 Full-row-rank convergence of multiplicative resource 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.