Suppose that there exists an integer l s.t. a$_{l}$$_{+2}$ − a$_{l}$ ≤ δ. Let Then The inequality in the fourth line holds because MTR implies stochastic dominance of Y$_{1}$ over Y$_{0}$ . This is illustrated in Figure A.3(a) and (b), where the sum of the lower bound on each triangle is equal to ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } and ∞ ∑ k = −∞ max { F$_{1}$ ($_{˜}$ a$_{k}$$_{+1}$ ) − F$_{0}$ ($_{˜}$ a$_{k}$ ) , 0 } , respectively. Figure B.1: a$_{k}$$_{+2}$ − a$_{k}$ > δ at the optimum Therefore, it is innocuous to assume a$_{k}$$_{+2}$ − a$_{k}$ > δ for every integer k at the optimum. 73 ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } = l − 1 ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } + max { F$_{1}$ ( a$_{l}$$_{+1}$ ) − F$_{0}$ ( a$_{l}$ ) , 0 } + max { F$_{1}$ ( a$_{l}$$_{+2}$ ) − F$_{0}$ ( a$_{l}$$_{+1}$ ) , 0 } + ∞ ∑ k = l +2 max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } ≤ l − 1 ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } + max { F$_{1}$ ( a$_{l}$$_{+2}$ ) − F$_{0}$ ( a$_{l}$ ) , 0 } + ∞ ∑ k = l +2 max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } = ∞ ∑ k = −∞ max { F$_{1}$ ($_{˜}$ a$_{k}$$_{+1}$ ) − F$_{0}$ ($_{˜}$ a$_{k}$ ) , 0 } . ˜ a$_{k}$ = a$_{k}$ for k ≤ l, ˜ a$_{k}$ = a$_{k}$$_{+1}$ for k ≥ l + 1 . (a) (b)