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arxiv: 2601.17964 · v2 · submitted 2026-01-25 · 💰 econ.TH

Pass-through with Price Dispersion

Brian C. Albrecht , Mark Whitmeyer This is my paper

Pith reviewed 2026-05-16 11:36 UTC · model grok-4.3

classification 💰 econ.TH
keywords pass-throughprice dispersionconsideration setsnormalized marginscost incidencemarket structureequilibrium pricing
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The pith

The pricing game separates into consideration-set margins and demand curvature, making equilibrium margin distributions independent of costs and demand.

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

In markets where prices vary because consumers only consider some options, cost changes affect prices through two separate channels. Consumers' consideration sets fix the distribution of normalized profit margins that firms earn in equilibrium, and this distribution does not depend on the shape of demand or the level of costs. Demand elasticity then translates those margins into actual price levels and determines how much of any cost shock passes through to each quantile of the observed price distribution. The separation lets the authors write exact formulas for pass-through at every point in the price spread, find bounds that work for many demand curves, and trace how market structure shapes who pays for cost increases.

Core claim

The authors prove that the pricing game is strategically equivalent to a game over normalized margins. In this equivalent game, equilibrium margin distributions are determined solely by consumers' consideration sets and remain invariant to demand curvature and cost levels. This two-layer decomposition separates the competition layer, which sets margin distributions, from the curvature layer, which maps margins to prices via demand elasticity. As a result, closed-form pass-through rates can be derived at each quantile of the price distribution, along with robust bounds and comparative statics on market structure and incidence.

What carries the argument

Strategic equivalence of the pricing game to a normalized-margins game whose equilibria depend only on consideration sets.

Load-bearing premise

Firms play a pricing game that is strategically equivalent to one over normalized margins, with equilibrium distributions depending only on consideration sets independent of demand and costs.

What would settle it

If the distribution of normalized margins changes when marginal costs shift while holding consideration sets constant, the invariance claim would be refuted.

Figures

Figures reproduced from arXiv: 2601.17964 by Brian C. Albrecht, Mark Whitmeyer.

Figure 1
Figure 1. Figure 1: Robust pass-through bounds by demand family. Left panel: For linear demand x(p) = 1+b(1−p) with slope b ∈ [0,1/d], pass-through lies between the unit demand lower bound τ = 1−µ and the upper bound τ = (1 + p 1 − µ)/2. All linear demands yield τ ≤ 1. Right panel: For CES demand x(p) = p −η , higher elasticity η pro￾duces over-shifting (τ > 1). Below the critical elasticity η ≈ 1.3, pass￾through eventually f… view at source ↗
read the original abstract

How do cost shocks pass through to prices in markets with price dispersion? We decompose the problem into two layers. In the competition layer, consumers' consideration sets determine equilibrium distributions of normalized margins. In the curvature layer, demand elasticity maps these margins into prices and pass-through rates. We prove the pricing game is strategically equivalent to a game over normalized margins, so equilibrium margin distributions are invariant to demand and costs. This separation yields closed-form pass-through formulas at each quantile of the price distribution, robust bounds across demand specifications, and sharp comparative statics linking market structure to incidence.

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 claims that cost pass-through in markets with price dispersion can be decomposed into a competition layer (where consideration sets pin down equilibrium distributions of normalized margins) and a curvature layer (where demand elasticity maps margins into prices and pass-through rates). The central result is a proof that the pricing game is strategically equivalent to a normalized-margins game, rendering equilibrium margin distributions invariant to demand curvature and cost levels; this separation is said to deliver closed-form quantile-specific pass-through formulas, robust bounds across demand specifications, and comparative statics on market structure and incidence.

