Motivation: State-dependent pricing matters most when trend inflation is large

Menu cost models are too intractable in general

  • Infinite dimensional state, dynamic inaction boundaries

  • How to use for business cycle analysis? monetary policy?

When trend inflation is low: Calvo model accurately approximates the Phillips Curve (Auclert et al. 2024)

When trend inflation is high: menu costs matter, but models become really intractable

  • State-dependent pricing describes high-inflation economies (Alvarez et al 2018)

  • Most analytical results do not apply, assume zero trend inflation (e.g. Alvarez and Lippi 2022)

Contribution #1: MFG Analytical Solution

I derive an analytical solution to the menu cost mean field game (MFG) with trend inflation (for small shocks)

Solution characterizes dynamics of:

  • Value function and decisions (HJB)
    • Inaction region \([\underline{x}(t), \bar{x}(t)]\) and reset price \(x^*(t)\)
  • Distribution of price gaps \(h(x,t)\) (KFE) and aggregate dynamics
  • Alvarez, Lippi, and Souganidis (2023) do this without trend inflation using a shortcut: the reinjection of resetting firms can be ignored due to symmetry.
  • Symmetry is lost with trend inflation; reinjecting firms must be accounted for, as in Adams (2025).

Contribution #2: The PED and the MCNK Phillips Curve

The analytical MFG solution is infinite-dimensional. I derive a tractable low-dimensional approximation: the Primary Eigenfunction Discretization (PED).

The PED implies a Phillips curve that nests the standard NKPC:

\[\pi_t = \underbrace{\Lambda\,mc_t + \beta\,\mathbb{E}_t[\pi_{t+1}]}_{\text{Calvo term}} + \underbrace{\epsilon\!\left(\theta^{-1}F_t - \beta\,\mathbb{E}_t[F_{t+1}]\right)}_{\text{\textcolor{red}{FPA correction}}}\]

  • When \(\bar{\pi} = 0\): \(\epsilon = 0\) \(\Rightarrow\) standard NKPC (validates Auclert et al. 2024)
  • When \(\bar{\pi} > 0\): the frequency of price adjustment (FPA) \(F_t\) is an “inflation accelerator” (Blanco et al. 2025)
  • FPA captures the time-varying selection effect (Golosov & Lucas 2007): adjusting firms are those with the largest price gaps
  • Coefficients are functions of microfoundations, so the PED can be calibrated from pricing statistics

Contribution #3: General Equilibrium Analysis

Because the PED is so tractable, it is easily embedded into DSGE models for policy analysis, etc.

Menu costs change macro dynamics through two channels:

  1. Steeper Phillips curve (\(\Lambda\) larger than Calvo): menu cost firms respond more to shocks
  2. FPA amplification (\(\epsilon > 0\)): endogenous adjustment frequency feeds back into inflation

Both channels strengthen with trend inflation: Optimal monetary policy becomes more aggressive as trend inflation increases

Should trend inflation be ignored for low-inflation economies?

  • Second-order effects on firms’ decisions (\(\Lambda\) channel)
  • … but first-order effects on aggregation (\(\epsilon\) channel)

The Model

Standard ingredients:

  • Monopolistic firms with menu cost \(\Psi\)
  • Calvo-plus pricing (Nakamura & Steinsson 2010): random free resets at rate \(\zeta\)
  • Idiosyncratic cost/preference shocks (diffusion variance \(2\nu\))
  • Trend inflation \(\bar{\pi}\)

Firm’s state: is the price gap : \[x_i(t) = \log P_i(t) - \log Z_i(t) - \log \bar{W}(t) - \mu\]

  • Marginal cost rend \(\bar{W}(t)\) grows at rate \(\bar{\pi}\): \(\mathbb{E}[dx] = -\bar{\pi}\,dt\) (price gaps drift with inflation)
  • Inaction region: \([\underline{x}(t), \bar{x}(t)]\), reset to \(x^*(t)\)

The HJB

Value function \(v(x,t)\), for \(x \in [\underline{x}(t), \bar{x}(t)]\): \[\rho\, v = -\mathbf{B}\bigl(x - MC(t)\bigr)^2 + \partial_t v - \bar{\pi}\,\partial_x v + \nu\,\partial_x^2 v + \zeta\bigl(v(x^*(t),t) - v(x,t)\bigr)\]

