= Positive-price convergence for increasing resource supplies
{title2=$\mu(t)\to\mu^*$}
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 $\sum_j\kappa_j\mu_j(\partial_jV)^2$. 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.
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