Bounding the Population Shares Affected by Treatments

The fraction of a population that is affected by a treatment (the “responders”) may be as important to identify as the average magnitude of the treatment effect. I show that if the distributions of potential outcomes with and without treatment are identified, then the total variation distance between them serves as the sharp lower bound on the share of responders. It can be computed for randomized control trials, instrumental variables, and other empirical designs. I demonstrate the usefulness of the approach in three examples of economic interest, related to behavioral biases in retirement savings, electoral fraud, and student cheating.