A large literature seeks to provide microfoundations of price setting for macro models. A challenge has been to develop a model in which monetary policy shocks have the highly persistent effects on real variables estimated by many studies. Nominal price stickiness has proved helpful but not sufficient without some form of "real rigidity" or "strategic complementarity." We embed a model with a real rigidity a la Kimball (1995), wherein consumers flee from relatively expensive products but do not flock to inexpensive ones. We estimate key model parameters using micro data from the U.S. CPI, which exhibit sizable movements in relative prices of substitute products. When we impose a significant degree of real rigidity, fitting the micro price facts requires very large idiosyncratic shocks and implies large movements in micro quantities.