We study mechanism design problems in quasi-linear environments where the en- velope theorem and revenue equivalence principle fail due to non-convex and non- differentiable valuations. Despite these obstacles, we obtain a characterization of in- centive compatibility based on the familiar Mirrlees representation of the indirect util- ity and a monotonicity condition on the allocation rule. These conditions pin down the range of possible payoffs as a function solely of the allocation rule, thus providing a revenue inequality. We illustrate the usefulness of our approach in three economic applications where standard techniques do not apply: we derive the optimal selling mechanism in a buyer-seller situation where the buyer has loss-averse preferences; we find a zero payment (hence budget-balanced) efficient mechanism in a public goods lo- cation model; and we consider a principal–agent model with ex post non-contractible actions available to the agent.
