The transitory shock of the financial crisis of 2008 pushed most economies to permanently lower-level growth paths than those prevalent before the crisis, which can be considered as a manifestation of hysteresis. It is well known that some fixed adjustment costs lead to hysteresis in aggregate output. This paper investigates within an agent-based model, whether the fixed costs of price adjustment (menu costs) lead to the same result. Hysteresis emerges in some simple variants of the model independently of firms being assumed boundedly or perfectly rational, but these model variants fit to the empirical data poorly. The model's empirical performance can be improved by assuming that firms are hit by idiosyncratic productivity shocks, but these shocks eliminate hysteresis generated by menu costs. However, hysteresis survives even in their presence, if it is generated by demand-supply interactions, i.e., positive feedbacks from the output gap to potential output. Our conclusion is that if one would like menu costs to serve as an at least as relevant explanation for the hysteretic dynamics of aggregate output as demand-supply interactions, one has to find an alternative assumption to replace idiosyncratic productivity shocks as a mechanism to assure good empirical fit for the model.