## Fisher Equilibrium Price with a Class of Concave Utility Functions (2004)

Venue: | IN THE PROCEEDINGS OF ESA 2004 |

Citations: | 13 - 1 self |

### BibTeX

@INPROCEEDINGS{Chen04fisherequilibrium,

author = {Ning Chen and Xiaotie Deng and Xiaoming Sun and Andrew Chi-chih Yao},

title = {Fisher Equilibrium Price with a Class of Concave Utility Functions },

booktitle = {IN THE PROCEEDINGS OF ESA 2004},

year = {2004},

pages = {169--179},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study efficient algorithms for computing equilibrium price in the Fisher model for a class of nonlinear concave utility functions, the logarithmic utility functions. A duality relation is derived between buyers and sellers under such utility functions. It is applied to design a polynomial time algorithm for calculating equilibrium price, for the special case when either the number of sellers or the number of buyers is bounded by a constant.

### Citations

129 |
The theory of linear economic models
- Gale
- 1960
(Show Context)
Citation Context ...orithm for the linear exchange model [8], the first nontrivial finite time algorithm for price equilibrium. Noticeably, Gale made an extensive study for equilibrium prices of linear utility functions =-=[11]-=-. The first (known to us) non-trivial polynomial time algorithm for equilibrium pricing models were derived by Eaves for Cobb-Douglas utility functions [9]. Recently, Deng, Papadimitriou, and Safra [4... |

98 | Market Equilibrium via a Primal-Dual-Type Algorithm
- Devanur, Papadimitriou, et al.
(Show Context)
Citation Context ...uently, a fair number of explorative works have followed for equilibrium and approximate equilibrium price under linear utility functions to study the problem with general input size. Devanur, et al. =-=[5]-=-, obtained a polynomial time algorithm for a price equilibrium for the Fisher model, in which agents are initialed with certain amount of money. Later, on the basis of the duality type of algorithm pr... |

61 |
A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities
- Jain
- 2004
(Show Context)
Citation Context ...proposed in [5], Jain et al. [12], Devanur and Vazirani [6], and Garg and Kapoor [14] showed different polynomial time approximation schemes for the linear utility functions in the Fisher model. Jain =-=[13]-=- gave a polynomial time algorithm to compute equilibrium price for linear utility functions in the AD model. Unfortunately, those results were for the linear utility functions, which still leaves open... |

57 |
Existence of an Equilibrium for a Competitive
- Arrow, Debreu
- 1954
(Show Context)
Citation Context ...60223004, 60321002) of National Natural Science Foundation of China. ⋆ ⋆ ⋆ The research of this author was supported in part by US National Science Foundation under grant CCR-0310466sArrow and Debreu =-=[1]-=- proved the existence of equilibrium price assuming goods are divisible. Their proof was based on the fixed point theorem and thus was not constructive. Fisher, on the other hand, made an effort to so... |

43 |
The computation of equilibrium prices
- Scarf
- 1982
(Show Context)
Citation Context ... hand, made an effort to solve the problem and designed a hydraulic apparatus and proved that it would result in an exact solution [10], which was recently implemented digitally by Brainard and Scarf =-=[2]-=-. On the algorithmic side, Scarf’s pioneering work [16] pointed to the possibility of obtaining the equilibrium price in the limit through an iterative combinatorial algorithm. In contrast to infinite... |

41 | Auction algorithms for market equilibrium
- Garg, Kapoor
- 2004
(Show Context)
Citation Context ...odel, in which agents are initialed with certain amount of money. Later, on the basis of the duality type of algorithm proposed in [5], Jain et al. [12], Devanur and Vazirani [6], and Garg and Kapoor =-=[14]-=- showed different polynomial time approximation schemes for the linear utility functions in the Fisher model. Jain [13] gave a polynomial time algorithm to compute equilibrium price for linear utility... |

