Optimal Allocation with Ex-post Verification and Limited Penalties

Proof of Lemma 2. Consider an allocation g(x) that satisfies (IC) and (F). We construct a monotonic g̃(x) that preserves constraints (IC) and (F), but increases the principal’s payoff. We have assumed that F has almost everywhere positive density, so F−1 exists. Define S(t) = ∣∣{y : g(F−1(y)) ≤ t}∣∣, t ∈ R+. Note that S is weakly increasing and satisfies S(t) ∈ [0, 1] for all t. Define g̃(x) = S−1(F (x)) for all x where S−1(F (x)) exists, and extend g̃ to [a, b] by right continuity. Observe that g̃ satisfies (F) by construction. In addition,