Dual congestion potential ($H(\mu )={\sum }_j{\int }_0^{{\mu }_j}{q}_j(u)du-{\sum }_r{w}_r\log (A^T\mu {)}_r$)

For strictly increasing continuous supply functions $q_j$ with $q_j(0)=0$, this potential is strictly convex on prices with positive route-price sums. Its minimum is interior for every used resource and gives the unique positive-price equilibrium. Multiplicative price dynamics can also have boundary equilibria of multiplicative price dynamics if zero prices are permitted.