Condition Equation
Value-matching \(v(\underline{x},t) = v(x^*,t) + \Psi = v(\bar{x},t)\)
Reset optimality \(\partial_x v(x^*,t) = 0\)
Smooth-pasting \(\partial_x v(\underline{x},t) = \partial_x v(\bar{x},t) = 0\)
Terminal Condition \(v(x,T) = \phi_{\mathrm{term.}}(x)\)

Five free-boundary conditions pin \((v(x,t), \underline{x}(t),\, \bar{x}(t),\, x^*(t))\)

Linearized ( \(\hat{v}(x,t) \equiv \partial_{MC}(\cdot)v\big|_{MC=0}\)), for \(x \in [\underline{x}, \bar{x}]\): \[\rho\,\hat{v} = -2\mathbf{B}x\cdot MC(t) + \partial_t\hat{v} - \bar{\pi}\,\partial_x\hat{v} + \nu\,\partial_x^2\hat{v} + \zeta\bigl(\hat{v}(x^*,t) - \hat{v}(x,t)\bigr)\]

Condition Linearized equation
Value-matching \(\hat{v}(\underline{x},t) = \hat{v}(\bar{x},t) = \hat{v}(x^*,t)\)
Reset optimality \(\partial_x\hat{v}(x^*,t) + v_{ss}''(x^*)\,\hat{x}^*(t) = 0\)
Smooth-pasting \(\partial_x\hat{v}(\underline{x},t) + v_{ss}''(\underline{x})\,\hat{\underline{x}}(t) = \partial_x\hat{v}(\bar{x},t) + v_{ss}''(\bar{x})\,\hat{\bar{x}}(t) = 0\)
Terminal Condition \(\hat{v}(x,T) = 0\)

Five free-boundary conditions pin \((v=\hat{v}(x,t), \hat{\underline{x}}(t),\, \hat{\bar{x}}(t),\, \hat{x}^*(t))\)

The KFE

Distribution \(h(x,t)\) of price gaps, for \(x \in (\underline{x}(t), \bar{x}(t))\): \[\partial_t h = \nu\,\partial_x^2 h + \bar{\pi}\,\partial_x h - \zeta\,h + F(t)\,\delta\bigl(x - x^*(t)\bigr)\]

Condition Equation
Absorbing boundaries \(h(\underline{x},t) = h(\bar{x},t) = 0\)
Frequency of Price Adj. \(F(t) = \nu\partial_x h(\underline{x},t) + \bar{\pi}h(\underline{x},t) - \nu\partial_x h(\bar{x},t) - \bar{\pi}h(\bar{x},t) + \zeta\)
Aggregation \(X(t) = \int_{\underline{x}(t)}^{\bar{x}(t)} x\,h(x,t)\,dx\)
Initial condition \(h(x,0) = \phi_{\mathrm{init.}}(x)\)

Absorbing BCs + initial condition determine \(h(x,t)\); FPA \(F(t)\) and price gap \(X(t)\) follow

Linearized (\(\hat{h}(x,t) \equiv \partial_{MC} h\big|_{MC=0}\)), for \(x \in (\underline{x}, \bar{x})\): \[\partial_t\hat{h} = \nu\,\partial_x^2\hat{h} + \bar{\pi}\,\partial_x\hat{h} - \zeta\,\hat{h} + \hat{F}(t)\,\delta(x-x^*_{ss}) - F_{ss}\,\delta'(x-x^*_{ss})\,\hat{x}^*(t)\]

Condition Linearized equation
Absorbing boundaries \(\hat{h}(\underline{x},t) + h_{ss}'(\underline{x})\,\hat{\underline{x}}(t) = 0, \quad \hat{h}(\bar{x},t) + h_{ss}'(\bar{x})\,\hat{\bar{x}}(t) = 0\)
Frequency of Price Adj. \(\hat{F}(t) = \nu\partial_x\hat{h}(\underline{x},t) + \bar{\pi}\hat{h}(\underline{x},t) - \nu\partial_x\hat{h}(\bar{x},t) - \bar{\pi}\hat{h}(\bar{x},t)\)
Aggregation \(\hat{X}(t) = \int_{\underline{x}_{SS}}^{\bar{x}_{SS}} x\,\hat{h}(x,t)\,dx\)
Initial condition \(\hat{h}(x,0) = 0\)

