When a monopolist asks consumers to choose a particular nonlinear tariff option, consumers do not completely know their type. Their valuations of the good and/or optimal quantity purchases are only fully realized after the optional tariff has been subscribed. In order to characterize the menu of optimal nonlinear tariffs when consumers’ demands are stochastic, I show that the increasing hazard rate property of distributions is preserved under convolution. This result, together with very weak assumptions on demand (common to standard nonlinear pricing), ensures that the continuum of optional nonlinear tariffs is characterized by quantity discounts. I test nonparametrically the theoretical prerequisites of the model using data directly linked to consumer types from the 1986 Kentucky telephone tariff experiment. I show that the distribution of actual calls second order stochastically dominates the distribution of expected calls, which fully supports the suggested type-varying theoretical model. Finally, I analyse possible welfare effects of the introduction of optional tariffs and their relative expected profitability using the empirical distribution of consumer types in two local exchanges with differentiated calling patterns. The evidence suggests that a menu of optional two-part tariffs dominates any other pricing strategy.
