As shown in Figure 6, A D = φ for A = φ, and A D = [ a + δ, ∞ ) for A = [ a, ∞ ) with a ∈ ( −∞ , ∞ ). Then, µ$_{0}$ ( A ) − µ$_{1}$ $^{(}$A D ) = 0 for A = φ , while µ$_{0}$ ( A ) − µ$_{1}$ $^{(}$A D ) = F$_{1}$ ( a + δ ) − F$_{0}$ ( a ) for A = [ a, ∞ ) . Therefore, the RHS in (6) reduces to which is equal to the Makarov lower bound. One can derive the Makarov upper bound in the same way. Now consider the support restriction Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) = 1. Note that this restriction is linear in the entire joint distribution π, since it can be rewritten as ∫ 1$_{C}$ ( y$_{0}$, y$_{1}$ ) dπ = 1. The linearity makes it possible to handle this restriction with penalty. In particular, since support restrictions hold with probability one, the corresponding penalty is infinite. Therefore, one can embed 1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ ) into the cost function with an infinite multiplier λ = ∞ as follows: The minimization problem (9) is well defined with λ = ∞ as noted in Remark 1. Note that for λ = ∞ , any joint distribution which violates the restriction Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) = 1 would cause infinite total costs in (9) and it is obviously excluded from the potential optimal joint distribution candidates. The optimal joint distribution should thus satisfy the restriction Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) = 1 to avoid infinite costs by not permitting any positive probability density for the region outside of the set C . Similarly, the upper bound on the DTE is written as To the best of my knowledge, this is the first paper that allows for { 0 , 1 , ∞} -valued costs. Although the econometrics literature based on the optimal transportation approach has used Lemma 3 for { 0 , 1 }− valued costs, the problem (9) cannot be solved using Lemma 3. In the next section, I develop a dual representation for (9) in order to characterize sharp bounds on the DTE. This section characterizes sharp DTE bounds under general support restrictions by developing a dual representation for problems (9) and (10). I use this characterization to derive sharp DTE bounds for various 16 sup π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ { 1 { y$_{1}$ − y$_{0}$ ≤ δ } − λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) } dπ (10) = 1 − inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ { 1 { y$_{1}$ − y$_{0}$ > δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) } dπ. inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ { 1 { y$_{1}$ − y$_{0}$ < δ } + λ (1 − 1$_{C}$ ( y$_{0}$, y$_{1}$ )) } dπ (9) sup a ∈$_{R}$ max [ F$_{1}$ ( a + δ ) − F$_{0}$ ( a ) , 0] , 3 Main Results