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.