Paper: arXiv 2507.01985

Authors: Aldric Labarthe, Yann Kerzreho

Abstract

We develop a framework for horizontal differentiation in which firms compete on a product manifold representing the feasible combinations of characteristics. This approach generalizes both the Hotelling line and Salop circle to any Riemannian space, allowing for a unified analysis of product space. We show that equilibrium existence and stability are governed by intrinsic geometric properties, specifically curvature, symmetry and dimension. We prove that negative curvature and high intrinsic dimension act as stabilizers of minimum differentiation equilibria, moving the analysis beyond the symmetry-induced instabilities found in simpler, fixed domains. By characterizing curvature as a measure of consumer heterogeneity, we demonstrate that intrinsic geometry is a fundamental determinant of competitive outcomes.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly novel and mathematically sophisticated theoretical framework, unifying spatial competition models using Riemannian geometry. However, it is purely theoretical and mathematical with no empirical data analysis or real-world validation included in the provided text.

Research Flowchart

  flowchart TD
    A["Research Goal: Generalize Horizontal<br>Differentiation to Riemannian Product Manifolds"] --> B["Framework Setup: Modeling Product Space<br>via Riemannian Geometry"]
    B --> C["Inputs: Curvature, Symmetry,<br>& Dimension Properties"]
    C --> D["Analysis: Evaluating Equilibrium Existence<br>& Stability via Intrinsic Geometry"]
    D --> E["Outcome 1: Negative Curvature & High<br>Dimension Stabilize Minimum Differentiation"]
    D --> F["Outcome 2: Intrinsic Curvature Identified<br>as Measure of Consumer Heterogeneity"]

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0 of 58 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.