marginal distributions. Unfortunately, both approaches deliver bounds that are often too wide to be infor- mative in practice. Since these two conditions are often plausible in practice, a natural way to tighten the bounds is considering both MTR and given marginal distributions of potential outcomes. However, methods of establishing informative bounds on the DTE under these two restrictions have remained unanswered. Specifically, in the existing copula approach it is technically challenging to find out the particular joint distributions that achieve the best possible bounds on the DTE under the two restrictions. In this paper, I propose a novel approach to circumvent these difficulties associated with identifying the DTE under these two restrictions. Methodologically, my approach involves formulating the problem as an optimal transportation linear program and embedding support restrictions on the potential outcomes including MTR into the cost function. A key feature of the optimal transportation approach is that it admits a dual formulation. This makes it possible to derive the best possible bounds from the optimization problem with respect to given marginal distributions but not the joint distribution, which is an advantage over the copula approach. Specifically, the linearity of support restrictions in the entire joint distribution allows for the penalty formulation. Since support restrictions hold with probability one, the corresponding multiplier on those constraints should be infinite. To the best of my knowledge, the dual representation of such an optimization problem with an infinite Lagrange multiplier has not been derived in the literature. In this paper, I develop a dual representation for { 0 , 1 , ∞} -valued costs by extending the existing result on duality for { 0 , 1 } -valued costs. My approach applies to general support restrictions on the potential outcomes as well as MTR. Such support restrictions encompass shape restrictions on the treatment response function that can be written as g ( Y$_{0}$, Y$_{1}$ ) ≤ 0 with probability one for any continuous function g : R → $_{R}$, including MTR, concave treatment response, and convex treatment response. 1 Moreover, considering support restrictions opens the way to identify the DTE in the Roy model of self-selection and the DTE conditional on some sets of potential outcomes. Numerous examples in applied economics fit into this setting because marginal distributions are point or partially identified under weak conditions and support restrictions are often implied by economic theory and plausible priors. The marginal distributions of the potential outcomes are point-identified in randomized ex- periments or under unconfoundedness. Even if selection depends on unobservables, they are point-identified for compliers under the local average treatment effects assumptions (Imbens and Rubin (1997), Abadie (2002)) and are partially identified in the presence of instrumental variables (Kitagawa (2009)). Also, MTR has been defended as a plausible restriction in empirical studies of returns to education (Manski and Pepper $^{1}$Let Y$_{d}$ = f ( t$_{d}$ ) where Y$_{d}$ is a potential outcome and t$_{d}$ is a level of inputs for multi-valued treatment status d. Concave treatment response and convex treatment response assume that the treatment response function f is concave and convex, respectively. 3