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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.

Use finite sets of links $J$, source-sink pairs $S$, and admissible routes $R$, with at least one route for each pair. Route $r$ carries aggregate flow $x_r$. The link-route incidence matrix has $A_{jr}=1$ if route $r$ uses link $j$ and zero otherwise; repeated traversals, if allowed, are counted with their multiplicity. The source-sink route incidence matrix has $H_{sr}=1$ if $r$ serves pair $s$. Thus $y=Ax$ gives link throughputs and $f=Hx$ gives source-sink demands. Write $R_s$ for the routes serving $s$ andA Wardrop equilibrium routes positive flow only on minimum-delay routes:This is a nonatomic equilibrium: an individual infinitesimal user's route change does not alter the aggregate link delays.

For fixed nonnegative demand $f$, the feasible route-flow set $\{x\ge 0:Hx=f\}$ is a nonempty product of finite-dimensional simplexes and hence a compact set. The Beckmann potentialis continuous, so it attains a minimum over this feasible set. Each primitive is a strictly convex function, since its derivative $D_j$ is continuous and strictly increasing. Its route-flow derivative is exactly the corresponding route delay:If a minimizer had positive flow on $r$ and a lower-delay alternative $q\in R_s$, transferring a small amount from $r$ to $q$ would give directional derivative $L_q-L_r<0$, contradicting minimality. Hence every minimizer is a Wardrop equilibrium.

Conversely, let $x$ be a Wardrop equilibrium and $z$ any feasible competitor. Convexity givesFor each $s$, subtract ${\lambda }_s$ inside this sum. The demand constraint makes ${\sum }_{r\in R_s}{\lambda }_s(z_r-x_r)=0$, and the remaining terms are nonnegative: used routes have $L_r-{\lambda }_s=0$, while unused routes have $x_r=0$, $z_r\ge 0$ and $L_r-{\lambda }_s\ge 0$. Thus $V(Az)\ge V(Ax)$. We have proved both existence and the optimization characterization.

Strict convexity in the link vector makes $y$ unique: two different minimizing link vectors would have a feasible midpoint of strictly smaller potential. Thus link delays and the minimum route delays at the Wardrop equilibrium are also unique. The route vector need not be unique, because different route decompositions may have the same $Ax$ and $Hx$. For example, two successive stages with two parallel links at each stage have four routes. With unit demand and link delay $D(u)=u$, the vectorsall give throughput $1/2$ on each of the four links and delay one on every route. This exhibits route-flow nonuniqueness at a Wardrop equilibrium. The equilibrium link throughputs are unique; the individual route flows need not be.

For elastic demand, make the usual physical assumptions explicit: delays are nonnegative on $[0,\infty )$, every route is nonempty, and $B_s:[0,\infty )\to [0,\infty )$ is finite, continuous and strictly decreasing. PutThe limit exists by monotonicity; it need not be zero. The inverse demand function $P_s$ is continuous, strictly decreasing, satisfies $P_s(M_s)=0$, and tends to infinity as $v\downarrow {\ell }_s$.

Choose any reference $b_s\in ({\ell }_s,M_s)$ and use elastic-demand utility with an interior reference:Each utility is strictly concave, with derivative $P_s$. A reference inside the range avoids a potentially divergent integral from zero. For instance, $B(\lambda )=(1+\lambda {)}^{-1/2}$ has inverse $v^{-2}-1$, whose integral from zero is infinite, although the displayed reference-point primitive is finite at every interior demand.

Give $G_s$ its limiting value at ${\ell }_s$, which may be $-\infty$, and extend it by $-\infty$ outside $[{\ell }_s,M_s]$. This encodes the physical demand range in the objective. After substituting $y=Ax$, $f=Hx$, minimizeIts effective feasible set is contained in $\{x\ge 0:{\ell }_s\le (Hx{)}_s\le M_s\}$, a nonempty compact set. The objective is lower semicontinuous, including a possible value $+\infty$ at a lower endpoint, and is finite when all demands equal their interior reference values. It therefore has a finite-valued minimizer.

