This paper studies the implications of inflation persistence (generated by backward-looking price setters) for monetary policy in a New Keynesian "input-output" model -- a model with sticky prices in both intermediate and final goods sectors. Optimal policy under commitment depends on the degree of inflation persistence in both sectors. Under discretion, speed-limit targeting -- targeting the change in the output gap -- outperforms price-level and inflation targeting in the presence of inflation persistence. If inflation persistence is low in the intermediate goods sector, price-level targeting outperforms in inflation targeting despite high inflation persistence in the final goods sector. Illus. This is a print on demand edition of an important, hard-to-find publication.