The Monopolist's problem is a simple model from economics which displays rich mathematical behaviour and lies at the intersection of optimal transport, free boundary problems, and convex analysis. In this talk I will discuss joint work-in-progress with Robert McCann and Kelvin Shuangjian Zhang. We employ new techniques based upon convex perturbations which we combine with the sweeping (balayage) property of measures in convex order (first used in this context by Rochet and Choné). Our techniques let us answer questions previously out of reach including regularity of the free boundary problem (where we highlight similarities to the obstacle problem), and confirming many aspects of McCann and Zhang's hypothesized form of the solution on the square.