Sticky Prices for Inflationary Economies: A Tractable Linear Approximation to Menu Cost Models with Trend Inflation

When inflation is low, the Calvo model is a good approximation of sticky prices. But when inflation is high, menu costs matter for macroeconomics. Drawing from recent work on mean field games, I derive an analytical solution to the menu cost model with trend inflation in response to small shocks. The solution includes dynamics of the value function, distribution of price gaps, and aggregate variables. Then, I consider a discrete time approximation that is tractable enough for use in standard DSGE models. Menu costs modify the usual Calvo Phillips curve with a single variable: the frequency of price adjustment. Accounting for the frequency matters in an inflationary economy; when trend inflation is zero, the term disappears. But surprisingly, the effect of trend inflation on the Phillips curve is first-order. The modified system is a function of the microfoundations and can be calibrated to match pricing statistics, a useful result even without trend inflation. Finally, I embed the price-setting block in an ot