Informational Inertia and Staggered Prices

We study a duopoly in which firms and consumers do not know which product better fits consumers' needs. Consumers begin with a default seller, privately know their switching costs, and may pay to acquire a private signal about product fit. Search determines which evidence is acquired; switching determines whether it appears in the observed purchase. A market may therefore remain divided yet informationally silent. We characterize the interval of silent price gaps and show that, at the symmetric prior and equal prices, silence obtains precisely when the switching-cost floor plus twice the comparison cost exceeds the value of a favorable signal. Prices lie on a finite grid, and firms receive independent Poisson reset opportunities. A martingale argument confines every ergodic invariant regime to a fixed belief and a finite policy-closed class of silent price pairs. A Calvo renewal equation then links stationary dispersion to nonmatching resets and yields identities for every positive price-gap moment. We give primitive conditions under which every invariant distribution of every stationary Markov equilibrium, whenever one exists, has positive dispersion; prohibitively costly comparison yields absorption at equal cap prices. Thus consumer frictions determine both the information in market activity and the price configurations compatible with stationarity.