I provide a microfounded theory for one of the oldest, but so far informal, explanations of price rigidity: the kinked demand curve theory. Assuming that some customers observe at no cost only the price of the store they happen to be at gives rise to a kink in firms’ demand curves: a price increase above the market price repels more customers than a price decrease attracts. The kink in turn makes a range of prices consistent with equilibrium, but an intuitive criterion—the adaptive rational-expectations criterion—selects a unique equilibrium where prices stay constant for a long time. The kinked-demand theory is consistent with price-setters’ account of price-rigidity as arising from the customer’ —not the firm’s—side, and can be tested against menu-cost models in micro data: it predicts that prices should be more likely to change if they have recently changed, and that prices should be more flexible in markets where customers can more easily compare prices. The kinked-demand theory has novel implications for monetary policy: its Phillips curve is strongly convex but does not contain any (present or past) expectations of inflation; its trade-off between output and inflation persists in the long-run; changes to the distribution of sectoral productivity shift the Phillips curve; and monetary shocks have a much longer-lasting real effect than in a menu-cost model, despite also being a model of state-dependent pricing.