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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.

This paper studies the design of a heterogeneous good by a public institution when allocation is delegated to a revenue-maximizing monopolist. The designer—a public institution—chooses a distribution of qualities to maximize utilitarian welfare, anticipating the monopolist’s optimal screening mechanism. We show that the monopolist’s screening incentives impose sharp constraints on achievable outcomes, generating two distinct distortions: First, under mild conditions, the set of excluded consumers is independent of the chosen quality distribution. Second, the designer optimally shades quality downward for intermediate types relative to the designer's first-best allocation. Exploiting a reduction to a convex optimization problem, we characterize the optimal quality distribution and identify conditions under which providing a uniform good is optimal.