Pass-through with Price Dispersion

TL;DR

The paper develops a dispersed-price framework for pass-through that decomposes cost transmission into a competition layer, governed by consideration-structure, and a curvature layer, driven by demand elasticity. By reformulating the pricing game in terms of normalized effective margins $\\mu(p;c)$, it proves a mu-isomorphism: equilibrium margin distributions depend only on how consumers consider firms, not on demand form or costs, while prices follow from a demand-specific map $\\phi(\\mu,c)$. This separation yields closed-form quantile pass-through, robust bounds across demand families, and clear comparative statics, with practical insights for merger analysis, welfare assessment, and empirical strategy focusing on consideration data. The framework applies to symmetric and asymmetric firm settings, extends to independent consideration, and provides tractable results for transaction-weighted pass-through and envelope bounds, offering a comprehensive toolkit for incidence analysis in dispersed-price markets.

Abstract

How do cost shocks pass through to prices in markets with price dispersion? Pass-through analysis typically assumes a single equilibrium price, but empirical studies consistently document substantial price variation, even for homogeneous products. This paper develops a tractable framework that decomposes the pass-through problem into two distinct tiers. The first is a competition layer where consumers' \textit{consideration sets} determine equilibrium distributions of normalized margins. The second is a curvature layer where demand elasticity determines how these margins translate into prices and pass-through rates. The key theoretical innovation is showing that the strategic pricing game with arbitrary downward-sloping demand is order-isomorphic to a baseline unit-demand game once reformulated in terms of normalized effective margins. This decomposition yields closed-form pass-through formulas, robust bounds across demand specifications, and clear comparative statics linking market structure to incidence.

Figures (1)

• Figure 1: Robust pass-through bounds by demand family. Left panel: For linear demand $x(p) = 1 + b(1-p)$ with slope $b \in [0, 1/d]$, pass-through lies between the unit demand lower bound $\tau = 1 - \mu$ and the upper bound $\tau = (1 + \sqrt{1-\mu})/2$. All linear demands yield $\tau \leq 1$. Right panel: For CES demand $x(p) = p^{-\eta}$, higher elasticity $\eta$ produces over-shifting ($\tau > 1$). Below the critical elasticity $\eta \approx 1.3$, pass-through eventually falls below one at high margins; above it, the invertibility condition binds before $\tau$ can fall to one.

Theorems & Definitions (49)

• Definition 1
• Example 2.1: Random Search
• Example 2.2: Spatial Markets
• Definition 2
• Definition 3
• Definition 4
• Lemma 3.1
• proof
• Definition 5
• Lemma 3.2