Using a standard model of nonuniform pricing, properties are derived for both free entry and limited entry Bertrand equilibrium in nonuniform strategies. These properties follow from the feasibility of Bertrand undercutting with nonuniform prices. When there is free entry, zero-profit minimum average cost production occurs in equilibrium. If, in addition, average cost attains its minimum at any quantity below market output, all prices collapse to a uniform price equal to minimum average cost. A necessary condition for existence of this equilibrium is derived and compared to similar conditions from uniform price theory. Analogies to uniform price theory depend on the rationing capabilities of nonuniform prices and the rationing assumptions employed in uniform price models. Without free entry, prices may not collapse to a uniform price in equilibrium. Also, positive profit may occur but all firms must earn equal profit and incur equal marginal cost, while consumers pay average outlay no greater than marginal cost.