•Data give digital monopolies market power from improved matching & reduced privacy.•While data improve matches, they allow extraction of consumer surplus.•Thus, consumer surplus is maximized for intermediate privacy.•With regulated prices, consumer surplus is maximized without privacy protection.•Price regulation may allow data’s social benefits without consumer harm. Increasing returns to scale in data gathering and processing give rise to a new form of monopoly, referred to here as digital monopoly. Digital monopolies create new challenges for regulators and antitrust authorities. We address two in this paper: market power arising from improved match values and from reduced privacy. The digital monopoly’s profit and social surplus always increase as privacy decreases. However, consumer surplus is non-monotone in privacy. Without privacy, the match value is perfect but completely extracted by the digital monopoly. In contrast, as privacy goes to infinity, match values and social surplus go to zero. With regulated prices, consumer surplus is maximized without privacy protection. As with natural monopolies, price regulation thus remains an appropriate tool in the digital age to capture the social benefits from increasing returns to scale without harming consumers.