Assumption 2: Income is a Monoton Instrumental Variable (MIV)
The second assumption is that the probability of bad health decreases weakly with income. Assuming this allows the derivation of additional bounds on P[H(0)=1] and P[H(1)=1].
To do so, decompose P[H(0)=1] and P[H(1)=1] by applying the Law of Total Probability with respect to the instrumental variable: