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This paper presents a simple "menu cost" example in which there is a clear distinction between price stickiness and downward rigidity of prices. While price stickiness may or may not exist in "menu cost" models, downward rigidity shows up whenever there is a reduction of the expected rate of inflation. This reduction changes the optimal target and threshold for each firm. For some, it also implies an immediate increase of their own price. This upward jump of prices in case of a disinflationary attempt is interpreted as downward rigidity since there are no symmetric forces when expected inflation increases.
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Daniel Tsiddon (1991) studied this question.
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