### 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 language | English |
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Journal | Discrete Applied Mathematics |

DOIs | |

Publication status | Accepted/In press - 2019 Jan 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics