Identifying the Cumulative Causal Effect of a Non-Binary Treatment from a Binary Instrument

The effect of a treatment may depend on the intensity with which it is administered. We study identification of ordered treatment effects with a binary instrument, focusing on the effect of moving from the treatment's minimum to maximum intensity. With arbitrary heterogeneity across units, standard IV assumptions (Angrist and Imbens, 1995) do not constrain this parameter, even among compliers. We consider a range of additional assumptions and show how they can deliver sharp, informative bounds. We illustrate our approach with two applications, involving the effect of (1) health insurance on emergency department usage, and (2) attendance in an after-school program on student learning.