  A monotone function interval is the set of monotone functions that lie
pointwise between two fixed monotone functions. We characterize the set of
extreme points of monotone function intervals and apply this to a number of
economic settings. First, we leverage the main result to characterize the set
of distributions of posterior quantiles that can be induced by a signal, with
applications to political economy, Bayesian persuasion, and the psychology of
judgment. Second, we combine our characterization with properties of convex
optimization problems to unify and generalize seminal results in the literature
on security design under adverse selection and moral hazard.
