### Abstract

We identify a class of economies for which tatonnement is equivalent to gradient descent. This is the class of economies for which there is a convex potential function whose gradient is always equal to the negative of the excess demand. Among other consequences, we show that a discrete version of tatonnement converges to the equilibrium for the following economies of complementary goods. i. Fisher economies in which all buyers have complementary CES utilities, with a linear rate of convergence. (In Fisher economies all agents are either buyers or sellers of non-numeraire goods, but not both.) This shows that tatonnement converges for the entire range of Fisher economies when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range. ii. Fisher economies in which all buyers have Leontief utilities, with an O(1/t) rate of convergence.

Original language | English (US) |
---|---|

Journal | Games and Economic Behavior |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Equilibria
- Gradient descent
- Market
- Tatonnement

### ASJC Scopus subject areas

- Finance
- Economics and Econometrics

### Cite this

*Games and Economic Behavior*. https://doi.org/10.1016/j.geb.2019.03.014

**Tatonnement beyond gross substitutes? Gradient descent to the rescue.** / Cheung, Yun Kuen; Cole, Richard; Devanur, Nikhil R.

Research output: Contribution to journal › Article

*Games and Economic Behavior*. https://doi.org/10.1016/j.geb.2019.03.014

}

TY - JOUR

T1 - Tatonnement beyond gross substitutes? Gradient descent to the rescue

AU - Cheung, Yun Kuen

AU - Cole, Richard

AU - Devanur, Nikhil R.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We identify a class of economies for which tatonnement is equivalent to gradient descent. This is the class of economies for which there is a convex potential function whose gradient is always equal to the negative of the excess demand. Among other consequences, we show that a discrete version of tatonnement converges to the equilibrium for the following economies of complementary goods. i. Fisher economies in which all buyers have complementary CES utilities, with a linear rate of convergence. (In Fisher economies all agents are either buyers or sellers of non-numeraire goods, but not both.) This shows that tatonnement converges for the entire range of Fisher economies when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range. ii. Fisher economies in which all buyers have Leontief utilities, with an O(1/t) rate of convergence.

AB - We identify a class of economies for which tatonnement is equivalent to gradient descent. This is the class of economies for which there is a convex potential function whose gradient is always equal to the negative of the excess demand. Among other consequences, we show that a discrete version of tatonnement converges to the equilibrium for the following economies of complementary goods. i. Fisher economies in which all buyers have complementary CES utilities, with a linear rate of convergence. (In Fisher economies all agents are either buyers or sellers of non-numeraire goods, but not both.) This shows that tatonnement converges for the entire range of Fisher economies when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range. ii. Fisher economies in which all buyers have Leontief utilities, with an O(1/t) rate of convergence.

KW - Equilibria

KW - Gradient descent

KW - Market

KW - Tatonnement

UR - http://www.scopus.com/inward/record.url?scp=85064275977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064275977&partnerID=8YFLogxK

U2 - 10.1016/j.geb.2019.03.014

DO - 10.1016/j.geb.2019.03.014

M3 - Article

AN - SCOPUS:85064275977

JO - Games and Economic Behavior

JF - Games and Economic Behavior

SN - 0899-8256

ER -