We study the welfare cost of inflation in a multi-sector sticky price model with production networks. We consider both time- and state-dependent (menu cost) pricing models with firm-specific productivity shocks, and match sectoral moments of the distribution of price changes in our quantitative exercises. We derive analytical expressions and sufficient statistics for the cost of inflation which illustrate how aggregate productivity and labor supply effects of inflation are determined by the interaction of heterogeneous input-output linkages and price stickiness across sectors. A key mechanism is that in a network economy, mis-allocative consequences of inflation are amplified by input-output linkages as they propagate across sectors. Quantitatively, even at low to moderate levels of inflation, we find that our calibrated production network model amplifies the costs of inflation significantly, around 5-15 times, compared to the one-sector model.