Multidimensional Screening, Affiliation, and Full Separation

We solve a class of two-dimensional screening problems in which one dimension has the standard features, while the other dimension is impossible to exaggerate and enters the agent's utility only through the message but not the true type. Natural applications are procurement and regulation where the producer's ability to produce quality and his costs of producing quantity are both unknown ; or selling to a budget constrained buyer. We show that under these assumptions, the orthogonal incentive constraints are necessary and suffcient for the full set of incentive constraints. Provided that types are affliated and all the conditional distributions of types have monotonic inverse hazard rates, the solution is fully separating in both dimensions.