TY - JOUR

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

AU - Yokote, Koji

N1 - Funding Information:
The author is grateful to the four referees of the journal, as well as Yukihiko Funaki, Yoko Kawada, Fuhito Kojima, Kazuo Murota, Yuta Nakamura, and Noriaki Okamoto for their valuable comments. This work was supported by JSPS, Japan Grant-in-Aid for Research Activity Start-up (Grant number: 17H07179 ) and Waseda University, Japan Grant for Special Research Projects (Project number: 2017S-006 ).

PY - 2020/3/31

Y1 - 2020/3/31

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

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U2 - 10.1016/j.dam.2019.08.022

DO - 10.1016/j.dam.2019.08.022

M3 - Article

AN - SCOPUS:85071916615

VL - 275

SP - 134

EP - 143

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -