The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities

Research output: Contribution to journalArticle

Abstract

The separation theorem in discrete convex analysis states that two disjoint discrete convex sets can be separated by a hyperplane with a 0–1 normal vector. We apply this theorem to markets with heterogeneous commodities and uncover the mathematical structure behind price adjustment processes. When p is not an equilibrium price vector, i.e., when aggregate demand and aggregate supply are disjoint, the separation theorem indicates the existence of overdemanded/underdemanded items. This observation yields a generalization of Hall's (1935) theorem and a characterization of equilibrium price vectors by Gul and Stacchetti (2000). Building on this characterization, we show that adjusting the prices of overdemanded/underdemanded items corresponds to Ausbel's (2006) auction. We further extend our approach to markets with continuous commodities and uncover a striking connection between auctions and classical taˆtonnement processes.

Original languageEnglish
JournalDiscrete Applied Mathematics
DOIs
Publication statusAccepted/In press - 2019 Jan 1

Keywords

  • Auction
  • Discrete convex analysis
  • Hall's theorem
  • Separation theorem
  • Walrasian taˆtonnement

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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