  We establish the monotonicity property for the mass of non-pluripolar
products on compact Kahler manifolds, and we initiate the study of complex
Monge-Ampere type equations with prescribed singularity type. Using the
variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and
uniqueness of solutions with small unbounded locus. We give applications to
Kahler-Einstein metrics with prescribed singularity, and we show that the
log-concavity property holds for non-pluripolar products with small unbounded
locus.
