capacity and toll setting are separate stages—as then firms want to limit the competition in the toll stage by setting lower capacities—or when firms set capacities one after the other in a Stackelberg fashion—as then firms want to limit their competitors' capacities by setting higher capacities. In our Stackelberg competition, the firms that act last have few if any capacity decisions to influence. Hence, they are more concerned with the toll competition substage, and set a higher volume/capacity ratio than the public operator. The firms that act first care more about their competitors' capacities they can influence: they set a lower ratio and have the largest capacities. The average volume/capacity ratio is below the public ratio, and hence the average private travel time is too short. Still, in our numerical model, for three or more firms, welfare is higher under Stackelberg competition than under Nash competition, because of the larger Stackelberg capacity expansion and lower tolls.