We develop a multisector, open economy, New Keynesian framework to evaluate how potentially binding capacity constraints, and shocks to them, shape inflation. We show that binding constraints for domestic and foreign producers shift domestic and import price Phillips Curves up, similar to reduced-form markup shocks. Further, data on prices and quantities together identify whether constraints bind due to increased demand or reductions in capacity. Applying the model to interpret recent US data, we find that binding constraints explain half of the increase in inflation during 2021-2022. In particular, tight capacity served to amplify the impact of loose monetary policy in 2021, fueling the inflation takeoff.