The feasibility of Bertrand undercutting with nonuniform prices is established and properties are derived for Bertrand equilibrium in nonuniform price strategies. With free entry, equilibrium entails zero-profit minimum average cost production. If there is more than one producing firm, all prices collapse to a minimum average cost uniform price. An existence condition is compared to conditions from uniform price theory. Without free entry equilibrium, prices may not collapse to a uniform price. Positive profit may occur but all firms earn equal profit and incur equal marginal cost, while consumers pay average outlay no greater than marginal cost.