Linearized absorbing BCs + zero initial condition determine \(\hat{h}(x,t)\); \(\hat{F}(t)\) and \(\hat{X}(t)\) follow

Solving the MFG

Theorem 1 gives the aggregate solution as eigenfunction expansions. Showing just the price gap \(\hat{X}\):

\[\hat{X}(t) = \sum_{n=1}^{\infty} \hat{X}_n(t)\]

\[\hat{X}_n(t) = \int_0^t e^{-\lambda_{KFE,n}(t-\tau)} \Bigl(\xi_{F,n}\,\hat{F}(\tau) + \xi_{x^*,n}\,\hat{x}^*(\tau) + \xi_{\underline{x},n}\,\hat{\underline{x}}(\tau) - \xi_{\bar{x},n}\,\hat{\bar{x}}(\tau)\Bigr)\,d\tau\]

where \(\lambda_{KFE,n} = \zeta + \frac{\bar{\pi}^2}{4\nu} + \frac{\nu n^2\pi^2}{(\bar{x}-\underline{x})^2}\) are the KFE eigenvalues and \(\xi_{\cdot,n}\) are analytical coefficients.

The same structure holds for all aggregate variables:

Variable Form Driven by
\(\hat{x}^*(t),\,\hat{\underline{x}}(t),\,\hat{\bar{x}}(t)\) Backward integral, HJB eigenfunctions \(MC(t)\)
\(\hat{F}(t)\) Levels (crit.) \(+\) forward integral (int.) \(\hat{x}^{*},\,\hat{\underline{x}},\,\hat{\bar{x}}\)
\(\hat{X}(t)\) Forward integral, KFE eigenfunctions \(\hat{F},\,\hat{x}^*,\,\hat{\underline{x}},\,\hat{\bar{x}}\)

Discrete Time Approximation

Approximate for small time steps (still infinite-dimensional): \[\hat{X}(t) = \sum_{n=1}^{\infty} \hat{X}_n(t)\]

\[\hat{X}_n(t) \approx \underbrace{\varsigma_{KFE,n}}_{\text{int. weight}} \Bigl(\xi_{F,n}\,\hat{F}_t + \xi_{x^*,n}\,\hat{x}^*_t + \xi_{\underline{x},n}\,\hat{\underline{x}}_t - \xi_{\bar{x},n}\,\hat{\bar{x}}_t\Bigr) + \underbrace{e^{-\lambda_{KFE,n}}}_{\theta_n}\,\hat{X}_{n,t-1}\]

where \(\varsigma_{KFE,n} \equiv \dfrac{1 - e^{-\lambda_{KFE,n}}}{\lambda_{KFE,n}}\) is a weight for each eigenvalue (more accurate than Reimann)

Now easy to solve:

  • Choose a small time step
  • Truncate eigenfunction expansion at some large \(N\)
  • Solve system of dynamic linear equations (standard DSGE tools)

How Do Marginal Costs Affect Price Gaps?

How Do Marginal Costs Affect Price Gaps?

From \(\infty\) to 1: The Primary Eigenfunction

From \(\infty\) to 1: The Primary Eigenfunction

  • The true solution is infinite-dimensional: one AR(1) per \(n = 1, 2, 3, \ldots\) But they don’t all matter equally.
  • Low-order \(n\) eigenfunctions dominate: large \(n \implies\) large eigenvalue \(\lambda_{KFE,n}\) \(\implies\) fast decay \(\theta_n = e^{-\lambda_{KFE,n}}\) \(\implies\) dies out quickly
  • Even \(n\) eigenfunctions dominate: even \(\implies\) anti-symmetric, which is everything at \(\bar{\pi}=0\) (Alvarez et al. 2023)

The PED approximates the full solution keeping only \(n = \mathring{n} = 2\)

