This paper disentangles fluctuations in disaggregate prices into macroeconomic and idiosyncratic components using a factor-augmented vector autoregression (FAVAR) in order to shed light on sectoral inflation dynamics in Switzerland. We find that disaggregated prices react only slowly to monetary policy and other macroeconomic shocks, but relatively quickly to idiosyncratic shocks. We document that there is a large heterogeneity across sectors in the reaction to monetary policy shocks and show that sectors with larger volatility of idiosyncratic shocks react more readily to monetary policy. This finding stands in contrast to the rational inattention model of price setting. We also find that sectors, which change prices infrequently, react less strongly but if they do change their prices, they adjust them by a large amount. This suggests that the source of sluggish response to aggregate shocks is heterogeneity in menu costs rather than rational inattention. Furthermore, even though prices respond with a significant delay to identified monetary policy shocks, we find no evidence of a price puzzle on average. For single sectors, however, we still find a hump-shaped response which can partially be explained by the fact that, by law, rents are tied to interest rates in Switzerland.