Optimal pricing policies for public transportation networks

In this paper we study the problem of finding an optimal pricing policy for the use of the public transportation network in a given populated area. The transportation network, modeled by a Borel set $\Sigma\subset{\mathbb R}^n$ of finite length, the densities of the population and of the services (or workplaces), modeled by the respective finite Borel measures $\varphi_0$ and $\varphi_1$, and the effective cost $A(t)$ for a citizen to cover a distance $t$ without the use of the transportation network are assumed to be given. The pricing policy to be found is then a cost $B(t)$ to cover a distance $t$ with the use of the transportation network (i.e., the ``price of the ticket for a distance $t$''), and it has to provide an equilibrium between the needs of the population (hence minimizing the total cost of transportation of the population to the services/workplaces) and that of the owner of the transportation network (hence maximizing the total income of the latter). We present a model for such a choice and discuss the existence as well as some qualitative properties of the resulting optimal pricing policies.