Menus costs impacts firms and mean that inflation can be persistent. During the last true inflation bout, there was significant research on the impact of menu costs on the effectiveness of monetary policy and the reason for why rational expectations could not explain the actual behavior of the economy. Menu costs create price stickiness which impacts firm profits and economic behavior. It was one of the micro foundations for New Keynesian thinking.
Menu cost arguments are simple. There are costs associated with firms changing prices. Hence, prices are often sticky. The presence of sticky prices means that policy surprises will have real effects. Sticky prices impact business profits and the response to policy changes.
Menu costs create asymmetric price changes to input cost changes. Initially there will be a slow response to a producer price increase if businesses view price increases are transitory. Firms may accept lower margins; however, over time, price will ratchet higher by more than the increase in inputs because the cost of menu changes is not trivial.
Prices may increase by more than a change to just offset input costs. On the other hand, once prices increase, they will be sticky at returning to their old levels even if input costs decrease. Think of the simple example of a restaurant menu. If new prices are printed, it is unlikely that the old price list will be returned even if inputs costs decrease. This may not occur with all markets. For example, the asymmetric price behavior is exactly the opposite for gasoline prices. See "Why don't gasoline prices move one-for-one with oil prices"
CPI will not immediately move in lockstep with producer prices, but if the PPI remains elevated for an extended period and there is the perception that these prices will not be quickly reversed, retail prices will start to ratchet higher to return margins to their old level. Of course, the question is whether consumers will accept these higher prices and whether all businesses will follow. This is the critical reason for focusing on inflationary expectations and the feedback to behavior. Of course, we are in a new world where many prices can be changed quickly through the internet. If inflation expectations change with lower menu costs price dynamics can be more volatile and less sticky.
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