where Proof. See Appendix A. The identifying power of MTR on the lower bound has an interesting graphical interpretation. As shown in Figure 7(a), the DTE under MTR corresponds to the probability of the region between two straight lines Y$_{1}$ = Y$_{0}$ and Y$_{1}$ = Y$_{0}$ + δ . Given marginal distributions, the Makarov lower bound is obtained by picking y ∗ ∈ R such that a rectangle [ y ∗ − δ, ∞ ) × ( −∞ , y $^{∗}$] yields the maximum Fr´echet lower bound among all rectangles below the straight line Y$_{1}$ = Y$_{0}$ + δ. As shown in Figure 7(b), under MTR the probability of any rectangle [ y − δ, ∞ ) × ( −∞ , y ] below the straight line Y$_{1}$ = Y$_{0}$ + δ is equal to that of the triangle between two straight lines Y$_{1}$ = Y$_{0}$ + δ and Y$_{1}$ = Y$_{0}$. Now one can draw multiple mutually disjoint triangles between two straight lines Y$_{1}$ = Y$_{0}$ and Y$_{1}$ = Y$_{0}$ + δ as in Figure 7(c). Since the probability of each triangle is equal to the probability of the rectangle extended to the right and bottom sides, the lower bound on each triangle is obtained by applying the Fr´echet lower bound to the extended rectangle. Then the improved lower bound is obtained by summing the Fr´echet lower bounds on the triangles. One of the key benefits of my characterization based on the optimal transportation approach is that it guarantees sharpness of the bounds. To show sharpness of given bounds in a copula approach, one should show what dependence structures achieve the bounds under fixed marginal distributions. This is technically difficult under MTR. However, the optimal transportation approach gets around this challenge by focusing on a dual representation involving given marginal distributions only. Now I provide a sketch of the procedure to derive the lower bound under MTR from Theorem 1. The proof of deriving the lower bound from Theorem 1 proceeds in two stpng. The first step is to show that the dual function ϕ is nondecreasing so that one can put A$_{k}$ = ( a$_{k}$, ∞ ) for a$_{k}$ ∈ [ −∞ , ∞ ] at the optimum. For any ( y$_{0}$, y$_{1}$ ) in the support of the optimal joint distribution, the dual function ϕ for the lower bound is written as ϕ ( y$_{0}$ ) = inf y$_{1}$ ≥ y$_{0}$ { 1 { y$_{1}$ − y$_{0}$ < δ } − ψ ( y$_{1}$ ) } . F U Δ ( δ ) =      1 + inf y ∈$_{R}$ { min ( F$_{1}$ ( y ) − F$_{0}$ ( y − δ )) , 0 } , for δ ≥ 0 , 0 , for δ < 0 . , F L Δ ( δ ) =      sup { a$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } , for δ ≥ 0 , 0 , for δ < 0 , , where A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for each integer k $^{}}$. 20