We introduce convex function intervals (CFIs): families of convex functions subject to prescribed constraints on their levels and slopes. We show that a large class of economic design problems can be formulated as linear programs on CFIs. Our main contributions are: (i) a geometric characterization of CFIs' extreme points; and (ii) a verification theorem providing tractable sufficient conditions to confirm the optimality of a candidate extreme point. Our framework unifies existing results in mechanism and information design and delivers closed-form expressions for optimal mechanisms in environments generally considered intractable: monopolistic screening and delegation with type-dependent participation constraints, persuasion with informativeness constraints, and contest design with limited disposal of prizes.

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.

We study how to optimally allocate bounded rewards to heterogeneous agents based on a measured score when: (i) the principal's preferences exhibit complementarities between rewards and an agent's worth, and (ii) agents can invest to enhance  (and, to some extent, manipulate) their scores. Our main result is that, in a large class of environments, it is optimal to use cutoff rules that promise the highest reward to agents with a score above the cutoff and nothing to others. In particular, intermediate rewards and randomization are useless. 

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.