  Two classical problems in economics, the existence of a market equilibrium
and the existence of social choice functions, are formalized here by the
properties of a family of cones associated with the economy. It was recently
established that a necessary and sufficient condition for solving the former is
the nonempty intersection of the family of cones, and one such condition for
solving the latter is the acyclicity of the unions of its subfamilies. We show
an unexpected but clear connection between the two problems by establishing a
duality property of the homology groups of the nerve defined by the family of
cones. In particular, we prove that the intersection of the family of cones is
nonempty if and only if every subfamily has acyclic unions, thus identifying
the two conditions that solve the two economic problems. In addition to their
applications to economics, the results are shown to extend significantly
several classical theorems, providing unified and simple proofs: Helly's
theorem, Caratheodory's representation theorem, the
Knaster-Kuratowski-Marzukiewicz theorem, Brouwer's fixed point theorem, and
Leray's theorem on acyclic covers.
