  We introduce the notion of non-monotone utilities, which covers a wide
variety of utility functions in economic theory. We then prove that it is
PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets
with linear and non-monotone utilities. Building on this result, we settle the
long-standing open problem regarding the computation of an approximate
Arrow-Debreu market equilibrium in markets with CES utility functions, by
proving that it is PPAD-complete when the Constant Elasticity of Substitution
parameter \rho is any constant less than -1.
