  The distribution of treatment effects (DTE) is often of interest in the
context of welfare policy evaluation. In this paper, I consider partial
identification of the DTE under known marginal distributions and support
restrictions on the potential outcomes. Examples of such support restrictions
include monotone treatment response, concave treatment response, convex
treatment response, and the Roy model of self-selection. To establish
informative bounds on the DTE, I formulate the problem as an optimal
transportation linear program and develop a new dual representation to
characterize the identification region with respect to the known marginal
distributions. I use this result to derive informative bounds for concrete
economic examples. I also propose an estimation procedure and illustrate the
usefulness of my approach in the context of an empirical analysis of the
effects of smoking on infant birth weight. The empirical results show that
monotone treatment response has a substantial identifying power for the DTE
when the marginal distributions of the potential outcomes are given.