The boundary exclusion for elastic Wardrop demand shows that every minimizing demand is interior. If $f_s={\ell }_s$ and the utility there is finite, adding a small flow $h$ on any route serving $s$ costs at most a bounded constant times $h$, while the utility gain is at least $h{P}_s({\ell }_s+h)$, whose ratio to $h$ tends to infinity. This decreases the objective. If that endpoint utility is $-\infty$, a finite minimizer cannot occur there in the first place. At $f_s=M_s>0$, choose a used route. Its delay is strictly positive, since its nonempty links have positive throughput and strictly increasing nonnegative delays. Removing a small amount of its flow saves a positive first-order cost, whereas the utility loss is $o(h)$ because $P_s(M_s)=0$. This also decreases the objective. Hence ${\ell }_s<f_s<M_s$ for every pair.

The interior-demand route derivatives are nowAt a minimizer they are zero on positive route flows and nonnegative on zero route flows, by increasing or decreasing a single route flow. Since $f_s>0$, some route is used, soThese are precisely the elastic-demand Wardrop equilibrium conditions. Conversely, those conditions give the same nonnegative directional derivative for every competitor; convexity then proves global minimality. The equivalent constrained form is the requested minimization of $V(y)-G(f)$ with $Hx=f$, $Ax=y$, $x\ge 0$. Strict convexity determines both $y$ and $f$ uniquely, while their route decomposition can still be nonunique.

Assume each link delay ${\ell }_e$ is continuous and nondecreasing. Define the Beckmann potential on feasible route flows byThe integral of a nondecreasing function is convex, so $\Psi$ is a convex function. The chain rule givesAt a minimum, move a small amount of flow from a used route $p$ to any other route $q$ for the same source-sink pair. This is a feasible direction, and its one-sided derivative is $L_q-L_p\ge 0$. Thus every used route has minimum cost, establishing the Wardrop equilibrium conditions.

Conversely, if $f$ satisfies the Wardrop equilibrium conditions, then for any feasible $g$,Here ${\sum }_p{L}_p{f}_p={\lambda }_k{d}_k$ on each pair, whereas ${\sum }_p{L}_p{g}_p\ge {\lambda }_k{d}_k$. The supporting inequality for a convex function then gives $\Psi (g)\ge \Psi (f)$. ThereforeThe feasible route-flow set is compact, and the Beckmann potential is continuous, so a minimizer exists. If each ${\ell }_e$ is strictly increasing, the potential is strictly convex as a function of link loads. Two minimizers with distinct link-load vectors would have a midpoint with strictly smaller potential, a contradiction. Thus equilibrium link loads are unique in this case; route-flow nonuniqueness at a Wardrop equilibrium can still occur because different route decompositions can have the same link loads.

The optimization of selfish route choice must be distinguished from the minimization of total travel timeFor differentiable delays, the marginal social cost on link $e$ is ${\ell }_e(v_e)+v_e{\ell }_e^{\prime }(v_e)$, rather than just ${\ell }_e(v_e)$. The marginal external cost toll ${\tau }_e(v_e)=v_e{\ell }_e^{\prime }(v_e)$ makes each traveler's perceived link cost equal to this derivative. If $D$ is convex, the tolled Wardrop equilibria minimize $D$: the corresponding Beckmann potential is exactly $D$, up to its zero reference value. Convexity of $D$ is an additional assumption and does not follow merely from nondecreasing delays.

An elastic-demand extension assigns pair $k$ an inverse demand function $P_k(d_k)$ and allows demand to vary. Subtracting ${\sum }_k{\int }_0^{d_k}{P}_k(u)du$ from the Beckmann potential produces the optimization formulation: its marginal condition equates minimum route delay with marginal willingness to travel on positive-demand pairs. With decreasing inverse demands, the subtracted utility is concave and the resulting objective remains convex. This explains why both fixed-demand route choice and an elastic-demand Wardrop equilibrium fit a common optimization framework.

Let $R_s$ be the finite, nonempty set of available routes for source-sink pair $s$, and write $x_r$ for a route flow. The link-route incidence matrix has $A_{jr}=1$ when route $r$ uses link $j$, and zero otherwise. The source-sink route incidence matrix has $H_{sr}=1$ when $r\in R_s$. Thus $y=Ax$ gives link throughputs and $f=Hx$ gives aggregate source-sink flows. A route's delay isA Wardrop equilibrium uses only routes of minimum delay for their source-sink pair:An infinitesimal user cannot improve its delay by changing route.

