We study the behavior of competing platforms in a two-sided market. A user derives two types of utility from joining a platform: the intrinsic membership benefit, which varies across users, and the network benefit, which is determined by the number (measure) of users from the other side joining the same platform. When multiple platforms compete in price to attract users, in each symmetric equilibrium, each platform earns a zero profit. We also construct the unique equilibrium in the case of platform monopoly. Our comparison of welfare levels attained in oligopoly and monopoly shows that when there are at least two platforms in the market, social welfare decreases as the number of platforms increases; and that social welfare is maximized when two platforms or a single platform operates, depending on the magnitudes of network externality and marginal cost.