where Note that at the optimum ϕ ( y$_{0}$ ) + ψ ( y$_{1}$ ) = 1 { y$_{1}$ − y$_{0}$ < δ } for any ( y$_{0}$, y$_{1}$ ) in the support of the optimal joint distribution. Therefore, dual functions ϕ and ψ can be written as follows: for any ( y$_{0}$, y$_{1}$ ) in the support of the optimal joint distribution, In my proof of Theorem 1, A$_{k}$ is defined as A$_{k}$ = { x ∈$_{R}$ : ϕ ( x ) > s + k } for the function ϕ , some s ∈ [0 , 1] , and each integer k. Since the dual function ϕ is continuous, if ϕ is nondecreasing then A$_{k}$ = ( a$_{k}$, ∞ ) for some a$_{k}$ ∈ [ −∞ , ∞ ] . Note that A$_{k}$ = φ for a$_{k}$ = ∞ . Also, since { A$_{k}$ } ∞ k = −∞ is a monotonically decreasing sequence of open sets, a$_{k}$ ≤ a$_{k}$$_{+1}$ for every integer k. In contrast, if ϕ is nonincreasing at the optimum then A$_{k}$ = ( −∞ , a$_{k}$ ) for a$_{k}$ ∈ [ −∞ , ∞ ] and a$_{k}$$_{+1}$ ≤ a$_{k}$ for each integer k . Note that A$_{k}$ = φ for a$_{k}$ = −∞ . In the next subsection, I will show that the function ϕ is monotone for economic examples considered in this paper and that sharp DTE bounds in each example are readily derived from monotonicity of ϕ . Remark 3 (Robustness of the sharp bounds) My sharp DTE bounds are robust for support restrictions in the sense that they do not rely too heavily on the small deviation of the restriction. I can verify this by showing that sharp bounds under Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) ≥ p converge to those under Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) = 1 , as p goes to one. The sharp lower bound under Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) ≥ p can be obtained with a multiplier ˜ λ$_{p}$ ≥ 0 as follows: Obviously, ˜ λ$_{0}$ = 0 . Furthermore, ˜ λ$_{p}$ ≤ ˜ λ$_{q}$ for 0 ≤ p < q ≤ 1 since F L Δ ( δ ) is nondecreasing in p. The proof of Theorem 1 can be easily adapted to the more general case in which the multiplier is given as a positive integer. If ˜ λ$_{p}$ = 2 K in (12) for some positive integer K , then the dual representation reduces to where { A$_{k}$ } K k = − ( K − 1) is monotonically decreasing. As K goes to infinity, this obviously converges to the dual representation for the infinite Lagrange multiplier, which is given in (11). 18 Φ$_{c}$ = { ( ϕ, ψ ) ; ϕ ( y$_{0}$ ) + ψ ( y$_{1}$ ) ≤ 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) with λ = ∞} . ϕ ( y$_{0}$ ) = inf y$_{1}$ :( y$_{0}$,y$_{1}$ ) ∈ C { 1 { y$_{1}$ − y$_{0}$ < δ } − ψ ( y$_{1}$ ) } . F L Δ ( δ ) = inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ $^{{}$1 { y$_{1}$ − y$_{0}$ < δ } + ˜ λ$_{p}$ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) $^{}}$dπ. (12) sup { A$_{k}$ } ∞ k = −∞ K ∑ − ( K − 1) max $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A C k $^{)}$, 0 $^{}}$,