Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation

Codenotti, Bruno; McCune, Benton; Penumatcha, Sriram; Varadarajan, Kasturi

doi:10.1007/11590156_41

Cited by 22 publications

(52 citation statements)

References 17 publications

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“…Devanur and Kannan [14] gave a polynomial-time algorithm for markets with piecewise-linear and concave (PLC) utilities and a constant number of goods. Codenotti, McCune, Penumatcha, and Varadarajan [15] gave a polynomialtime algorithm for CES markets when the elasticity of substitution s ≥ 1/2.…”

Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

“…We know that an (approximate) market equilibrium in any Fisher's economy with CES utilities can be found in polynomial time [4,15,12,21,7,22]. In fact, Ye [21] proved that if every utility function is the minimum of a collection of homogeneous linear functions, then one can find a Fisher equilibrium in polynomial time.…”

Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

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Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Chen

Teng

2009

Algorithms and Computation

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Abstract. It is a common belief that computing a market equilibrium in Fisher's spending model is easier than computing a market equilibrium in Arrow-Debreu's exchange model. This belief is built on the fact that we have more algorithmic success in Fisher equilibria than ArrowDebreu equilibria. For example, a Fisher equilibrium in a Leontief market can be found in polynomial time, while it is PPAD-hard to compute an approximate Arrow-Debreu equilibrium in a Leontief market.In this paper, we show that even when all the utilities are additively separable, piecewise-linear, and concave functions, finding an approximate equilibrium in Fisher's model is complete in PPAD. Our result solves a long-term open question on the complexity of market equilibria. To the best of our knowledge, this is the first PPAD-completeness result for Fisher's model.

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Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

Section: Market Equilibria: Fisher's Model Vs Arrow-debreu's Modelmentioning

confidence: 99%

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Chen

Teng

2009

Algorithms and Computation

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“…The set of equilibria in these markets can be disconnected [18,6]. Furthermore, no efficient algorithm is known for computing the equilibrium prices in these markets, except in the case of proportional endowments, where the set of equilibria is convex [5].…”

Section: Introductionmentioning

confidence: 99%

Leontief economies encode nonzero sum two-player games

Codenotti

Saberi

Varadarajan

et al. 2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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We consider Leontief exchange economies, i.e., economies where the consumers desire goods in fixed proportions. Unlike bimatrix games, such economies are not guaranteed to have equilibria in general. On the other hand, they include suitable restricted versions which always have equilibria.We give a reduction from two-player games to a special family of Leontief exchange economies, which are guaranteed to have equilibria, with the property that the Nash equilibria of any game are in one-to-one correspondence with the equilibria of the corresponding economy.Our reduction exposes a potential hurdle inherent in solving certain families of market equilibrium problems: finding an equilibrium for Leontief economies (where an equilibrium is guaranteed to exist) is at least as hard as finding a Nash equilibrium for two-player nonzero sum games.As a corollary of the one-to-one correspondence, we obtain a number of hardness results for questions related to the computation of market equilibria, using results already established for games [17]. In particular, among other results, we show that it is NP-hard to say whether a particular family of Leontief exchange economies, that is guaranteed to have at least one equilibrium, has more than one equilibrium.Perhaps more importantly, we also prove that it is NPhard to decide whether a Leontief exchange economy has an equilibrium. This fact should be contrasted against the known PPAD-completeness result of [30], which holds when the problem satisfies some standard sufficient conditions that make it equivalent to the computational version of Brouwer's Fixed Point Theorem.On the algorithmic side, we present an algorithm for finding an approximate equilibrium for some special Leontief economies, which achieves quasi-polynomial time whenever each trader does not demand too much more of any good than some other good.

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“…The market problem is to find equilibrium prices. 2 The symbol * marking a variable will be used to denote the value of the variable at equilibrium.…”

Section: The Market Problemsmentioning

confidence: 99%

Tatonnement in ongoing markets of complementary goods

Cheung

Cole

Rastogi

2012

Proceedings of the 13th ACM Conference on Electronic Commerce

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This paper continues the study, initiated by Cole and Fleischer in [6], of the behavior of a tatonnement price update rule in Ongoing Fisher Markets. The prior work showed fast convergence toward an equilibrium when the goods satisfied the weak gross substitutes property and had bounded demand and income elasticities.The current work shows that fast convergence also occurs for the following types of markets:• All pairs of goods are complements to each other, and• the demand and income elasticities are suitably bounded.In particular, these conditions hold when all buyers in the market are equipped with CES utilities, where all the parameters ρ, one per buyer, satisfy −1 < ρ ≤ 0.In addition, we extend the above result to markets in which a mixture of complements and substitutes occur. This includes characterizing a class of nested CES utilities for which fast convergence holds.An interesting technical contribution, which may be of independent interest, is an amortized analysis for handling asynchronous events in settings in which there are a mix of continuous changes and discrete events.

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Market Equilibrium for CES Exchange Economies: Existence, Multiplicity, and Computation

Cited by 22 publications

References 17 publications

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Spending Is Not Easier Than Trading: On the Computational Equivalence of Fisher and Arrow-Debreu Equilibria

Leontief economies encode nonzero sum two-player games

Tatonnement in ongoing markets of complementary goods

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