In this paper, I study monetary non-neutrality in a frictional product market. The model incorporates the idea that goods are in general not monopolistically supplied; rather, consumers can purchase the same good from many outlets. I find that incorporating this feature into a menu cost model increases the degree of monetary non-neutrality. The reason is that competition between stores makes their prices complementary, so that firms face strong penalties for setting prices far away from their competitors. It makes pass-through of cost shocks heterogeneous and decreases the strength of the selection effect. I also use the model to study cyclical changes in monetary non-neutrality. I find that the data are consistent with a decline in the competitiveness of markets in recessions, which makes monetary policy less effective at stimulating output.