Two duopolists compete in price on the market for a homogeneous product. They can use a 'profiling technology' that allows them to identify the willingness-to-pay of their consumers with some probability. If both firms have profiling technologies of the exact same precision, or if one firm cannot use any profiling technology, then the Bertrand paradox continues to prevail. Yet, if firms have technologies of different precisions, then the price equilibrium exhibits both price discrimination and price dispersion, with positive expected profits. Increasing the precision of both firms’ technologies does not necessarily harm consumers.