Quality at Stale Prices
Posted prices determine both demand and what a sale reveals about a seller. We study a long-lived firm whose commitment type always supplies high quality and whose strategic type chooses hidden quality when customers arrive. Purchases are public, individual non-purchases are not, and a public-information pricing desk can reset a finite-menu price only at Poisson opportunities. We derive the purchase-only Bayesian filter and an exact cost threshold for zero strategic quality. The threshold ranges over every inherited current price, not merely the price chosen at an immediate reset. We identify conditions under which the largest quality gain occurs at a stale price: silence has lowered reputation, the desk would now cut the price, and that reset would eliminate the incentive to supply quality. Our main theorem shows that a nonzero full-state stationary equilibrium bifurcates from this threshold. If the cost shortfall is $δ$, quality has height $O(δ)$, support width $O(\sqrtδ)$, and total mass $O(δ^{3/2})$; value and pricing feedback are therefore lower order. Price rigidity can sustain quality by preserving the demand environment in which a sale is both difficult and reputationally valuable.