Topological Methods in Economics: From Equilibrium Existence to Topological Data Analysis

Jakub Ryłow

No 2026-9, Working Papers from Faculty of Economic Sciences, University of Warsaw

Abstract: This paper surveys how the three branches of topology — differential, algebraic, and point-set — enter mathematical economics, and organises classical and contemporary results within a single conceptual frame. The classical layer rests on four pillars: Sard’s theorem and the preimage theorem, which underwrite Smale’s convexity-free proof of Walrasian equilibrium existence and Debreu’s local-uniqueness theorem for generic economies; the Brouwer and Kakutani fixed-point theorems behind the Arrow–Debreu existence proof; the Chichilnisky–Heal homotopy obstruction that ties continuous anonymous social choice to contractibility of preferred cones, and through it to the existence of competitive equilibrium; and the outer Hausdorff metric of Berliant and ten Raa, which makes location theory well-posed on the infinite-dimensional commodity space of land parcels. The contemporary layer turns the same apparatus toward data. The L¹ norm of H₁ persistence landscapes computed on rolling windows of asset returns rises three to eight weeks before the 2008 and 2015–2016 market crashes, outperforming the VIX in lead time; Ollivier–Ricci curvature on equity correlation networks supplies a complementary leading indicator of systemic fragility. Optimal transport recasts identifiability in matching markets as uniqueness of a dual Kantorovich plan, governed by the topology of the type-space supports. The manifold hypothesis makes nonparametric instrumental-variable convergence rates depend on intrinsic rather than ambient dimension. Together these results trace a coherent arc from fixed-point theorems to topological data analysis; a Morse-theoretic synthesis on the state manifold of an economy, linking equilibria, tâtonnement dynamics, and saddle-point crises, is stated as a research programme rather than a theorem.

Keywords: differential topology; persistent homology; topological data analysis; Walrasian equilibrium; optimal transport; social choice; Ollivier–Ricci curvature; manifold hypothesis (search for similar items in EconPapers)
JEL-codes: B16 C02 C14 D50 G01 (search for similar items in EconPapers)
Pages: 37 pages
Date: 2026
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.wne.uw.edu.pl/download_file/7139/0 First version, 2026 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:war:wpaper:2026-9

Access Statistics for this paper

More papers in Working Papers from Faculty of Economic Sciences, University of Warsaw Contact information at EDIRC.
Bibliographic data for series maintained by Jacek Rapacz ().