Summary: Typically, work on mechanism design has assumed that all private information can be captured in a single scalar variable. This paper explores one way in which this assumption can be relaxed in the context of the multiproduct nonlinear pricing problem. It is shown that the firm will choose to exclude some low value consumers from all markets. A class of cases that allow explicit solution is derived by making use of a multivariate form of ``integration by parts''. In such cases the optimal tariff is cost-based.