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arxiv: 2508.12073 · v3 · submitted 2025-08-16 · 💰 econ.TH

Closed-Form of Two-Agent New Keynesian Model with Price and Wage Rigidities

Kenji Miyazaki This is my paper

Pith reviewed 2026-05-18 22:56 UTC · model grok-4.3

classification 💰 econ.TH
keywords two-agent new keynesian modelmonetary policy transmissionprice and wage rigiditieshousehold heterogeneityclosed-form solutionrotemberg rigiditiesis curve slopewelfare loss function
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The pith

Monetary transmission in a two-agent New Keynesian model amplifies when heterogeneity-induced IS-slope effects combine with strong price stickiness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives a closed-form solution for a two-agent New Keynesian model that includes both price and wage rigidities of the Rotemberg type. It shows that monetary policy effects on output strengthen when the share of hand-to-mouth households creates a steeper IS curve and when prices adjust slowly, but these gains disappear if only wages are sticky. The solution reveals the joint role of the hand-to-mouth share, price duration, and wage duration in shaping how policy moves the economy. A modified welfare loss function is also derived to show how profit distribution across household types changes the value of inflation stabilization.

Core claim

The paper analytically demonstrates that, in a Two-Agent New Keynesian model with Rotemberg-type price and wage rigidities, monetary transmission can be amplified when two mechanisms are sufficiently strong: the heterogeneity-induced IS-slope effect and the price-stickiness channel. We also show when amplification weakens or disappears, most notably under pure wage stickiness, where the price channel shuts down and the heterogeneity-driven term vanishes. The closed-form solution makes transparent how price stickiness, wage stickiness, and the share of hand-to-mouth households jointly shape amplification.

What carries the argument

The heterogeneity-induced IS-slope effect interacting with the price-stickiness channel in the closed-form solution derived from microeconomic foundations.

If this is right

  • Monetary policy gains more traction on output when both the heterogeneity-induced IS-slope effect and price stickiness are strong.
  • Amplification vanishes under pure wage stickiness because the price channel shuts down and the heterogeneity term disappears.
  • The share of hand-to-mouth households directly scales the size of the amplification through its effect on the IS curve slope.
  • The modified aggregate welfare loss function assigns greater weight to inflation stabilization once distributional effects from firm profits are included.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form expressions could be used to compare policy rules across different degrees of household heterogeneity without numerical simulation.
  • Extensions to open economies might show how trade openness interacts with the same two mechanisms to alter the required strength of price stickiness for amplification.
  • Central banks could monitor the share of hand-to-mouth households as a state variable that changes the marginal benefit of reducing price rigidity.

Load-bearing premise

Household heterogeneity between savers and hand-to-mouth types can be modeled from micro foundations without imposing common restrictive assumptions on relative wages or labor supply across types.

What would settle it

Empirical data showing identical monetary policy responses to interest rate shocks in economies with high price stickiness versus high wage stickiness, even when hand-to-mouth household shares are large, would contradict the claimed amplification.

Figures

Figures reproduced from arXiv: 2508.12073 by Kenji Miyazaki.

Figure 1
Figure 1. Figure 1: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p044_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p045_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p046_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p047_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p048_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Impulse response functions of monetary shock with [PITH_FULL_IMAGE:figures/full_fig_p049_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Impulse response functions of technology shock with [PITH_FULL_IMAGE:figures/full_fig_p050_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Impulse response functions of technology shock with [PITH_FULL_IMAGE:figures/full_fig_p051_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Impulse response functions of technology shock with [PITH_FULL_IMAGE:figures/full_fig_p052_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between simple and general adjustment cost [PITH_FULL_IMAGE:figures/full_fig_p053_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison between simple and general adjustment cost [PITH_FULL_IMAGE:figures/full_fig_p054_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The effect of ϕ on the two roots of characteristic function Notes: The blue dotted curve represents  γκ1 κ2 + 1 − γδp  (ξ−1)  ξ − γ γκ1+(1−γδp)κ2  . The red dashed line represents (ϕ − 1)ξ. The yellow solid line represents (ϕ ′ − 1)ξ with ϕ ′ > ϕ. The two intersections of the blue dotted curve and the red dashed line are the initial roots of characteristic function. The inter￾sections of the blue dott… view at source ↗
Figure 14
Figure 14. Figure 14: The effect of δp on the two roots of characteristic function Notes: The blue dotted curve represents  γκ1 κ2 + 1 − γδp  (ξ−1)  ξ − γ γκ1+(1−γδp)κ2  . The red dashed line represents (ϕ − 1)ξ. The yellow solid curve represents  γκ1 κ2 + 1 − γδ′ p  (ξ − 1)  ξ − γ γκ1+(1−γδ′ p )κ2  with δ ′ p > δp. The two intersec￾tions of the blue dotted curve and the red dashed line are the initial roots of charact… view at source ↗
Figure 15
Figure 15. Figure 15: The effect of κ1, κ2, κ2/κ1 on the two roots of characteristic function Notes: The blue dotted curve represents  γκ1 κ2 + 1 − γδp  (ξ−1)  ξ − γ γκ1+(1−γδp)κ2  . The red dashed line represents (ϕ − 1)ξ. The yellow solid curve repre￾sents  γκ′ 1 κ ′ 2 + 1 − γδ′ p  (ξ − 1)  ξ − γ γκ′ 1+(1−γδp)κ ′ 2  with κ ′ 1 < κ1, κ ′ 2 < κ2, and κ ′ 2 /κ′ 1 < κ2/κ1. The two intersections of the blue dashed curve a… view at source ↗
read the original abstract

