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

# When is multidimensional screening a convex program?

created by figalli on 03 Dec 2010

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BibTeX]

*Accepted Paper*

**Inserted:** 3 dec 2010

**Journal:** J. Econom. Theory

**Year:** 2010

**Abstract:**

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 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.*

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