We study the nonlinear pricing of goods whose usage generates revenue for the seller and of which buyers can freely dispose. The optimal price schedule is a multi-part tariff, featuring tiers within which buyers pay a marginal price of zero. We apply our model to digital goods, for which advertising, data generation, and network effects make usage valuable, but monitoring legitimate usage is infeasible. Our results rationalize common pricing schemes including free products, free trials, and unlimited subscriptions. The possibility of free disposal harms producer and consumer welfare and makes both less sensitive to changes in usage-based revenue and demand.

This article proposes a model of non-Bayesian social learning in networks that accounts for heuristics and biases in opinion aggregation. The updating rules are represented by non-linear opinion aggregators from which we extract two extreme networks capturing strong and weak links. We provide graph-theoretic conditions for these networks that characterize opinions’ convergence, consensus formation, and efficient or biased information aggregation. Under these updating rules, agents may ignore some of their neighbours’ opinions, reducing the number of effective connections and inducing long-run disagreement for finite populations. For the wisdom of the crowd in large populations, we highlight a trade-off between how connected the society is and the non-linearity of the opinion aggregator. Our framework bridges several models and phenomena in the non-Bayesian social learning literature, thereby providing a unifying approach to the field.

We consider general Bayesian persuasion problems where the receiver’s utility is single-peaked in a one-dimensional action. We show that a signal that pools at most two states in each realization is always optimal and that such “pairwise” signals are the only solutions under a nonsingularity condition on utilities. Our core results provide conditions under which the induced receiver action is single-dipped or single-peaked on each set of nested signal realizations. We also provide conditions for the optimality of either full disclosure or negative assortative disclosure, where all signal realizations are nested. Methodologically, our results rely on novel duality and complementary slackness theorems. Our analysis extends to a general problem of assigning one-dimensional inputs to productive units, which we call “optimal productive transport.” This problem covers additional applications including matching with peer effects (assigning workers to firms, students to schools, or residents to neighborhoods), robust option pricing (assigning future asset prices to price distributions), and partisan gerrymandering (assigning voters to districts).