We develop a multisector sticky-price DSGE model that can endogenously deliver differential responses of prices to aggregate and sectoral shocks. Input-output production linkages and a (standard) monetary policy rule contribute to a slow response of prices to aggregate shocks. In turn, labor market segmentation at the sectoral level induces within-sector strategic substitutability in price-setting decisions, which helps the model deliver a fast response of prices to sector-specific shocks. We estimate the model using aggregate and sectoral price and quantity data for the United States and find that it accounts well for a range of sectoral price facts.