Ironing, sweeping, and multidimensional screening

We provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of Mussa and Rosen (1978). These results are also directly applicable to the multidimensional nonlinear pricing problems studied by Wilson (1993) and Armstrong (1996). We establish that bunching is robust in these multidimensional screening problems, even with very regular distributions of types. This comes from a strong conflict between participation constraints and second order incentive compatibility conditions. We consequently design a new technique, the sweeping procedure, for dealing with bunching in multidimensional contexts. This technique extends the ironing procedure of Mussa and Rosen (1978) to several dimensions. We illustrate it on two examples: we first solve a linear quadratic version of the bidimensional nonlinear pricing problem, where consumers' types are exponentially distributed. The solution involves pure bundling for consumers with low demands. The second example is the bidimensional version of the Mussa and Rosen problem when consumers' types are uniformly distributed on a square. The solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum, but only limited choice for lower qualities. This seems to be a quite general pattern for multidimensional screening problems. The sweeping procedure is potentially applicable to other multidimensional screening problems.