The introduction of digital price tags and e-commerce facilitates the implementation of price adjustments and thereby diminishes the degree of nominal rigidity in an economy. Is this phenomenon welfare-improving? We address this question using a multi-sector New Keynesian model with information frictions and dispersed beliefs. Increased price flexibility may decrease welfare through the dispersed belief channel and the amplified spillover effects. Dispersed beliefs create a novel channel through which the welfare cost of inflation in a sector increases with price flexibility, altering the optimal inflation index stabilization policy. A monetary policy that stabilizes the optimal inflation index mitigates this paradox.