This paper analytically demonstrates that, in a Two-Agent New Keynesian model with Rotemberg-type price and wage rigidities, monetary transmission can be amplified when two mechanisms are sufficiently strong: the heterogeneity-induced IS-slope effect and the price-stickiness channel. We also show when amplification weakens or disappears, most notably under pure wage stickiness, where the price channel shuts down and the heterogeneity-driven term vanishes. The framework features household heterogeneity between savers and hand-to-mouth households and is derived from microeconomic foundations while avoiding restrictive assumptions on relative wages or labor supply across types that are common in prior analytical work. The closed-form solution makes transparent how price stickiness, wage stickiness, and the share of hand-to-mouth households jointly shape amplification. We further derive a modified aggregate welfare loss function that quantifies how heterogeneity, operating through distributional effects from firm profits, changes the relative importance of stabilizing inflation. Overall, the tractable yet micro-founded analytical framework clarifies the interaction between household heterogeneity and nominal rigidities and identifies sufficient conditions under which monetary policy gains or loses traction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives a closed-form solution for a two-agent New Keynesian model with Rotemberg price and wage rigidities. It analytically shows that monetary transmission is amplified when the heterogeneity-induced IS-slope effect and the price-stickiness channel are both sufficiently strong, and that amplification weakens or disappears under pure wage stickiness (where the price channel shuts down and the heterogeneity-driven term vanishes). The framework is micro-founded, avoids common restrictive assumptions on relative wages or labor supply across saver and hand-to-mouth types, and includes a modified aggregate welfare loss function that accounts for distributional effects from firm profits.

Significance. If the closed-form derivations hold without hidden restrictions from linearization or firm optimization, the paper supplies a transparent analytical tool for tracing how household heterogeneity, price stickiness, and wage stickiness jointly determine monetary policy traction. The explicit sufficient conditions for amplification and the welfare extension are potentially useful for policy analysis and for reconciling heterogeneous-agent models with observed transmission mechanisms.

major comments (2)
  1. [IS-curve derivation under wage rigidity only] The section deriving the IS curve and heterogeneity term under pure wage stickiness: the claim that the heterogeneity-driven term vanishes cleanly requires an explicit step showing that the equilibrium mapping from aggregate hours to type-specific hours (via firm labor demand and Rotemberg wage adjustment costs) does not re-introduce an implicit restriction on relative labor supply equivalent to those the paper seeks to avoid. Without this step, the vanishing result risks being an artifact of the chosen linearization rather than a general feature of the micro-foundations.
  2. [Welfare loss derivation] The derivation of the modified aggregate welfare loss function: the incorporation of distributional effects from firm profits needs to be shown to alter the relative weight on inflation stabilization in a manner that remains robust when the hand-to-mouth share or the degree of wage stickiness varies; the current presentation leaves unclear whether this modification is first-order or higher-order in the linearization.
minor comments (2)
  1. Clarify the notation for the share of hand-to-mouth households and the relative wage variables when they first appear; inconsistent use across the closed-form expressions and the welfare section would hinder readability.
  2. Consider adding a compact table that juxtaposes the closed-form IS slope and amplification factor under the four regimes (flexible prices/wages, price stickiness only, wage stickiness only, both rigid) to make the sufficient conditions for amplification immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The two major comments identify areas where additional explicit steps would strengthen the presentation. We address each point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [IS-curve derivation under wage rigidity only] The section deriving the IS curve and heterogeneity term under pure wage stickiness: the claim that the heterogeneity-driven term vanishes cleanly requires an explicit step showing that the equilibrium mapping from aggregate hours to type-specific hours (via firm labor demand and Rotemberg wage adjustment costs) does not re-introduce an implicit restriction on relative labor supply equivalent to those the paper seeks to avoid. Without this step, the vanishing result risks being an artifact of the chosen linearization rather than a general feature of the micro-foundations.

