This paper studies the implications for inflation dynamics of introducing non-price competition into a New Keynesian model featuring both nominal rigidities, which are in the form of staggered prices, and real rigidities, which are in the form of strategic complementarities in price setting. Under very general assumptions, we show that the presence of non-price competition among firms dampens the overall degree of real rigidities in the economy, thereby increasing the sensitivity of inflation to movements in real marginal costs. Because of this property, our analysis provides additional insights to the existing theories on real rigidities, showing that the strong linkage between real rigidities and the sensitivity of inflation to real marginal costs that has been found in previous works is not robust across alternative assumptions about inter-firms competition.