= Optimal-flow calibrated congestion tolling
{title2=$T_j(u)=y_j^*D_j\prime(y_j^*)+\varepsilon_j(u-y_j^*)_+^2$}
Let $y^*$ be a globally delay-minimizing feasible vector of link <throughputs>. For continuously differentiable strictly increasing delays, calibrated tolls
$$
T_j(u)=y_j^*D_j'(y_j^*)+\varepsilon_j(u-y_j^*)_+^2\qquad(\varepsilon_j\geq0)
$$
make the marginal social costs at $y^*$ equal the perceived route costs there. First-order route-exchange conditions make $y^*$ a tolled <Wardrop equilibrium>. The nonnegative nondecreasing penalty vanishes at the optimum; choosing $\varepsilon_j>0$ 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 $y^*$, even when the total-delay objective is not <convex>.
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