Information design through scarcity and social learning

We show that a firm may benefit from strategically creating scarcity for its product, in order to trigger herding behaviour from consumers in situations where such behaviour is otherwise unlikely. We consider a setting with social learning, where consumers observe sales from previous cohorts and update beliefs about product quality before making their purchase. Imposing a capacity constraint directly limits sales but also makes information coarser for consumers, who react favourably to a sell-out because they infer only that demand must exceed capacity. Consumer learning is then limited even with large cohorts and unbounded private signals, because the firm acts strategically to influence the consumers' learning environment. Our results suggest that in suitable environments capacity constraints can serve as a useful tool to implement optimal information design in practice: if private signals are not too precise and capacity can be changed over time, then in large markets the firm's optimal choice of capacity delivers the same expected sales as the Bayesian persuasion solution.