The Economics of Convex Function Intervals

We introduce convex function intervals (CFIs): families of convex functions satisfying given level and slope constraints. CFIs naturally arise as constraint sets in economic design, including problems with type-dependent participation constraints and two-sided (weak) majorization constraints. Our main results include: (i) a geometric characterization of the extreme points of CFIs; (ii) sufficient optimality conditions for linear programs over CFIs; and (iii) methods for nested optimization on their lower level boundary that can be applied, e.g., to the optimal design of outside options. We apply these results to four settings: screening and delegation problems with type-dependent outside options, contest design with limited disposal, and mean-based persuasion with informativeness constraints. We draw several novel economic implications using our tools. For instance, we show that better outside options lead to larger delegation sets, and that posted price mechanisms can be suboptimal in the canonical monopolistic screening problem with nontrivial, type-dependent participation constraints.