  A principal wishes to transact business with a multidimensional distribution
of agents whose preferences are known only in the aggregate. Assuming a twist
(= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can
choose only pure strategies, we identify a structural condition on the
preference b(x,y) of agent type x for product type y -- and on the principal's
costs c(y) -- which is necessary and sufficient for reducing the profit
maximization problem faced by the principal to a convex program. This is a key
step toward making the principal's problem theoretically and computationally
tractable; in particular, it allows us to derive uniqueness and stability of
the principal's optimum strategy -- and similarly of the strategy maximizing
the expected welfare of the agents when the principal's profitability is
constrained. We call this condition non-negative cross-curvature: it is also
(i) necessary and sufficient to guarantee convexity of the set of b-convex
functions, (ii) invariant under reparametrization of agent and/or product types
by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang's
necessary and sufficient condition (A3w) for continuity of the correspondence
between an exogenously prescribed distribution of agents and of products. We
derive the persistence of economic effects such as the desirability for a
monopoly to establish prices so high they effectively exclude a positive
fraction of its potential customers, in nearly the full range of non-negatively
cross-curved models.
