We analyze competition on nonlinear prices in homogeneous goods markets with consumer search. In equilibrium firms offer two-part tariffs consisting of a linear price and lump-sum fee. The equilibrium production is socially efficient as the linear price of equilibrium two-part tariffs equals to the production marginal cost. Firms thus compete in lump-sum fees, which are dispersed in equilibrium. We show that sellers enjoy higher profit, whereas consumers are worse-off with two-part tariffs than with linear prices. The competition softens because with two-part tariffs firms can make effective per-consumer demand less elastic than the actual demand.