3.2 Economic Examples 3.2.1 Monotone Treatment Response In this subsection, I derive sharp bounds on the DTE for concrete economic examples from the general characterization in Theorem 1. As economic examples, MTR, concave treatment response, convex treatment response, and the Roy model of self-selection are discussed. Since the seminal work of Manski (1997), it has been widely recognized that MTR has an interesting identifying power for treatment effects parameters. MTR only requires that the potential outcomes be weakly monotone in treatment with probability one: His bounds on the DTE under MTR are obtained as follows: for δ < 0 , F$_{Δ}$ ( δ ) = 0 , and for δ ≥ 0 , where p = Pr ( D = 1) , and y L 0 is the support infimum of Y$_{0}$ while y U 1 is the support supremum of Y$_{1}$. He did not impose any other condition such as given marginal distributions of Y$_{0}$ and Y$_{1}$ . Note that MTR has no identifying power on the DTE in the binary treatment setting without additional information. Since MTR restricts only the lowest possible value of Y$_{1}$ − Y$_{0}$ as zero, the upper bound is trivially obtained as one for any δ ≥ 0. Similarly, MTR is uninformative for the lower bound, since MTR does not restrict the highest possible value of Y$_{1}$ − Y$_{0}$ . 10 Furthermore, when the support of each potential outcome is given as $_{R}$, they yield completely uninformative upper and lower bounds [0 , 1] . However, I show that given marginal distribution functions F$_{0}$ and F$_{1}$ , MTR has substantial identifying power for the lower bound on the DTE. Corollary 1 Suppose that Pr ( Y$_{1}$ = Y$_{0}$ ) = 0 . Under MTR, sharp bounds on the DTE are given as follows: for any δ ∈$_{R}$ , $^{10}$Note that Y$_{1}$ is observed for the treated and Y$_{0}$ is observed for the untreated groups. For the treated, the highest possible value is Y − Y L 0 , while it is Y U 1 − Y for the untreated. The lower bound is achieved when Pr( Y$_{0}$ = y L 0 | D = 1) = 1 and ( Y$_{1}$ = y U 1 | D = 0) = 1 . 19 F L Δ ( δ ) ≤ F$_{Δ}$ ( δ ) ≤ F U Δ ( δ ) , Pr $^{(}$Y − y L 0 ≤ δ | D = 1 $^{)}$p + Pr $^{(}$y U 1 − Y ≤ δ | D = 0 $^{)}$(1 − p ) ≤ F$_{Δ}$ ( δ ) ≤ 1 , Pr ( Y$_{1}$ ≥ Y$_{0}$ ) = 1 .