 Increasing-supply resource-price dynamics

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.