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.

We study how to optimally segment monopolistic markets with a redistributive objective. We characterize optimal redistributive segmentations and show that they (i) induce the seller to price progressively, i.e., richer consumers pay higher prices than poorer ones, and (ii) may not maximize consumer surplus, instead granting extra profits to the monopolist. We further show that optimal redistributive segmentations are implementable via price-based regulation.

Rewards (e.g., prizes, labels, money) are often allocated to agents based on a measured characteristic or aggregate—commonly called a score. This paper studies how a principal should optimally allocate rewards when: (i) the underlying characteristic and reward are complements in her preferences, and (ii) agents can strategically invest to enhance their scores (and, to some extent, manipulate them). Our main result shows that, in a broad class of environments, optimal mechanisms take the form of simple cutoff rules that grant the highest reward to agents whose scores exceed a specified threshold while offering nothing below this threshold. Notably, intermediary rewards and randomization provide no additional benefit to the principal. This finding establishes a stronger theoretical foundation and provides a potential explanation for the prevalence of binary ``pass or fail'' rules in environments where agents can strategically invest in response to screening mechanisms.

We study a persuasion problem in which a sender designs an information structure to induce a non-Bayesian receiver to take a particular action. The receiver, who is privately informed about his preferences, is a wishful thinker: he is systematically overoptimistic about the most favorable outcomes. We show that wishful thinking can lead to a qualitative shift in the structure of optimal persuasion compared to the Bayesian case, whenever the sender is uncertain about what the receiver perceives as the best-case outcome in his decision problem.