Calculus of Variations and Geometric Measure Theory

A. Figalli - Y. H. Kim - R. J. McCann

When is multidimensional screening a convex program?

created by figalli on 03 Dec 2010


Accepted Paper

Inserted: 3 dec 2010

Journal: J. Econom. Theory
Year: 2010


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, quasi-linear utilities, {\dummy and that agents can choose only pure strategies}, we identify a structural condition on the value $b(x,y)$ of product type $y$ to agent type $x$ --- 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 andor 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.