Nonlinear pricing with random participation

The canonical selection contracting programme takes the agent's participation decision as deterministic and Finds the optimal contract, typically satisfying this constraint for the worst type. Upon weakening this assumption of known reservation values by introducing independent randomness into the agents' outside options, we find that some of the received wisdom from mechanism design and nonlinear pricing is not robust and the richer model which allows for stochastic participation affords a more general empirical specification. We develop a multidimensional methodology for addressing this class of problems, providing two important applications to nonlinear pricing. First, with nonlinear pricing by a monopolist the familiar "no-distortion-at-the-top" result persists, but in tandem with the surprising conclusion that there is either no distortion at the bottom or bunching. Second, in a simple model of product differentiated duopolists competing with nonlinear pricing we show that, generally. the duopoly outcome is qualitatively similar to the monopoly outcome, However, when marginal costs are symmetric and competition is sufficiently intense, distortions disappear and the equilibrium outcome takes a remarkably simple form: efficient quality allocations with cost-plus-fee pricing.