How does firm heterogeneity in financial constraints shape the transmission of monetary policy to prices and inflation? Using detailed micro data on Swedish firms and high-frequency monetary policy surprises for the Riksbank, we document a pronounced heterogeneity: Smaller, financially constrained firms adjust prices significantly less than larger firms in response to changes in monetary policy. This result is consistent with models of customer markets and financial constraints: Because the external finance premium rises after a monetary contraction, constrained firms lower prices less to maintain cash flows, resulting in a loss of market share. Additional evidence supports this channel. After a monetary policy shock, smaller firms exhibit a more negative response of their sales. They also have a higher share of bank debt, and their debt is significantly more responsive to monetary policy. Further evidence shows that alternative channels like the working capital channel or differences in marginal costs cannot explain the differential price response of small and large firms. Our results demonstrate that financial constraints significantly weaken the transmission of monetary policy to inflation and imply that the strength and distribution of financial constraints across firms is a key determinant of policy effectiveness.