The monopolist's problem of deciding what types of products to
manufacture and how much to charge for each of them,
knowing only statistical information about the preferences of
an anonymous field of potential buyers, is one of the basic
problems analyzed in economic theory. The solution to this problem when
the space of products and of buyers can each be parameterized by a single
variable (say quality X, and income Y) garnered Mirrlees (1971) and
Spence (1974) their Nobel prizes in 1996 and 2001.
The multidimensional version of this question is a largely open problem
in the calculus of variations (see Basov's book "Multidimensional
Screening".)
I will describe recent work with A Figalli and YH Kim, identifying structural
conditions on the value b(X,Y) of product X to buyer Y which reduce this
problem to a convex program in a Banach space leading to uniqueness and
stability results for its solution, confirming robustness of certain economic
phenomena observed by Armstrong (1996) such as the desirability for the
monopolist to raise prices enough to drive a positive fraction of buyers out
of the market, and yielding conjectures about the robustness of other phenomena
observed Rochet and Chone (1998), such as the clumping together of products
marketed into subsets of various dimension. The passage to several dimensions
relies on ideas from differential geometry / general relativity, optimal transportation,
and nonlinear PDE.
