This paper documents that households with higher marginal propensities to consume (MPCs) consume goods with more flexible prices. Consequently, they face more cyclical and volatile inflation, and experience higher inflation following expansionary monetary policy shocks. We embed this MPC‐price stickiness relationship into a tractable multi‐sector two‐agent New Keynesian (TANK) model and show that it dampens monetary policy effectiveness by about 15% relative to a homogeneous‐basket benchmark. Introducing heterogeneous consumption baskets also generates an inefficient flexible‐price equilibrium, leading to a novel trade‐off between stabilization and redistribution, which alters the optimal monetary policy prescription.