arxiv: 2604.21423 · v1 · submitted 2026-04-23 · 💰 econ.GN · q-fin.EC

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Demand Curvature and Pass-Through in Differentiated Oligopoly

Paul S. Koh

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Pith reviewed 2026-05-08 13:24 UTC · model grok-4.3

classification 💰 econ.GN q-fin.EC

keywords pass-throughdifferentiated oligopolydemand curvaturemultiproduct ownershiptax incidencemerger analysissufficient statisticssmall-share limit

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The pith

A general representation of the pass-through matrix decomposes price responses into demand curvature, substitution, and multiproduct ownership in differentiated oligopoly.

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

This paper derives a formula showing how costs pass through to prices when products are differentiated and firms may own several products at once. The formula separates the effect into demand curvature, how consumers substitute between products, and the ownership structure of the firms. This matters for understanding who bears the burden of taxes or cost increases and how mergers change pricing incentives. It also gives a simple way to approximate these effects using statistics from estimated demand systems. In the limit of small market shares, the shape of demand at the tail further determines the pass-through.

Core claim

The paper derives a general representation of the pass-through matrix that decomposes equilibrium price responses into the roles of demand curvature, substitution, and multiproduct ownership. This extends the classic insight in single-product monopoly to multiproduct settings in which diversion and ownership also matter. A tractable first-order approximation is developed yielding a sufficient-statistics characterization for empirically relevant demand systems, and the small-share limit is characterized showing how common demand specifications impose tail restrictions that shape pass-through.

What carries the argument

The pass-through matrix decomposed by demand curvature, substitution patterns, and ownership structure.

If this is right

Pass-through in single-product monopoly depends only on demand curvature.
In multiproduct oligopoly, substitution and ownership additionally determine pass-through.
The first-order approximation allows computation of pass-through from a few demand statistics.
Common demand systems restrict pass-through in the small-share limit through their tail properties.
The framework applies to tax incidence, cost shocks, and merger analysis.

Where Pith is reading between the lines

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

If the decomposition is accurate, empirical work on pass-through can rely on local demand estimates rather than full market simulations.
Merger simulations could incorporate this to predict changes in pass-through rates post-merger.
Extensions to other competitive settings like quantity competition or dynamic models may be possible.
The characterization suggests testing demand systems by their implied pass-through in small-share regimes.

Load-bearing premise

The derivations assume standard Nash price competition with twice-differentiable demand and that higher-order terms can be ignored in the approximation under common functional forms.

What would settle it

Numerically solving for equilibrium prices before and after a small cost shock in a specific oligopoly model with multiproduct firms and comparing the resulting pass-through matrix to the decomposed formula would test the claim.

Figures

Figures reproduced from arXiv: 2604.21423 by Paul S. Koh.

**Figure 1.** Figure 1: Finite Neumann approximation: matrix error view at source ↗

**Figure 2.** Figure 2: Exact vs. approximate price responses Notes: The figure compares exact price responses to those implied by the first-order Neumann approximation and the small-share approximation under three tax experiments: a uniform tax, a firm-level tax, and a singleproduct tax. The horizontal axis reports average product share, which varies across simulated markets through shifts in mean utilities. The vertical axis r… view at source ↗

read the original abstract

This paper studies cost pass-through in differentiated-product oligopoly. I derive a general representation of the pass-through matrix that decomposes equilibrium price responses into the roles of demand curvature, substitution, and multiproduct ownership. This extends the classic insight in single-product monopoly to multiproduct settings in which diversion and ownership also matter. I then develop a tractable first-order approximation that yields a sufficient-statistics characterization for empirically relevant demand systems. Finally, I characterize the small-share limit and show how common demand specifications impose tail restrictions that shape pass-through. The results provide a practical framework for applied work on tax incidence, merger analysis, and related questions in imperfect competition.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean decomposition of pass-through in multiproduct oligopoly that extends monopoly logic and offers practical approximations.

read the letter

The main thing to know about this paper is that it derives a general representation of the pass-through matrix in differentiated oligopoly, decomposing price responses into demand curvature, substitution patterns, and multiproduct ownership. This extends the single-product monopoly insight to settings where diversion and ownership matter, which covers most real markets. It does this well by starting from the Nash price competition first-order conditions and using the demand Jacobian and Hessian to get the decomposition without additional assumptions. The tractable first-order approximation then provides a sufficient-statistics characterization for empirically relevant demand systems, which should be handy for applied researchers. The small-share limit section shows how common specifications impose tail restrictions that influence pass-through. The derivations look solid and follow standard technical conditions like twice continuous differentiability and interior equilibrium. There is no indication of circularity or post-hoc fitting. The only soft spot worth noting is that the approximation and small-share results may be sensitive to the demand functional form; if higher-order terms dominate or if the demand is far from the usual ones, the characterizations could require more qualification. But the core general representation does not depend on those. This paper is for empirical industrial organization economists and those working on tax incidence or merger analysis in imperfectly competitive markets. A reader looking for a theoretical framework to apply to estimated demand systems will find it useful. I recommend sending it to peer review. The contribution is focused and the tools it offers have clear applied value.

Referee Report

0 major / 2 minor

Summary. The paper derives a general representation of the pass-through matrix for cost pass-through in differentiated-product oligopoly. This representation decomposes equilibrium price responses into the roles of demand curvature, substitution patterns, and multiproduct ownership. It extends the single-product monopoly insight to multiproduct settings. The paper also develops a tractable first-order approximation for sufficient-statistics characterization in empirically relevant demand systems and characterizes the small-share limit, showing how common demand specifications impose tail restrictions that shape pass-through.

Significance. If the derivations hold, this provides a practical framework for applied work on tax incidence, merger analysis, and imperfect competition. The decomposition isolates key economic forces and generalizes classic results. Strengths include the use of standard oligopoly primitives for the general representation and the separation of the exact decomposition from the approximation and limit cases. This could be useful for empirical researchers needing to understand pass-through determinants.

minor comments (2)

[Abstract] The abstract is concise but could briefly mention the key assumptions underlying the general representation to set expectations for readers.
The paper would benefit from a table summarizing the main results, such as the pass-through decomposition formula and the approximation, for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper and for recommending minor revision. The referee's summary accurately reflects the manuscript's focus on the pass-through matrix decomposition, its extension of single-product results, and the development of approximations and limit characterizations for applied settings.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from primitives

full rationale

The central result is a matrix decomposition of the pass-through operator obtained directly from the first-order conditions of Nash-Bertrand competition once the demand Jacobian and Hessian are defined. This follows from standard differentiated oligopoly primitives (twice-differentiable demand, interior equilibrium, invertible Jacobian) without fitting parameters to the target pass-through quantities or invoking self-citations as load-bearing support. The subsequent first-order approximation and small-share limit are explicitly optional characterizations that do not underpin the general representation. No step reduces by construction to its own inputs, and no uniqueness theorem or ansatz is smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review limits visibility into explicit free parameters or invented entities; standard IO assumptions are invoked but not enumerated.

axioms (2)

domain assumption Firms play static Nash equilibrium in prices
Implicit in any oligopoly pass-through derivation; required for equilibrium price responses to be well-defined.
standard math Demand is twice continuously differentiable
Needed to define curvature terms and first-order approximations.

pith-pipeline@v0.9.0 · 5396 in / 1398 out tokens · 43068 ms · 2026-05-08T13:24:05.556851+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 1 canonical work pages

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