We establish the monotonicity property for the mass of non-pluripolar products on compact Kähler manifolds, and we initiate the study of complex Monge-Ampère 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 Kähler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for nonpluripolar products with small unbounded locus.Without the non-zero mass condition X θ n φ > 0 this characterization cannot hold (see Remark 3.3). The equivalence between (i) and (iii) in the above theorem shows that P θ [u] is the same potential for any u ∈ E(X, θ, φ), and equals to P θ [φ]. Given this and the inclusion E(X, θ, φ) ⊂ E(X, θ, P θ [φ]), one is tempted to consider only potentials φ in the image of the operator ψ → P θ [ψ], when studying the classes of relative full mass E(X, θ, φ). These potentials seemingly play the same role as V θ , the potential with minimal V * K,θ is supported on K, we thus haveand the desired inequality holds in this case. If M := M φ (K) ≥ 1, then by (12) we have φ ≤ M −1 V * K,φ + (1 − M −1 )φ ≤ φ + 1, and by definition of the relative capacity we can write: