Figure 10: a$_{k}$$_{+1}$ ≤ a$_{k}$ + δ v.s. a$_{k}$$_{+1}$ = a$_{k}$ + δ 3.2.2 Concave/Convex Treatment Response Recall the setting of Example 2 in Subsection 2.1. Let W denote the outcome without treatment and let Y$_{0}$ and Y$_{1}$ denote the potential outcomes with treatment at low-intensity, and with treatment at high-intensity, respectively. Let t$_{d}$ denote the level of input for each treatment status for d = 0 , 1 , while t$_{W}$ is a level of input without the treatment with t$_{W}$ < t$_{0}$ < t$_{1}$ . Either ( W, Y$_{0}$ ) or ( X, Y$_{1}$ ) is observed for each individual, but not ( W, Y$_{0}$, Y$_{1}$ ). Given W = w, the distribution of Y$_{1}$ − Y$_{0}$ under concave treatment response corresponds to the probability of the intersection of { Y$_{1}$ − Y$_{0}$ ≤ δ } , { Y$_{0}$ − w t$_{0}$ − t$_{W}$ ≥ Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ $^{}}$, and { Y$_{1}$ ≥ Y$_{0}$ ≥ w } in the support space of ( Y$_{0}$, Y$_{1}$ ). Similarly, given W = w, the distribution of Y$_{1}$ − Y$_{0}$ under convex treatment response corresponds to the probability of the intersection of { Y$_{1}$ − Y$_{0}$ ≤ δ } , { Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ ≥ Y$_{0}$ − w t$_{0}$ − t$_{W}$ $^{}}$, and { Y$_{1}$ ≥ Y$_{0}$ ≥ w } in the support space of ( Y$_{0}$, Y$_{1}$ ). Note that { Y$_{0}$ − w t$_{0}$ − t$_{W}$ ≥ Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ } and { Y$_{1}$ − Y$_{0}$ t$_{1}$ − t$_{0}$ ≥ Y$_{0}$ − w t$_{0}$ − t$_{W}$ } correspond to the regions below and above the straight line Y$_{1}$ = t$_{1}$ − t$_{W}$ t$_{0}$ − $_{t$_{W}$}$Y$_{0}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − $_{t$_{W}$}$w , respectively. Corollary 2 derives sharp bounds under concave treatment response and convex treatment response from Theorem 1. Corollary 2 Take any w in the support of W such that the conditional marginal distributions of Y$_{1}$ and Y$_{0}$ given W = w are both absolutely continuous with respect to the Lebesgue measure on $_{R}$. Let F$_{0}$$_{,W}$ ( ·| w ) and F$_{1}$$_{,W}$ ( ·| w ) be conditional distribution functions of Y$_{0}$ and Y$_{1}$ given W = w , respectively. (i) Under concave treatment response, sharp bounds on the DTE are given as follows: for any δ ∈$_{R}$ , F L Δ ( δ ) ≤ F$_{Δ}$ ( δ ) ≤ F U Δ ( δ ) 24