Inflation-targeting central banks around the world often state their inflation objectives with regard to the consumer price index (CPI). Yet the literature on optimal monetary policy based on models with nominal rigidities and more than one sector suggests that CPI inflation is not always the best choice from a social welfare perspective. We revisit this issue in the context of an estimated multi-sector New-Keynesian small open economy model where sectors are heterogeneous along multiple dimensions. With key parameters of the model estimated using data from an inflation targeting economy, namely Canada, we particularly focus on (i) the role of sector-specific real rigidities, specially in the form of factor mobility costs, and (ii) welfare implications of targeting alternative price indices. Our estimations reveal considerable heterogeneity across sectors, and in several dimensions. Moreover, in contrast to existing studies, our welfare analysis comparing simple optimized policy rules based on alternative sectoral inflation rates provides support for CPI-based targeting policies by central banks. Capital mobility costs matter importantly in this regard.