    Authors: We appreciate this suggestion for greater transparency. In the model, the representative firm’s labor demand is uniform across household types, and Rotemberg wage adjustment costs are levied on the aggregate nominal wage. Type-specific hours are then determined by the aggregate labor supply condition together with the hand-to-mouth budget constraint; no additional cross-type restriction on relative labor supply is imposed beyond these standard micro-foundations. The heterogeneity term in the IS curve therefore vanishes because wage rigidity operates symmetrically on the aggregate wage that enters both agents’ labor-supply decisions. To make this mapping fully explicit, we will insert a short appendix subsection that derives the equilibrium hours allocation step by step from the firm’s first-order condition and the wage Phillips curve. This addition will clarify that the result is not an artifact of linearization. revision: partial

  2. Referee: [Welfare loss derivation] The derivation of the modified aggregate welfare loss function: the incorporation of distributional effects from firm profits needs to be shown to alter the relative weight on inflation stabilization in a manner that remains robust when the hand-to-mouth share or the degree of wage stickiness varies; the current presentation leaves unclear whether this modification is first-order or higher-order in the linearization.

    Authors: We agree that the order and robustness of the modification should be stated more clearly. Firm profits are distributed only to savers; this enters the aggregate welfare loss at first order through the consumption gap of hand-to-mouth households, which appears in the linearized resource constraint. The resulting extra term is proportional to the hand-to-mouth share and is independent of the wage-stickiness parameter. We will expand the welfare section to (i) derive the first-order term explicitly, (ii) show that the relative weight on inflation stabilization changes continuously with the hand-to-mouth share, and (iii) verify that the modification remains first-order for any finite degree of wage rigidity. These clarifications will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no load-bearing reductions to inputs or self-citations identified

full rationale

The paper presents an analytical closed-form derivation of a two-agent New Keynesian model with Rotemberg price and wage rigidities, starting from standard microeconomic household and firm optimization problems. The key results on amplification via heterogeneity-induced IS-slope and price-stickiness channels, as well as the vanishing of the heterogeneity term under pure wage stickiness, are obtained by solving the linearized equilibrium system under alternative parameter restrictions on rigidities. No equations are shown to be self-definitional, no fitted parameters are relabeled as predictions, and no uniqueness theorems or ansatzes are imported via self-citation in a load-bearing way. The framework explicitly states avoidance of common restrictive assumptions on relative wages and labor supply, and the reported properties follow directly from the model's equilibrium conditions rather than being imposed by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of fitted parameters, axioms, or new entities; the model relies on standard NK primitives plus household heterogeneity and Rotemberg adjustment costs.

pith-pipeline@v0.9.0 · 5709 in / 1123 out tokens · 48228 ms · 2026-05-18T22:56:20.445792+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. When Redistribution Becomes a State Variable: Monetary-Fiscal Stabilization with Type-Specific Sticky Wages

    econ.GN 2026-05 unverdicted novelty 6.0

    In a TANK model with type-specific sticky wages, the cross-type wage gap emerges as a second-order state variable requiring history-dependent transfers for RANK-equivalent stabilization.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    Before advancing to the complete IS-PC-MP framework, the relationship to existing literature merits examination

    also yields δp = 0, suggesting that household heterogeneity becomes irrelevant for aggregate dynamics—a finding that challenges the equivalence result proposed by Bilbiie and Trabandt (2025). Before advancing to the complete IS-PC-MP framework, the relationship to existing literature merits examination. The heterogeneity adjustment term −δpπp t can be rew...

    work page 2025

  2. [2]

    BothΩ p andΩ x decrease with the monetary policy response coefficientϕ

    work page

  3. [3]

    1The latter is derived from ∂Ωp ∂ηp/ψp = (φ+γ)(ϕ−ρm) {(φ+γ)(ϕ−ρm) + (1−ρm)(1−(φ+γ){λ/(1−λ)}δc)γηp/ψp}2

    Suppose ϕ+γκ1−1 κ2 −γδp >0.(10) Then: • Ωp andΩ x increase with the share of hand-to-mouth house- holdsλ. 1The latter is derived from ∂Ωp ∂ηp/ψp = (φ+γ)(ϕ−ρm) {(φ+γ)(ϕ−ρm) + (1−ρm)(1−(φ+γ){λ/(1−λ)}δc)γηp/ψp}2. 20 • Ωx decreases with goods and labor market competitiveness parametersψj and increases with adjustment cost parameters ηj > forj∈{p,w}

    work page

  4. [4]

    In addition,Ω p increases withψj and decreases withηj forj∈{p,w}

    Without heterogeneity (λ= 0), the condition (10) is not required. In addition,Ω p increases withψj and decreases withηj forj∈{p,w}. Appendix A contains the formal proof. The underlying economic intuition becomes clearer through an aggregate demand-aggregate supply (AD-AS) lens. The TANK Phillips curve (6) serves as our AS curve. With a more restrictive as...

