A theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem

Summary This paper provides simple proofs of a theorem on open coverings of a simplex and Scarf's core existence theorem through Brouwer's fixed point theorem.I want to thank Daniel Abrams, Yakar Kannai, Herbert Scarf, Nicholas Yannelis, Jingang Zhao, and an anonymous referee for their helpful comments. I also gratefully acknowledge the financial support from the Alfred P. Sloan Foundation.     

A simple proof of K-K-M-S theorem

Summary We give a simple proof of the K-K-M-S theorem based on the Kakutani fixed point theorem, the separation theorem for convex sets and the Berge maximum theorem.The author is grateful to Gerard Debreu for his advice and to Tatsuro Ichiishi, Michael Todd, Rajiv Vohra, Nicholas Yannelis, Lin Zhou and two anonymous referees for their comments. The original version of this note was written in the fall 1992 while the author was visiting the Department of Economics, University of California at Berkeley.     

On a covering theorem

Summary. A simple proof of Reny and Wooders' recent strengthening of Shapley's extension of the Knaster-Kuratowski-Mazurkiewicz lemma on a closed cover of a simplex is given. The proof uses Ky Fan's coincidence theorem. Received: May 26, 2000; revised version: January 22, 2001     

On a theorem of Gorman

This letter reconsiders Gorman's conditions for integration of a system of exactly aggregatable demand functions and shows that the system can be integrated using some standard results in Lie group theory.     

Characterization of the extreme core allocations of the assignment game

A characterization of the extreme core allocations of the assignment game is given in terms of the reduced marginal worth vectors. For each ordering in the player set, a payoff vector is defined where each player receives his or her marginal contribution to a certain reduced game played by his or her predecessors. This set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is proved to be the set of extreme points of the core of the assignment game. Therefore, although assignment games are hardly ever convex, the same characterization of extreme core allocations is valid for convex games.     

Comparative statics in the large based on a fixed point algorithm

The purpose of this paper is to present a theorem on comparative statics in the large by making use of theorems on a fixed point algorithm, and also to show its economic applications focused on Hicksian laws. We provide a unified approach to comparative statics in the large, and obtain laws of demand globally valid for generalized gross-substitute systems.     

When is the core equivalence theorem valid?

Summary An answer to the question of the title is provided. It is shown that under some conditions the core equivalence theorem holds true (if and) only if the price space is a vector lattice.Research of both authors was supported in part by a Chrysler Corporation grant to IUPUI.     

On a master-slave Bertrand game model

A master-slave Bertrand game model is proposed for upstream and downstream monopolies owned by different parties, in which the upstream monopolist's output is used as the main factor of production by the downstream monopolist who is a small purchaser of the upstream monopolist's output. The bifurcation of the Bertrand-Nash equilibrium is analyzed with Schwarzian derivative. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents (LLEs), bifurcation, time series diagrams and phase portraits. With the modified straight-line stabilization method, chaos control is used to improve the aggregate profits of the two oligopolists. Lastly the welfare impacts of price fluctuations and chaos controls are briefly discussed.     

The core of an m-sided assignment game

A generalization of Shapley-Shubik's (1972, Int. J. Game Theory, 1, 111–130) housing market in which there are m types of agents instead of two is considered. These games can have empty cores. A subclass of such games with nonempty cores is presented.     

A limit theorem for an approximate core of a large but finite economy

We present a formulation of an approximate core that can be sustained as an approximate equilibrium for a ‘large enough’ finite exchange economy whose traders need not have transitive, continuous or convex preferences.     

Inessentiality of large groups and the approximate core property: an equivalence theorem

Summary Inessentiality of large groups or, in other words, effectiveness of small groups, means that almost all gains to group formation can be realized by partitions of the players into groups bounded in absolute size. The approximate core property is that all sufficiently large games have nonempty approximate cores. I consider these properties in a framework of games in characteristic function form satisfying a mild boundedness condition where, when the games have many players, most players have many substitutes. I show that large (finite) games satisfy inessentially of large groupsif and only if they satisfy the approximate core property.This paper focuses on a part of another paper, Inessentiality of Large Coalitions and the Approximate Core Property; Two Equivalence Theorems, previously circulated as a University of Bonn Sonderforschungsbereich Discussion Paper.The author is indebted to Roger Myerson for a very stimulating comment and the term inessentiality of large coalitions. She is also indebted to Robert Anderson, Sergiu Hart, and Aldo Rustichini for helpful conversations. She is especially indebted to Robert Aumann for many stimulating and helpful discussions on research leading to this paper. The author gratefully acknowledges the financial support of the SSHRC and the hospitality and support of the University of Bonn through Sonderforschungsbereich 303.     

On core stability, vital coalitions, and extendability

If a TU game is extendable, then its core is a stable set. However, there are many TU games with a stable core that are not extendable. A coalition is vital if there exists some core element x such that none of the proper subcoalitions is effective for x. It is exact if it is effective for some core element. If all coalitions that are vital and exact are extendable, then the game has a stable core. It is shown that the contrary is also valid for matching games, for simple flow games, and for minimum coloring games.     

The limit theorem on the core of a production economy in vector lattices with unordered preferences

Summary. We prove Aliprantis, Brown, and Burkinshaw's (1987) theorem on the equivalence of Edgeworth production equilibria and pseudo-equilibria in a more general setting. We consider production economies with unordered preferences and general consumption sets in a vector lattice commodity space. We adapt the approach of Mas-Colell and Richard (1991) and prove our theorem by applying a separating hyperplane argument in the space of all allocations. We also generalize Podczeck's (1996) important result on the relationship between continuous and discontinuous equilibrium prices to the case of production. Received: April 18, 1997; revised version: February 6, 1998     

A Lagrangean approach to the limit theorem on the core of an economy

Conclusion It is shown that an allocation which remains in the core under replication gives rise to an utility allocation with the following property: The utility level of no class can be increased without decreasing the utility level of at least one other class even if the number of consumers is allowed to vary continuously. An examination of the first order conditions the corresponding maximum problem with constraints leads to the conclusion that the allocation can be sustained by a price vector as a competitive equilibrium. This is the limit theorem on the core, proved in this paper with a simple Lagrangean technique.The notion of Pareto efficiency with a variable number of consumers proves useful for other purposes as well. For an extensive discussion of that point the reader is referred to Schweizer (1981).Ours, of course, is not the first calculus approach to the limit theorem [see e. g. L. Johansen (1978)]. But, as far as I know, the method of differentiating the Lagrangean simply with respect to the number of consumers is novel and leads to a straightforward proof of the theorem which can be reproduced in class-rooms even at the undergraduate level.The author wants to thank Joseph Greenberg and Werner Hildenbrand as well as an anonymous referee for helpful comments.     

应用系统理论模型分析社会管理的均衡点——以广西贺州为例

许多发达的国家和地区运用系统理论及系统模型方法构造出经济、科技、社会协调发展模式,并设计出相应的战略对策,在实际应用中收到满意的经济效益、社会效益,甚至是政治效益。应用系统理论及其系统模型方法分析研究社会管理的综合均衡问题,是一个颇具价值的课题。     

A bibliographical note on a theorem of Hardy,Littlewood, and Polya

A well-known theorem of Hardy Littlewood and Polya together with its more recent generalizations is stated here. Some of these results were applied recently in the economic literature. The economic interpretation of the mathematical theorems is mentioned and references are included.