We study a dynamic reputation model with a fixed posted price where only purchases are public. A long-lived seller chooses costly quality; each buyer observes the purchase history and a private signal. Under a Markov selection, beliefs split into two cascades - where actions are unresponsive and investment is zero - and an interior region where the seller invests. The policy is inverse-U in reputation and produces two patterns: Early Resolution (rapid absorption at the optimistic cascade) and Double Hump (two investment episodes). Higher signal precision at fixed prices enlarges cascades and can reduce investment. We compare welfare and analyze two design levers: flexible pricing, which can keep actions informative and remove cascades for patient sellers, and public outcome disclosure, which makes purchases more informative and expands investment.
Lukyanov et al. (Thu,) studied this question.