From Technical Feasibility to Substitutability: A Geometric Theory of Differentiation

TL;DR

This paper develops a geometric framework modeling product differentiation as a Riemannian manifold, linking the set's curvature to market competition and technological substitution.

Contribution

It introduces a novel geometric approach to analyze how the structure of feasible products influences market outcomes and technological divergence.

Findings

Negative curvature increases technological divergence and reduces competition.

Positive curvature compresses technological distances and enhances competition.

High dimensionality and negative curvature stabilize minimum differentiation.

Abstract

We study horizontal differentiation when the set of feasible products is a structured subset of the Lancasterian characteristics space. Modeling this set as a compact Riemannian manifold, we show that intrinsic geometry governs substitutability and thereby determines market outcomes. We establish that production constraints induce sectional curvature, which controls the elasticity of technological substitution. Negative curvature amplifies technological divergence and attenuates competitive pressure, whereas positive curvature compresses technological distances and intensifies competition. This mapping yields a characterization of spatial competition in which equilibrium existence and stability are determined by geometric primitives. In particular, we show that sufficiently negative curvature and high dimensionality stabilize minimum differentiation, while continuous symmetries preclude it. The analysis provides a microfoundation linking technological constraints, through the geometry of the feasible set, to endogenous regimes of market power.