Elastic-demand Wardrop equilibrium

In an elastic-demand Wardrop equilibrium, demand $f_s$ responds to the minimum route delay ${\lambda }_s$ through $f_s=B_s({\lambda }_s)$. An inverse demand function supplies the utility term that must be subtracted from the Beckmann potential.

For nonnegative strictly increasing link delays and nonempty routes, let $M=B(0)$ and $L={\lim }_{\lambda \to \infty }B(\lambda )$. The inverse demand tends to infinity as flow decreases to $L$, so increasing flow slightly there gains more utility than the bounded link-cost increment. At $M$ the inverse demand is zero, whereas a used route has positive delay, so reducing flow improves the objective. A minimizer consequently has $L<f<M$ and satisfies the ordinary interior Wardrop equilibrium conditions. Lower-semicontinuous endpoint extensions permit minimization on the compact demand interval even when utility diverges at $L$.

A strictly decreasing demand function has a decreasing inverse on its range. The utility primitive $\int P_s(u)du$ is a concave function. A positive interior reference value avoids assuming that an improper integral from zero is finite.

An interior reference point defines the utility of a continuous strictly decreasing demand function without requiring integrability of its inverse at zero. The inverse demand is decreasing, so the utility is strictly concave. For example $B(\lambda )=(1+\lambda {)}^{-1/2}$ has inverse $f^{-2}-1$, whose integral from zero diverges. The interior-reference utility still has the required derivative and is finite at every interior demand.