 Optimal-flow calibrated congestion tolling

Let $y^{\ast }$ be a globally delay-minimizing feasible vector of link throughputs. For continuously differentiable strictly increasing delays, calibrated tollsmake the marginal social costs at $y^{\ast }$ equal the perceived route costs there. First-order route-exchange conditions make $y^{\ast }$ a tolled Wardrop equilibrium. The nonnegative nondecreasing penalty vanishes at the optimum; choosing ${\epsilon }_j>0$ makes the toll genuinely traffic-dependent. The perceived link delays remain strictly increasing, so their Beckmann potential gives unique equilibrium link loads. Every tolled Wardrop equilibrium therefore has the globally optimal $y^{\ast }$, even when the total-delay objective is not convex.