From Technical Feasibility to Substitutability: A Geometric Theory of 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.