 Solution

When link delays are separable and nondecreasing, Wardrop equilibria minimize the Beckmann potentialover 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,For differentiable delays its marginal cost is ${\ell }_j(y_j)+y_j{\ell }_j^{\prime }(y_j)$, not simply ${\ell }_j(y_j)$. Thus an equilibrium optimizing the Beckmann potential generally fails to minimize total delay. A marginal external cost toll $y_j{\ell }_j^{\prime }(y_j)$ 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.