This paper studies a general equilibrium model with multiple stages of production and asynchronized price setting that provides a new explanation for the observed persistent real effects of monetary shocks. The key feature of the model is a vertical chain-of-production structure. In this model, the effects of monetary shocks on price adjustment are gradually dampened via the interactions of firms through their input-output relations and the timing of their price decisions. The model predicts that prices adjust by a smaller amount and less rapidly at later stages than at earlier stages, which is supported by empirical evidence. More importantly, an increase in the total number of stages in the model leads to not only uniformly larger and longer-lasting real effects but also flatter paths of aggregate output response. With sufficiently many stages, the price level adjustment becomes arbitrarily close to zero and the aggregate output tends to carry the full burden of adjustment. Thus, the chain-of-production mechanism goes a long way in propagating the shocks.