33 | Efficient computation of equilibrium prices for markets with Leontief utilities
- Codenotti, Varadarajan
- 2004
(Show Context)
Citation Context ...iven price vector the above algorithm computes Γ (i) in O(m 3 ) time. Now, for any given graph G, we define for each buyer i the “bias factor” λi as follows: λi = ei + � j∈Γ (i) ɛi,jpj � j∈Γ (i) ai,j =-=(3)-=- Note that in the last iteration of the algorithm to determine the transaction graph, we have η = 1/λi > 0, and for any j ∈ Γ (i), xi,j = ai,j ηpj − ɛi,j = ai,j pj λi − ɛi,j > 0. At any time during th... |

27 | The spending constraint model for market equilibrium: algorithmic, existence and uniqueness results - Devanur, Vazirani - 2004 |

21 | An improved approximation scheme for computing Arrow-Debreu prices for the linear case
- Devanur, Vazirani
- 2003
(Show Context)
Citation Context ...ilibrium for the Fisher model, in which agents are initialed with certain amount of money. Later, on the basis of the duality type of algorithm proposed in [5], Jain et al. [12], Devanur and Vazirani =-=[6]-=-, and Garg and Kapoor [14] showed different polynomial time approximation schemes for the linear utility functions in the Fisher model. Jain [13] gave a polynomial time algorithm to compute equilibriu... |

17 |
The Computation of Economic Equilibria (with collaboration of T
- Scarf
- 1973
(Show Context)
Citation Context ...d a hydraulic apparatus and proved that it would result in an exact solution [10], which was recently implemented digitally by Brainard and Scarf [2]. On the algorithmic side, Scarf’s pioneering work =-=[16]-=- pointed to the possibility of obtaining the equilibrium price in the limit through an iterative combinatorial algorithm. In contrast to infinitely convergent algorithms of Scarf’s paradigm, Eaves der... |

12 |
A finite algorithm for the linear exchange model
- Eaves
- 1976
(Show Context)
Citation Context ...um price in the limit through an iterative combinatorial algorithm. In contrast to infinitely convergent algorithms of Scarf’s paradigm, Eaves derived a finite algorithm for the linear exchange model =-=[8]-=-, the first nontrivial finite time algorithm for price equilibrium. Noticeably, Gale made an extensive study for equilibrium prices of linear utility functions [11]. The first (known to us) non-trivia... |

2 |
Eaves, Finite Solution of Pure Trade Markets with Cobb-Douglas Utilities
- Curtis
- 1985
(Show Context)
Citation Context ...ilibrium prices of linear utility functions [11]. The first (known to us) non-trivial polynomial time algorithm for equilibrium pricing models were derived by Eaves for Cobb-Douglas utility functions =-=[9]-=-. Recently, Deng, Papadimitriou, and Safra [4] proposed a comprehensive study of complexity issues for the equilibrium price problem in the Arrow-Debreu model. Complexity and algorithmic results for t... |

1 |
On the Complexity of Equilibria, STOC 2002, 67-71
- Deng, Papadimitriou, et al.
- 2003
(Show Context)
Citation Context ...1]. The first (known to us) non-trivial polynomial time algorithm for equilibrium pricing models were derived by Eaves for Cobb-Douglas utility functions [9]. Recently, Deng, Papadimitriou, and Safra =-=[4]-=- proposed a comprehensive study of complexity issues for the equilibrium price problem in the Arrow-Debreu model. Complexity and algorithmic results for the equilibrium price are obtained. In addition... |

1 |
Introduction to Optimization”, Opt
- Polyak
- 1987
(Show Context)
Citation Context ...marginal utility functions, u (1) i,j (t), are all constant. Given a price vector P = (p1, . . . , pm) > 0, each buyer i has a unique optimal allocation � x∗ i,1 (P ), . . . , x∗i,m (P )� (see, e.g., =-=[15]-=-), that maximizes the utility function ui of buyer i: max ui(xi,1, . . . , xi,m) (1) s.t. p1xi,1 + · · · + pmxi,m = ei xi,j ≥ 0, ∀ 1 ≤ j ≤ msThe Fisher equilibrium problem is to find a price vector P ... |