### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

**The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities.** / Yokote, Koji.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The discrete separation theorem and price adjustment directions in markets with heterogeneous commodities

AU - Yokote, Koji

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Auction

KW - Discrete convex analysis

KW - Hall's theorem

KW - Separation theorem

KW - Walrasian taˆtonnement

UR - http://www.scopus.com/inward/record.url?scp=85071916615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071916615&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2019.08.022

DO - 10.1016/j.dam.2019.08.022

M3 - Article

AN - SCOPUS:85071916615

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -