  We prove a conjecture of Boucksom-Demailly-P\u{a}un-Peternell, namely that on
a projective manifold $X$ the cone of pseudoeffective classes in
$H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in
$H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincar\'e pairing. This is done by
establishing a conjectured transcendental Morse inequality for the volume of
the difference of two nef classes on a projective manifold. As a corollary the
movable cone is seen to be equal to the closure of the cone of balanced
metrics. In an appendix by Boucksom it is shown that the Morse inequality also
implies that the volume function is differentiable on the big cone, and one
also gets a characterization of the prime divisors in the non-K\"ahler locus of
a big class via intersection numbers.
