In this work, we study a bidimensional adverse selection problem in the framework of nonlinear pricing by a monopolist, where the firm produces a one-dimensional product and customers' preferences are described by two dimensions of uncertainty.

We prove that it is sufficient to consider, for each type of customer, incentive compatibility constraints over a one-dimensional set rather than the entire two-dimensional set as required by definition. For this purpose, we introduce a pre-order among types to compare their marginal valuation of consumption and we also take account possible shape of isoquants. As a consequence, the discretized problem is computationally tractable for relative fine discretizations.

Due to we extend the ideas applied in the unidimensional case with finite types when single-crossing condition is satisfied, our main assumption is the validity of single-crossing over each direction of uncertainty. Thus, we are able to have well-educated insights of the solution for a large class of valuation function and types' distributions.