For fixed $f_s\ge 0$, the feasible route flows form a nonempty compact convex set: for each $s$, $x_r\ge 0$ and ${\sum }_{r\in R_s}{x}_r=f_s$. Define the Beckmann potentialIt is continuous, so the extreme value theorem gives a minimizer. Since every $D_j$ is strictly increasing, $V$ is a strictly convex function of $y$. Its route-flow gradient is $\partial V(Ax)/\partial x_r=d_r(x)$.

If an occupied route $r\in R_s$ has greater delay than another route $q\in R_s$, moving a small amount of flow from $r$ to $q$ gives a negative directional derivative, contradicting minimality. Conversely, at a Wardrop equilibrium, for any feasible $\tilde{x}$,The first-order condition for convex optimization therefore proves optimality. ConsequentlyIf two minimizers had different $y$, their midpoint would give a strictly smaller Beckmann potential. Thus the equilibrium link throughputs are unique. The route flows need not be unique, since $V(Ax)$ need not be a strictly convex function of $x$.

For a concrete example, take two successive stages of parallel links: $a,b$ in the first stage and $c,d$ in the second, with one source-sink pair and four routes $ac,ad,bc,bd$. Put $D_j(y)=y$ on every link and total flow $f=2$. All allocationshave the same link throughputs $(1,1,1,1)$ and every route has delay two. These distinct allocations are all Wardrop equilibria.

For elastic demand, use the standard physical convention that $D_j\ge 0$ and $B_s:[0,\infty )\to [0,\infty )$ is finite, continuous, and strictly decreasing. Set $M_s=B_s(0)$ and ${\ell }_s={\lim }_{\lambda \to \infty }{B}_s(\lambda )$, and define the inverse demand function $P_s=B_s^{-1}$ on $({\ell }_s,M_s]$. It is continuous and strictly decreasing, with $P_s(M_s)=0$ and $P_s(u)\to \infty$ as $u\downarrow {\ell }_s$. Choose any reference value $f_s^0\in ({\ell }_s,M_s)$ and setEach summand of $-G$ is a strictly convex function. The reference values only add a constant to the objective. In the usual case ${\ell }_s=0$, one may use a lower limit zero whenever the improper integral is finite; that integrability is not guaranteed merely by continuity of $B_s$.

Minimize $\Psi$ over the compact feasible set $x\ge 0$, ${\ell }_s\le (Hx{)}_s\le M_s$, interpreting the objective at $f_s={\ell }_s$ by its one-sided limit, which can be $+\infty$. This extension is sequentially lower semicontinuous, and an interior-demand feasible point has finite value, so a minimizer exists. It cannot have $f_s={\ell }_s$: adding a small amount on any route for $s$ has bounded marginal link cost, whereas its marginal demand benefit $P_s(f_s)$ tends to infinity. Thus $f_s>{\ell }_s$.

For ${\ell }_s<f_s<M_s$, the route-flow partial derivative is $d_r-P_s(f_s)$. The first-order conditions for convex optimization giveIf $f_s=M_s$, reducing any occupied route cannot improve the objective only if that route has zero delay; nonnegative link delays then give ${\lambda }_s=0=P_s(M_s)$. Therefore the conditions in every case areConversely these conditions give the nonnegative directional derivative for every feasible competitor and hence minimize the convex function $\Psi$. Allowing $f$ explicitly yields the stated form with $Hx=f$ and $Ax=y$, with inverse-demand endpoint conventions understood. In particular, $G$ is defined on its demand range, rather than evaluating $B_s^{-1}$ outside that range.

Strict convexity in the pair $(y,f)$ proves that the equilibrium aggregate source-sink flows and link throughputs are unique. Individual route flows may still be nonunique. Nonnegative, finite demand, nonnegative delays, finite route sets, and a usable route for each source-sink pair are the implicit modeling assumptions needed for the existence assertions.