一个具有总加价漂移的可处理菜单成本模型

Atractable menu costmodel with an aggregatemarkup drift by Ko Munakata Monetary and Economic Department January 2026 JEL classification: E23, E31 Keywords: menu cost, Phillips curve, trend inflation,frequency of price changes BISWorking Papers are written by members of the Monetary and EconomicDepartment of the Bank for International Settlements, and from time to time by othereconomists, and are published by the Bank. The papers are on subjects of topicalinterest and are technical in character. The views expressed in this publication arethose of the authors and do not necessarily reflect the views of the BIS or its membercentral banks. This publication is available on the BIS website (www.bis.org). A tractable menu cost model with an aggregate markup drift∗ Ko Munakata† January 20, 2026 Abstract This paper extends the menu cost model of Gertler and Leahy (2008) by introduc-ing a drift in the aggregate markup. Assuming that the drift is always negative andnot large, consistent with moderate and positive trend inflation, the paper analyticallycharacterizes firms’ value function and markup distribution. It derives explicit equa-tions sufficient to close the model in general equilibrium, making the calculation ofimpulse responses to aggregate shocks as easy as in conventional representative-agentNew Keynesian models. In addition, the paper shows two implications of the model.First, the model replicates the empirically observed positive correlation between theinflation rate and the frequency of price changes. Second, the model yields an explicitequation representing the Phillips curve, with additional terms that make the inflationrate more responsive to aggregate shocks. JEL Classification: E23, E31 Keywords: Menu cost; Phillips curve; Trend inflation; Frequency of price changes 1Introduction Among models of price stickiness in monetary economics, menu cost models match theevidence from empirical studies of micro prices better than conventional time-dependentmodels, such as the Calvo model (Calvo (1983)).For example, menu cost models, whencombined with a firm-level idiosyncratic shock, often generate a positive correlation be-tween the inflation rate and the frequency of firms’ price changes, as found in empiricalstudies.1In pursuit of better understanding of inflation and macroeconomic dynamics, anincreasing number of researches have examined the implications of menu cost models.2 However, menu cost models are still far less popular than the time-dependent modelsin macroeconomic policy analysis in general, primarily for technical reasons. The formermodels feature state dependence in firms’ price setting: firms are not exogenously or ran-domly given a chance to change price; instead, they do so if and only if their current price(markup) is far enough from the reset value. This state dependence, while key for the em-pirical success of menu cost models, generates non-linearity in firms’ policy rule (i.e., therule that firms follow in choosing whether and how much they change their prices) and anon-trivial price (markup) distribution. Solving the non-linear optimization problem whilekeeping track of such a distribution is numerically demanding. Moreover, most menu costmodels do not yield an explicit equation representing the Phillips curve, a simple relation-ship between the inflation rate and the real marginal cost for firms, which policy analysisoften relies on.3 To address the issues, this paper extends the menu cost model of Gertler and Leahy(2008) (GL). The key assumption of this model is that idiosyncratic shocks hit firms only occasionally and follow a uniform distribution with a wide support (Assumption 1 in Section3.1).Further assuming zero trend inflation and adopting a first-order perturbation, GLderive a Phillips curve equation that looks identical to that in the Calvo model exceptfor the coefficient value on the real marginal cost. This paper instead analyzes nonlineardynamics of firms’ value function and markup distribution by employing two assumptions.First, firms’ aggregate markup always decreases relative to the three markup values thatcharacterize firms’ policy rule, unless they are hit by an idiosyncratic shock (Assumption2). The monotonically decreasing markup, which we call a negative drift in the markup, isconsistent with positive trend inflation and represents the steady erosion of firms’ markupdue to the monotonic increase in the aggregate nominal marginal cost. Second, the drift isso small that the firms that have just adjusted their prices are going to keep the prices fora long time, unless they are hit by an idiosyncratic shock (Assumption 3). Using the twoassumptions, this paper derives explicit equations sufficient to close the model in generalequilibrium. These equations make the calculation of impulse responses to aggregate shocksas easy as in conventional representative-agent New Keynesian DSGE models. This paperthen shows two implications of the model.First, the m