Pick any integer k . If b$_{k}$ = −∞ , then If b$_{k}$ > b$_{k}$$_{+1}$ > −∞ , then also If b$_{k}$ > b$_{k}$$_{+1}$ = −∞ , then Consequently, by Theorem 1, the sharp upper bound under MTR can be written as Since monotonicity of ϕ can be shown very similarly as in the proof of Corollary 1, I do not provide the proof. As given in Corollary 2, the sharp lower bound under concave treatment response is identical to the sharp lower bound under MTR and the proof is also the same. The sharp upper bound under convex treatment response is equal to the Makarov upper bound by the same token as the upper bound under MTR. Thus, I do not provide their proofs. Also, since the sharp lower bound under convex treatment response is derived very similarly to the sharp upper bound under concave treatment response, I provide a proof only for the sharp upper bound under concave treatment response. Consider a concave treatment response restriction Pr { Y$_{0}$ − w t$_{0}$ − t$_{W}$ ≥ Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ , Y$_{1}$ ≥ Y$_{0}$ ≥ w } = 1 for any w in the support of W and ( t$_{1}$, t$_{0}$, t$_{W}$ ) ∈$_{R}$ 3 s.t. t$_{W}$ < t$_{0}$ < t$_{1}$. The support satisfying { Y$_{0}$ − w t$_{0}$ − t$_{W}$ ≥ Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ , Y$_{1}$ ≥ Y$_{0}$ ≥ w } corresponds to the intersection of the regions below the straight line Y$_{1}$ = t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ Y$_{0}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w and above the straight line Y$_{1}$ = Y$_{0}$ as shown in Figure A.5. Note that t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ > 1 and the two straight lines intersect at ( w, w ). 68 Proof of Corollary 2 max $^{{}$µ$_{0}$ ( B$_{k}$ ) − µ$_{1}$ $^{(}$B D k $^{)}$, 0 } = 0 . max $^{{}$µ$_{0}$ ( B$_{k}$ ) − µ$_{1}$ $^{(}$B D k $^{)}$, 0 } = 0 . max $^{{}$µ$_{0}$ ( B$_{k}$ ) − µ$_{1}$ $^{(}$B D k $^{)}$, 0 } = max { F$_{0}$ ( b$_{k}$ ) − F$_{1}$ ( b$_{k}$ + δ ) , 0 } . F U Δ ( δ ) = 1 − sup { B$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$ ( B$_{k}$ ) − µ$_{1}$ $^{(}$B D k $^{)}$, 0 } = 1 − sup b$_{k}$ max { F$_{0}$ ( b$_{k}$ ) − F$_{1}$ ( b$_{k}$ + δ ) , 0 } = 1 + inf y max { F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) , 0 } .