= Solution
When link delays are separable and nondecreasing, <Wardrop equilibria> minimize the <Beckmann potential>
$$
\boxed{V(h)=\sum_j\int_0^{(Ah)_j}\ell_j(u)\,du}
$$
over the feasible route-flow polytope. Its <derivative> with respect to $h_r$ is $c_r(h)$. Since the <integrals> are convex, the first-order inequality for a global minimum is exactly the <Wardrop equilibrium> inequality in part (a). Alternatively, the <Karush-Kuhn-Tucker conditions> equate used-route <derivatives> to the multiplier for their demand constraint, with larger <derivatives> on unused routes. Continuity gives existence on the compact feasible set. Strictly increasing link delays give unique link flows, although different route decompositions can still give <route-flow nonuniqueness at a Wardrop equilibrium>.
The system objective is instead total travel time,
$$
T(y)=\sum_j y_j\ell_j(y_j).
$$
For differentiable delays its marginal cost is $\ell_j(y_j)+y_j\ell_j'(y_j)$, not simply $\ell_j(y_j)$. Thus an equilibrium optimizing the Beckmann potential generally fails to minimize total delay. A <marginal external cost toll> $y_j\ell_j'(y_j)$ makes travelers face the social marginal cost and implements a system optimum under the appropriate convexity assumptions.
These formulations clarify the modeling distinction: nonatomic users optimize their own paths; a planner optimizes total delay. For nonseparable or nonmonotone costs, the variational description can remain applicable while the <Beckmann potential> argument, existence or uniqueness needs new hypotheses.
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