In what I think is an insightful and potentially important paper, "Endogenous Information, Menu Costs and Inflation Persistence" (nber wp #14184, ungated version here ) Yuriy Gorodnichenko sets up a model where firms make "state dependent decisions on both pricing and acquisition of information". What he is able to do is produce more persistent effects of shocks without having to put some real rigidities in the model. Because here, when a firm changes it's price, it reveals it's information to other firms for free, so information embedded in a price change is a public good and firms postpone making them. As he puts it, "the information externality and menu costs reinforce each other in delaying price adjustment. As a result, the response of inflation to nominal shocks is both sluggish and hump shaped".
Yes! No Calvo model for price changes, no unmotivated quadratic costs of adjusting any or all real quantities (capital and labor), and he still gets the all important delayed hump inflation response!