Learning in nonlinear pricing with unknown utility functions

Optimal screening is one of the basic models of contracting under incomplete information, and we study the problem in a quality pricing application. We present a simple numerical method for solving the pricing problem when the firm has limited information about the buyers' utility functions. In the method, the firm learns the optimal price schedule as the demand data is collected. We examine what the firm can learn about the preferences by observing the sales, and how the revealed information can be used in adjusting the quality-price bundles to increase the profit. We analyze the properties of the solution and derive the first-order optimality conditions under different assumptions. We show that the problem can be solved by making use of these optimality conditions together with the buyers' marginal valuations. The firm can estimate the marginal valuations either by offering linear tariffs or by selling test bundles near the current solution.