Figure A.5: Support under concave treatment response The function ϕ can be readily shown to be nonincreasing. Thus, at the optimum B$_{k}$ = ( −∞ , b$_{k}$ ) with b$_{k}$$_{+1}$ ≤ b$_{k}$ and b$_{k}$ ∈ [ −∞ , ∞ ] for every integer k . By Theorem 1, for δ > 0 , B D k is written as Note that Y$_{1}$ = Y$_{0}$ + δ and Y$_{1}$ = t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ Y$_{0}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w intersect at ( t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + y$_{−}$$_{1}$, t$_{1}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w $^{)}$. I consider the following three cases: a) b$_{k}$$_{+1}$ ≤ b$_{k}$ ≤ t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w, b) b$_{k}$$_{+1}$ ≤ t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w ≤ b$_{k}$, and c) t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w ≤ b$_{k}$$_{+1}$ ≤ b$_{k}$. Case a) b$_{k}$$_{+1}$ ≤ b$_{k}$ ≤ t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w If b$_{k}$$_{+1}$ ≤ b$_{k}$ ≤ t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w, as illustrated in Figure A.5(a), for any y$_{0}$ < b$_{k}$$_{+1}$ ≤ t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w, there exists Figure. A.6: B D k for B$_{k}$ = ( −∞ , b$_{k}$ ) and B$_{k}$$_{+1}$ = ( −∞ , b$_{k}$$_{+1}$ ) B D k = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ < b$_{k}$ s.t. 0 ≤ y$_{1}$ − y$_{0}$ < δ and ( t$_{0}$ − t$_{W}$ ) y$_{1}$ − ( t$_{1}$ − t$_{W}$ ) y$_{0}$ ≤ − ( t$_{1}$ − t$_{0}$ ) w } ∪ { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ < b$_{k}$$_{+1}$ s.t. y$_{1}$ − y$_{0}$ ≥ δ and ( t$_{0}$ − t$_{W}$ ) y$_{1}$ − ( t$_{1}$ − t$_{W}$ ) y$_{0}$ ≤ − ( t$_{1}$ − t$_{0}$ ) w } . 69 (a) (b) (c)