    work page

  5. [5]

    Ωp ∞andΩ x ∞decrease with the monetary policy response coefficientϕ

    work page

  6. [6]

    greedflation conundrum

    Ωp ∞andΩ x ∞increase with the share of hand-to-mouth house- holdsλ. Appendix B provides the detailed proof. Notably, the restrictive condition (11) becomes unnecessary for long-run analysis. This suggests that even when heterogeneity dampens immediate policy effectiveness, cumulative effects may actually strengthen over time as distributional channels com...

    work page doi:10.1111/ecin.12424 2019

  7. [7]

    Notice that γ γκ1 + (1−γδp)κ2 is increasing inδp and decreasing inκ1 andκ2

    Preliminary: Transformation of the Characteristic Function The characteristic function is rewritten as f(ξ) = (γκ1 κ2 + 1−γδp ) (ξ−1) ( ξ− γ γκ1 + (1−γδp)κ2 ) −(ϕ−1)ξ. Notice that γ γκ1 + (1−γδp)κ2 is increasing inδp and decreasing inκ1 andκ2. The characteristic function is also rewritten as f(ξ) = (γκ1 κ2 + 1−γδp ) ξ2− ( ϕ−1 +γ κ2 + γκ1 κ2 + 1−γδp ) ξ+γ ...

    work page

  8. [8]

    That is, ∂ξ1 ∂ϕ<0, ∂ξ2 ∂ϕ>0

    Effect ofϕ Figure 13 shows that increasingϕlowersξ1 and raisesξ2. That is, ∂ξ1 ∂ϕ<0, ∂ξ2 ∂ϕ>0. This occurs becauseϕdirectly increasesΞ b in the characteristic equation. As a result, bothΩp andΩ x decrease with increasingϕ

    work page

  9. [9]

    That is, ∂ξ1 ∂δp >0, ∂ξ2 ∂δp >0

    Effect ofδp Figure 14 demonstrates that, givenκ1 andκ2, increasingδp affects both roots. That is, ∂ξ1 ∂δp >0, ∂ξ2 ∂δp >0. Since the denominator ofΩ p andΩ x are respectivelyΞ a(ξ2−ρm)and (κ2/κ1)Ξa(ξ2−ρm), it is sufficient to show that increasingδp lowers(ξ2−ρm) for proving ∂Ωp ∂δp >0, ∂Ωx ∂δp >0 35 givenκ1 andκ2. Now, Ξaξ2 = Ξb + √ Ξ 2 b−4ΞaΞc 2 . Since Ξ...

    work page

  10. [10]

    Given thatδp increases withλ: dΩp dλ>0, dΩx dλ>0

    Effect ofλ Since λdoes not directly affectκ1 or κ2, its impact operates solely through δp. Given thatδp increases withλ: dΩp dλ>0, dΩx dλ>0

    work page

  11. [11]

    That is, ∂ξ1 ∂ηj >0, ∂ξ2 ∂ηj <0, ∂ξ1 ∂ψj <0, ∂ξ2 ∂ψj >0, for j = w,p

    Effects ofηj andψj (j=p,w) •Direct Effects (holdingδp constant) 36 Figure 15 demonstrates that, givenδp, an increase inηp or ηw, or a decrease inψp orψw lowersκ1, κ2 andκ2/κ1, elevatingξ1 and diminishingξ2. That is, ∂ξ1 ∂ηj >0, ∂ξ2 ∂ηj <0, ∂ξ1 ∂ψj <0, ∂ξ2 ∂ψj >0, for j = w,p. This change lowers{γ+ (κ2/κ1)(1−γδp)}(ξ2−ρm)in the denominator ofΩ x. That is, ∂...

    work page

  12. [12]

    Inthiscase, thecondition ϕ+γ{(κ1−1)/κ2−δp}≥ 0becomes ϕ+γ(κ1−1)/κ2≥0, which is automatically satisfied given that ϕ>0, γ >0, κ1≥1, andκ2 > 0

    Special Case:λ= 0 Whenλ= 0, wehaveδp = 0. Inthiscase, thecondition ϕ+γ{(κ1−1)/κ2−δp}≥ 0becomes ϕ+γ(κ1−1)/κ2≥0, which is automatically satisfied given that ϕ>0, γ >0, κ1≥1, andκ2 > 0. Furthermore,dΩp/dηj =∂Ωp/∂ηj < 0and dΩp/dψj =∂Ωp/∂ψj >0. B Proof of Proposition 2 First, the analysis proves the following double sum: S= ∞∑ n=0 n∑ i=0 ξn−i 1 ρi m Consider t...

    work page