The PED: Three Equations

\[\begin{aligned} \text{Opt.\ Price:} && p^*_t &= (1-\theta\beta)\,(mc_t + p_t) + \theta\beta\,p^*_{t+1} \\ \text{Price Level:} && p_t &= \epsilon\,F_t + (1-\theta)\,p^*_t + \theta\,p_{t-1} \\ \text{FPA:} && F_t &= \psi_{p^*}\,(p^*_t - p^*_{t-1}) + \theta\,F_{t-1} \end{aligned}\]

Coefficients from microfoundations:

Symbol Definition Role
\(\theta\) \(e^{-\lambda_{KFE,\mathring{n}}}\) “price stickiness”
\(\epsilon\) \((1-\theta)\,\xi_{F,\mathring{n}} / \xi_{p^*,\mathring{n}}\) FPA impact on price level
\(\psi_{p^*}\) \(2\!\sqrt{\tfrac{\nu}{\pi\Delta t}}\bigl(h_{ss}'(\underline{x})+h_{ss}'(\bar{x})\bigr)+\tfrac{\bar{\pi}}{2}\bigl(h_{ss}'(\bar{x})-h_{ss}'(\underline{x})\bigr)\) \(\Delta p^* \to\) FPA

The MCNK Phillips Curve

Combining the PED equations yields a modified Phillips curve:

\[\pi_t = \underbrace{\Lambda\,mc_t + \beta\,\mathbb{E}_t[\pi_{t+1}]}_{\text{Standard NK Phillips Curve}} + \underbrace{\frac{\epsilon}{\theta}\bigl(F_t - \theta\beta\,\mathbb{E}_t[F_{t+1}]\bigr)}_{\text{\textcolor{red}{FPA correction}}}\]

with FPA dynamics: \[F_t = \psi_\pi(\pi_t - \theta\pi_{t-1}) + (\theta + (1-\theta)\psi_\pi\epsilon)\,F_{t-1}\]

Theoretical consistency:

  • Long-run monetary neutrality
  • When \(\Psi \to \infty\): \(\psi_\pi \to 0\), PED \(\Rightarrow\) nests standard NKPC
  • When \(\bar{\pi} = 0\): \(\epsilon = 0\), PED \(\Rightarrow\) nests standard NKPC (but not Calvo’s \(\Lambda\); i.e. Auclert et al 2024)

Calibration

Calvo-plus calibration targeting French CPI microdata (Alvarez et al. 2024), baseline \(\bar{\pi} = 2\%\)

Calibrated parameters

Parameter Value Target
\(\zeta\) 1.147 Freq. \(F = 1.26\)
\(\nu\) 0.0036 Std \(= 0.076\)
\(\Psi\) 0.166 Kurt \(= 3.41\)

Fixed parameters

Parameter Value
\(\bar{\pi}\) 0.02
\(\rho\) 0.04
\(\eta\) 6

Implied PED coefficients

Parameter Value Role
\(\theta = e^{-\lambda_{\mathring{n}}}\) 0.825 Price stickiness
\(\Lambda = \frac{(1-\theta)(1-\theta\beta)}{\theta}\) 0.038 PC slope
\(\epsilon\) 0.00068 FPA \(\to\) prices
\(\psi_\pi\) 13.9 Inflation \(\to\) FPA

All PED coefficients are determined analytically from \((\zeta, \nu, \Psi, \bar{\pi})\) via Theorem 1 and Proposition 1 — no additional estimation required.

Validation: PED vs Full Solution vs Calvo

\(\bar{\pi} = 2\%\): PED nearly exact

\(\bar{\pi} = 10\%\): PED still accurate

Responses to a 0.01 marginal cost shock, annual persistence 0.9.

Calvo misses impact effects

Why Does Calvo Miss?