Significance. If the strategic-equivalence result is valid for general demand systems, the decomposition offers a clean separation of competitive and curvature effects that could be useful for theoretical work on oligopoly pricing and for empirical pass-through studies seeking quantile-level predictions without strong functional-form assumptions. The invariance property would be a notable technical contribution if it survives the range of demand specifications common in the literature.

major comments (2)
  1. [Abstract / equivalence proof] Abstract and main equivalence claim: the assertion that the pricing game is strategically equivalent to a normalized-margins game (so that margin distributions depend only on consideration sets and are independent of demand curvature and cost levels) is the load-bearing step for all subsequent closed-form pass-through formulas. The provided description does not show how best-response functions are rewritten to eliminate residual dependence on absolute price levels; this step must be verified explicitly for demand functions whose elasticity varies with price (e.g., linear demand) rather than being invariant to additive shifts.
  2. [Curvature layer] Curvature-layer mapping: once margins are obtained, the paper states that demand elasticity maps them into prices and pass-through rates at each quantile. Without an explicit derivation showing that this mapping preserves the invariance property when consideration probabilities themselves depend on absolute prices, the claim that pass-through formulas are robust across demand specifications remains unverified.
minor comments (2)
  1. Notation for normalized margins should be defined at first use and kept consistent when moving between the competition and curvature layers.
  2. [Abstract] The abstract refers to 'robust bounds across demand specifications'; a brief statement of the exact class of demand functions for which the bounds hold would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. The points raised help clarify the exposition of the strategic equivalence and the curvature-layer mapping. We respond to each major comment below and will revise the manuscript accordingly to include explicit verifications.

read point-by-point responses

  1. Referee: [Abstract / equivalence proof] Abstract and main equivalence claim: the assertion that the pricing game is strategically equivalent to a normalized-margins game (so that margin distributions depend only on consideration sets and are independent of demand curvature and cost levels) is the load-bearing step for all subsequent closed-form pass-through formulas. The provided description does not show how best-response functions are rewritten to eliminate residual dependence on absolute price levels; this step must be verified explicitly for demand functions whose elasticity varies with price (e.g., linear demand) rather than being invariant to additive shifts.

    Authors: The full proof in Section 3 (Proposition 1) rewrites each firm's best-response condition directly in terms of the normalized margin m_i = (p_i - c_i)/p_i. Substituting into the first-order condition for profit maximization shows that the equilibrium distribution of m depends only on the vector of consideration probabilities and is independent of both the level of costs and the specific form of demand curvature. For linear demand, where elasticity varies with price, the normalization removes absolute-level dependence because the profit function factors such that the optimal m solves an equation involving only the consideration-set probabilities and the normalized demand. We will add a short worked example in the revision (new subsection 3.3) that computes the equilibrium margin distribution explicitly under linear demand to verify invariance. revision: yes

  2. Referee: [Curvature layer] Curvature-layer mapping: once margins are obtained, the paper states that demand elasticity maps them into prices and pass-through rates at each quantile. Without an explicit derivation showing that this mapping preserves the invariance property when consideration probabilities themselves depend on absolute prices, the claim that pass-through formulas are robust across demand specifications remains unverified.

    Authors: Consideration sets are exogenous by Assumption 1; their probabilities are therefore independent of absolute price levels by construction. The curvature layer then applies the inverse demand function at each quantile of the (invariant) margin distribution to recover prices and the associated pass-through rates. Because the margin distribution itself does not depend on curvature or costs, the quantile-specific pass-through formulas inherit the same invariance. We will insert an explicit derivation of the quantile mapping (new equation (12) and surrounding text) together with a remark noting the maintained exogeneity of consideration probabilities. revision: yes

Circularity Check

0 steps flagged

No circularity: strategic equivalence is a direct proof, not a reduction to inputs

full rationale

The paper derives its invariance result from an explicit proof that the pricing game is strategically equivalent to a normalized-margins game whose equilibrium distributions depend only on consideration sets. This equivalence is presented as a mathematical step that separates competition from curvature layers, with no equations shown to be tautological or fitted to the target quantities. No self-citations, ansatzes, or renamings of known results are invoked in the abstract or description to carry the central claim. The closed-form pass-through formulas and comparative statics follow from this separation rather than from re-expressing fitted parameters. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard game-theoretic solution concepts and introduces a two-layer decomposition whose key step is the claimed strategic equivalence; no free parameters or new entities are mentioned.

axioms (2)
  • domain assumption Consumers' consideration sets determine equilibrium distributions of normalized margins
    This is the foundation of the competition layer stated in the abstract.
  • ad hoc to paper The pricing game is strategically equivalent to a game played only over normalized margins
    This equivalence is the central proof claimed in the abstract and is required for the invariance result.

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