Multiplicative resource-price dynamics (${\dot{\mu }}_j={\kappa }_j{\mu }_j[(Ax{)}_j-C_j]$)

For route prices $p=A^T\mu$, logarithmic users demand $x_r=w_r/p_r$, while resource prices adjust asThis model raises prices under excess demand and lowers them under spare capacity. Positive capacities and nonempty routes are standard well-definedness assumptions. Positive initial prices approach the dual optimum; zero prices are invariant and can create boundary equilibria of multiplicative price dynamics that overload a resource.

Routes with positive logarithmic utility weights choose rates inversely proportional to their total resource price. Each resource changes its price multiplicatively according to demand minus its increasing supply. For continuous strictly increasing $q_j$ with $q_j(0)=0$, the potential $V={\sum }_r{w}_r\log ({\sum }_{j\in r}{\mu }_j)-{\sum }_j{\int }_0^{{\mu }_j}{q}_j(u)du$ is strictly concave and has nonnegative derivative along the dynamics. Positive-price convergence for increasing resource supplies requires positivity on used resources; zero-price faces are invariant.

For finite nonempty routes and positive weights, the potential of increasing-supply resource-price dynamics has a unique maximum. Increasing supplies give a linear penalty at large prices, so its superlevel sets are compact, and logarithmic route terms exclude zero route totals. Used resources have uniformly positive demand on a trapped set; a sufficiently small positive price therefore increases, giving a positive lower bound. The potential derivative is ${\sum }_j{\kappa }_j{\mu }_j({\partial }_{j}V{)}^2$. It has finite integral and is uniformly continuous, so it tends to zero by uniformly continuous integrable functions vanish at infinity. Every limit point then satisfies the unique stationary equations. Unused resource prices decay to zero. Only continuity and strict increase of the supply functions are needed.

In multiplicative resource-price dynamics, a zero price remains zero even if demand exceeds supply there. For one route using two resources, unit weight and supplies $q_j(u)=u$, both $(0,1)$ and $(1,0)$ are equilibria, while the unique positive equilibrium is $(1/\sqrt{2},1/\sqrt{2})$. Thus uniqueness on the positive price domain cannot be extended to every nonnegative price trajectory without an extra boundary rule.

For a complementary-slackness optimum ${\mu }^{\ast }$, defineinterpreting a zero-reference summand as ${\mu }_j/{\kappa }_j$. This is a nonnegative generalized reverse-relative-entropy expression for positive resource prices. Along multiplicative resource-price dynamics, with $p=A^T\mu$ and $g=Ax-C$,Its sublevel bounds keep all route prices away from zero and permit the LaSalle invariance principle, including optima with some zero individual prices.

For multiplicative resource-price dynamics with positive capacities, nonempty routes and strictly positive initial prices, full row rank of the link-route incidence matrix makesstrictly concave. There is a unique optimizing price vector, possibly on the boundary, and the relative-entropy Lyapunov function for resource prices proves convergence to it. Row-rank deficiency still gives unique optimal route prices and rates, but resource prices may depend on the initial state. Full column rank alone and unrestricted zero initial prices do not yield the same uniqueness theorem.