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

### 抄録

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.

元の言語 English Discrete Applied Mathematics https://doi.org/10.1016/j.dam.2019.08.022 Accepted/In press - 2019 1 1

### Fingerprint

Separation Theorem
Auctions
Disjoint
Convex Analysis
Normal vector
Theorem
Hyperplane
Convex Sets
Market

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Applied Mathematics

### これを引用

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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.",
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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.

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KW - Walrasian taˆtonnement

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