= Multiplicative resource-price dynamics
{title2=$\dot\mu_j=\kappa_j\mu_j[(Ax)_j-C_j]$}
For route prices $p=A^T\mu$, logarithmic users demand $x_r=w_r/p_r$, while resource prices adjust as
$$
\dot\mu_j=\kappa_j\mu_j[(Ax)_j-C_j].
$$
This 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.
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