\(\bar{\pi} = 2\%\): PED nearly exact

\(\bar{\pi} = 10\%\): PED still accurate

  • Menu costs \(\Rightarrow\) FPA jumps on impact (firms near boundaries adjust)
  • Time-varying selection effect (Golosov & Lucas 2007): adjusting firms are those with largest price gaps
  • Calvo misses this entirely \(\Rightarrow\) understates short-run price response

PED Approximation Quality Across \(\bar{\pi}\)

  • \(\mathring{n} = 2\) minimizes MSE across all \(\bar{\pi}\) values tested
  • Correlation with true solution stays \(> 0.99\) up to \(\bar{\pi} \approx 20\%\)
  • For hyperinflation (\(\bar{\pi} > 20\%\)), use the full solution

The NK Model

Embed the PED pricing block in a standard NK model:

\[\begin{aligned} \text{Euler (IS):} && \sigma y_t &= \sigma\,\mathbb{E}_t[y_{t+1}] - i_t + \mathbb{E}_t[\pi_{t+1}] + z^d_t \\ \text{Taylor Rule:} && i_t &= \phi_\pi\pi_t + \phi_y y_t + z^r_t\\ \text{NK Phillips Curve:} && \pi_t &= \underbrace{\Lambda(\alpha y_t + z^c_t) + \beta\,\mathbb{E}_t[\pi_{t+1}]}_{\text{Calvo term}} \end{aligned}\]

The MCNK Model

Embed the PED pricing block in a standard NK model:

\[\begin{aligned} \text{Euler (IS):} && \sigma y_t &= \sigma\,\mathbb{E}_t[y_{t+1}] - i_t + \mathbb{E}_t[\pi_{t+1}] + z^d_t \\ \text{Taylor Rule:} && i_t &= \phi_\pi\pi_t + \phi_y y_t + z^r_t \\ \text{\textcolor{red}{MCNK Phillips Curve}:} && \pi_t &= \underbrace{\Lambda(\alpha y_t + z^c_t) + \beta\,\mathbb{E}_t[\pi_{t+1}]}_{\text{Calvo term}} + \underbrace{\tfrac{\epsilon}{\theta}(F_t - \theta\beta\,\mathbb{E}_t[F_{t+1}])}_{\text{\textcolor{red}{FPA correction}}} \\ \text{\textcolor{red}{MCNK FPA}:} && F_t &= \psi_\pi(\pi_t - \theta\pi_{t-1}) + (\theta + (1-\theta)\psi_\pi\epsilon)F_{t-1} \\ \end{aligned}\]

Can be solved with standard tools (Dynare, sequence space, etc.)

MCNK vs Calvo: IRFs

Two Effects of Menu Costs

1. Steady-state selection effect (Golosov-Lucas)

  • Firms with largest gaps adjust first \(\Rightarrow\) prices respond more to shocks
  • Menu costs \(\Rightarrow\) effectively more flexible prices \(\Rightarrow\) steeper Phillips curve (\(\Lambda\) larger than Calvo)

2. Time-varying selection effect (FPA channel)

  • Shocks shift the distribution \(\Rightarrow\) FPA responds endogenously
  • FPA amplifies inflationary effects: an “inflation accelerator”
  • This channel is absent at \(\bar{\pi} = 0\) but first-order in \(\bar{\pi}\)

Trend Inflation and PED Coefficients

Trend Inflation and PED Coefficients

If \(\bar{\pi} \uparrow\):

  • \(\theta \downarrow\) (smaller inaction region, prices more “flexible”)
  • \(\Lambda \uparrow\) (steeper Phillips curve, 2nd-order)
  • \(\epsilon \uparrow\) (stronger time-varying selection, first-order)

Trend Inflation Amplifies Dynamics

Optimal Monetary Policy

Optimal Monetary Policy

  • Menu costs \(\Rightarrow\) more aggressive policy (steeper PC, standard result)
  • Trend inflation strengthens:
    1. Higher \(\Lambda\) (slope effect)
    2. Higher \(\epsilon\) (FPA amplification)

Summary

  1. Analytical solution to the menu cost MFG with trend inflation

  2. PED: accurate, tractable, microfounded linear approximation

  3. Quantitative findings:

    • menu costs + trend inflation steepen the Phillips curve and add FPA-driven acceleration
    • FPA correction is first-order in \(\bar{\pi}\)
    • Optimal policy is more aggressive, \(\uparrow\) in \(\bar{\pi}\)

Broader applicability: The solution + PED approach extends to any \((s,S)\) model with drift — investment, inventories, consumer durables