Curvature, Minimality and Uniqueness of Equilibrium

2026-06-03

AI Summary

This study investigates the relationship between the geometric properties of the equilibrium manifold in smooth pure exchange economies and the uniqueness of equilibrium prices. By introducing a local parametrization of the equilibrium manifold and leveraging tools from differential geometry—specifically curvature and minimal surface theory—the authors circumvent the explicit construction of a normal frame to establish, for the first time and without restrictions on the number of goods or agents, that the equilibrium manifold is intrinsically flat if and only if equilibrium prices are locally constant (hence globally unique). In the two-good case, they replace asymptotic assumptions prevalent in prior literature with a minimality condition, thereby deriving a clean equivalence between entropy and uniqueness and eliminating auxiliary hypotheses required by earlier results, thus refining and extending the curvature–uniqueness theorem to higher dimensions.

Abstract

For a smooth pure exchange economy with fixed aggregate resources, we study two geometric conditions on the equilibrium manifold $E(r)$ endowed with the metric induced from its Euclidean ambient space. First, for arbitrary numbers of commodities and consumers, we prove that intrinsic flatness forces equilibrium prices to be locally constant. Together with Balasko's uniqueness--constancy criterion, this yields a necessary and sufficient condition: $E(r)$ is intrinsically flat if and only if the normalized equilibrium price is unique for every economy with aggregate resources $r$. This extends the curvature--uniqueness theorem of \cite{LoiMatta2018} and completes the higher-dimensional direction pursued in \cite{LoiMattaUccheddu2023}. Second, in the two-commodity case, we show that minimality of $E(r)$ already forces local constancy of the price map. Under the uniform-distribution interpretation of \cite{LoiMatta2021}, this gives the minimal-entropy/uniqueness equivalence without the additional asymptotic assumption used there. Both arguments rely on the same local parametrization of $E(r)$ and avoid the explicit construction of a normal frame.

Problem

• curvature
• minimality
• uniqueness
• equilibrium manifold
• price uniqueness

Innovation

• intrinsic flatness
• minimality
• equilibrium uniqueness
• equilibrium manifold
• price constancy