Herding Prices: Informational Inertia and Price Dispersion

We study a duopoly in which firms and consumers are initially uncertain which product better fits a common need. Consumers learn from past purchases and private signals, but buying away from a privately observed default entails a switching cost. Two assignment-specific cutoffs partition the price gap into dominance, partially informative, signal-revealing, and inertial regions. In the inertial region, each consumer remains with her default seller: both firms continue to make sales, yet purchases reveal no private information. With fixed prices, we derive a closed-form discounted-demand kernel and, under a bounded-range condition, the unique symmetric atomless price equilibrium. We then allow switching costs to be privately heterogeneous above a common positive floor. This perturbation removes consumer ties and impossible purchase histories without removing informational inertia; if the floor is zero, by contrast, exact inertia disappears. The deterministic model is recovered as heterogeneity vanishes. Finally, firms receive symmetric Poisson opportunities to reset prices. On any finite price grid, the regularized Calvo game has a stationary Markov equilibrium on its countable reachable state space. Thus switching costs affect not only demand elasticity but also how much private information the sequence of purchases transmits.