Pass-through with Price Dispersion

1. Problem Statement

The paper addresses a gap in the economic literature regarding cost pass-through (the extent to which cost shocks are reflected in consumer prices) in markets characterized by price dispersion.

• The Gap: Standard pass-through literature (e.g., Weyl and Fabinger, 2013) assumes a unique equilibrium price (monopoly or symmetric oligopoly). However, empirical evidence shows that homogeneous products often trade at a distribution of prices due to consumer search frictions, limited information, or platform algorithms.
• The Challenge: Existing models for price dispersion (e.g., Varian, 1980; Burdett and Judd, 1983) characterize why prices vary but fail to provide a tractable framework for analyzing how cost shocks transmit through these price distributions. The interaction between demand curvature, cost levels, and mixed-strategy equilibria makes standard comparative statics intractable.

2. Methodology and Framework

The authors develop a tractable framework based on the consideration-set model (Armstrong and Vickers, 2022).

• Model Setup:

  • There are $n$ firms and a unit mass of consumers.
  • Consumers have consideration sets ($S$): subsets of firms they observe. They purchase from the cheapest firm in their set, provided the price is below a reservation value (normalized to 1).
  • Firms compete by choosing price distributions (mixed strategies).
  • Demand is downward-sloping ($x(p)$), and firms share a common marginal cost $c$.

• The Core Innovation: The $\mu$-Isomorphism
  The authors introduce a change of variables to decompose the pricing game into two distinct layers:

  1. The Competition Layer: Firms compete over normalized effective margins ($\mu$), defined as:
     $$\mu(p; c) \equiv \frac{(p-c)x(p)}{(1-c)x(1)}$$
     This variable maps the effective profit per customer to the unit interval $[0, 1]$.
  2. The Curvature Layer: The mapping from normalized margins back to prices, $\phi(\mu, c)$, which depends on the specific demand function $x(\cdot)$ and cost $c$.

  The Isomorphism Theorem (Theorem 3.3): The authors prove that the pricing game is strategically equivalent to a game played over normalized margins $\mu$ with unit demand and zero marginal cost.

  • Key Result: The equilibrium distribution of normalized margins ($F^\mu$) depends only on the market structure (the consideration sets $\{\alpha_S\}$) and is invariant to the demand function curvature and the cost level $c$.
  • Implication: Cost changes do not alter the strategic distribution of market power ($\mu$); they only shift the price distribution via the mechanical translation function $\phi$.

3. Key Contributions

1. Decomposition of Pass-Through: The paper separates pass-through analysis into a competition layer (determined by market structure) and a curvature layer (determined by demand elasticity). This allows researchers to solve for equilibrium margins once and apply them to any demand specification.
2. Closed-Form Quantile Pass-Through: The authors derive explicit formulas for pass-through rates at every quantile ($u$) of the price distribution, rather than just an aggregate average.
3. Transaction-Weighted Pass-Through: They derive a sufficient statistic for welfare-relevant pass-through (what consumers actually pay) under unit demand, showing it depends solely on the consideration structure.
4. Robust Bounds: The paper provides "envelope" bounds for pass-through rates that hold across entire families of demand functions (linear, constant elasticity, etc.) without needing precise parameter estimates.

4. Key Results

A. Equilibrium Characterization

• Symmetric Markets: The equilibrium margin quantile function is $\mu(u) = \rho / H(1-u)$, where $\rho$ is the captive-to-reach ratio and $H$ is the probability generating function of competitor counts.
• Asymmetric Duopoly: Firms with higher captive shares (higher $\rho$) maintain higher margins at every quantile. The firm with more captives prices at the monopoly margin ($\mu=1$) with positive probability (an atom), while the weaker firm mixes continuously.
• Independent Consideration: In markets where consumers consider firms independently, a hierarchical structure emerges. Higher-reach firms (larger $\lambda$) maintain higher margins and have flatter CDFs.

B. Pass-Through Formulas

• Quantile Pass-Through ($\tau^Q$):
  $$\tau^Q_i(u; c) = \phi_c(\mu_i(u), c) = \frac{x(p)(1-p)}{(1-c)[x(p) + (p-c)x'(p)]}$$
  This formula reveals that pass-through varies across the price distribution. Consumers buying at low prices (low margins) face different pass-through rates than those buying at high prices.
• Transaction-Weighted Pass-Through ($\tau^{trans}$):
  Under unit demand, the industry pass-through rate is:
  $$\tau^{trans} = 1 - \bar{\mu}^{trans}$$
  where $\bar{\mu}^{trans}$ is the average normalized margin among transacting consumers. This implies that pass-through is simply one minus the average market power.

C. Comparative Statics and Bounds

• Ordering: If a market structure change increases the margin distribution (e.g., fewer informed consumers), prices increase at all quantiles. If demand is not too convex, pass-through rates decrease (firms absorb more of the shock).
• Universal Bounds: Even without knowing the exact demand, pass-through is bounded below by $1 - \mu$. Firms with low market power ($\mu \approx 0$) must pass through at least 100% of costs; firms with high market power ($\mu \approx 1$) can pass through close to 0%.
• Demand Families:
  • Linear Demand: Pass-through is strictly below 1 (no over-shifting).
  • Constant Elasticity (CES): Over-shifting ($\tau > 1$) is possible if the implied price is sufficiently low relative to the elasticity parameter.

5. Significance and Applications

• Policy Analysis: The framework allows policymakers to estimate the incidence of taxes or cost shocks using data on consideration sets (e.g., platform clicks, geographic proximity) rather than requiring full structural estimation of demand elasticity.
• Merger Analysis: Mergers alter the consideration structure. The authors show that a merger can shift the incidence of future cost shocks toward consumers even if it has no immediate effect on price levels, by changing the distribution of margins.
• Empirical Relevance:
  • Gasoline Markets: Explains why informed consumers (who buy at low prices) face higher pass-through than uninformed consumers.
  • Online Retail: Demonstrates how platform algorithms (which shape consideration sets) determine who bears the burden of input cost increases.
• General Equilibrium Extension: The paper extends the model to endogenous consideration, showing how cost shocks can alter marketing strategies and thus the consideration structure, adding a second channel for pass-through effects.

Conclusion

Albrecht and Whitmeyer provide a powerful "decomposition" tool that renders the complex problem of pass-through in dispersed-price markets tractable. By isolating the strategic competition (market structure) from the mechanical translation (demand curvature), they offer closed-form solutions, robust bounds, and a unified framework for analyzing incidence in heterogeneous markets. This bridges the gap between industrial organization theory on price dispersion and the literature on tax incidence.