(2000)), the effect of funds for low-ability pupils (Haan (2012)), the impact of the National School Lunch Program on children’s health (Gundersen et al. (2011)), and various medical treatments (Bhattacharya et al. (2005, 2012)). Researchers sometimes have plausible information on the shape of treatment response functions from economic theory or from empirical results in previous studies. For example, based on di- minishing marginal returns to production, one may find it plausible to assume that the marginal effect of improved maize seed adoption on productivity diminishes as the level of adoption increases, holding other inputs fixed. Also, one may want to assume that the marginal adverse effect of an additional cigarette on infant birth weight diminishes as the number of cigarettes increases as shown in Hoderlein and Sasaki (2013). In the empirical literature, concave treatment response has been assumed for returns to schooling (Okumura (2010)) and convex treatment response for the effect of education on smoking (Boes (2010)). 2 A considerable amount of the literature has used the Roy model to describe people’s self-selection ranging from immigration to the U.S. (Borjas (1987)) to college entrance (Heckman et al. (2011)). Also, heterogeneity in treatment effects for unobservable subgroups defined by particular sets of potential outcomes has been of central interest in various empirical studies. Heterogeneous peer effects and tracking impacts (Duflo et al. (2011)) and heterogeneous class size effects (Ding and Lehrer (2008)) by the level of students’ performance, and the heterogeneity in smoking effects by potential infant’s birth weight (Hoderlein and Sasaki (2013)) have also been discussed in the literature focusing on heterogeneous average effects. I apply my method to an empirical analysis of the effects of smoking on infant birth weight. I propose an estimation procedure and illustrate the usefulness of my approach by showing that MTR has a substantial identifying power for the distribution of smoking effects given marginal distributions. As a support restriction, I assume that smoking has nonpositive effects on infant birth weight. Smoking not only has a direct impact on infant birth weight, but is also associated with unobservable factors that affect infant birth weight. To overcome the endogenous selection problem, I make use of the tax increase in Massachusetts in January 1993 as a source of exogenous variation. I point-identify marginal distributions of potential infant birth weight with and without smoking for compliers, which indicate pregnant women who changed their smoking status from smoking to nonsmoking in response to this tax shock. To estimate the marginal distributions of potential infant birth weight, I use the instrumental variables (IV) method presented in Abadie et al. (2002). Furthermore, I estimate the DTE bounds using plug-in estimators based on the estimates of marginal distribution functions. As a by-product, I find that the average adverse effect of smoking is more severe for women with a higher tendency to smoke and that smoking women with some college and college graduates are less likely to give births to low birth weight infants than other smoking women. In the next section, I give a formal description of the basic setup, notation, terms and assumptions $^{2}$All of these studies considered ATE or marginal distributions of potential outcomes only. 4