This paper revisits the New Keynesian framework, theoretically and quantitatively, in an economy with multiple sectors and input‐output linkages. Analytical expressions for the Phillips curve and welfare, derived as a function of primitives, show that the slope of all sectoral and aggregate Phillips curves is decreasing in intermediate input shares, while productivity fluctuations endogenously generate an inflation‐output tradeoff—except when inflation is measured according to the novel divine coincidence index. Consistent with the theory, the divine coincidence index provides a better fit in Phillips curve regressions than consumer prices. Monetary policy can no longer achieve the first‐best, resulting in a welfare loss of 2.9% of per‐period GDP under the constrained‐optimal policy, which increases to 3.8% when targeting consumer inflation. The constrained‐optimal policy must tolerate relative price distortions across firms and sectors in order to stabilize the output gap, and it can be implemented via a Taylor rule that targets the divine coincidence index.