Positive-price convergence for increasing resource supplies ($\mu (t)\to {\mu }^{\ast }$)

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.