Abstract We give a full analytic characterization of a large class of sticky-price models where the firm’s price-setting behavior is described by a generalized hazard function. Such a function allows for a vast variety of empirical hazards to be fitted. This setup is microfounded by random adjustment costs, as in Caballero and Engel (1999), or by information frictions, as in Woodford (2009). We establish two main results. First, we show how to identify all the primitives of the model, including the distribution of the fundamental adjustment costs and the implied generalized hazard function, using the distribution of price changes. Second, we derive a sufficient statistic for the aggregate effect of a monetary shock: given an arbitrary generalized hazard function, the cumulative impulse response of output to a once-and-for-all monetary shock is proportional to the ratio of the kurtosis of the steady-state distribution of price changes over the frequency of price adjustment. We prove that Calvo’s model yields the upper bound and Golosov and Lucas’s model the lower bound on this measure in the class of random menu cost models.