Many seemingly different economic problems share a common mathematical structure: they involve the maximization of a functional over sets of monotonic functions that are either majorized by, or majorize, a given function. This paper presents new, simpler proofs for the main characterization results of the extreme points of sets defined by monotonicity and majorization constraints obtained by Kleiner, Moldovanu, and Strack (2021). It demonstrates how these characterizations can be applied to a broad range of economic applications, including auction and information design, contest design, optimal delegation, optimal stopping, and decision problems under risk. The paper concludes with an overview of recent related work extending these characterizations to settings with additional constraints, multidimensional state spaces, and alternative stochastic orders.