Finite Exchange Economy Research Articles

Balanced contributions and fairness in exchange economies

This paper presents a new way to fairly distribute welfare gains derived from trade in exchange economies. The imposed fairness condition is based on the balanced marginal contributions condition satisfied by the Shapley value in transferable utility games. The solution is defined for cardinal finite exchange economies. Then, we point out that solutions for ordinal economies can be created by mapping finite ordinal exchange economies onto cardinal economies and then applying the original solution.

Indivisible commodities and the nonemptiness of the weak core

We consider a sufficient condition for the nonemptiness of the weak core in a finite exchange economy where every commodity is available only in integer quantities. We show that if the aggregate upper contour set is discretely convex, then the weak core is nonempty. In addition, we give two sufficient conditions for the aggregate upper contour set to be discretely convex. One is that every upper contour set of every agent is M ♮ -convex. The other is that the number of commodities is two and every agent’s preference relation is weakly monotone and discretely convex.

A COUNTABLE ECONOMY: AN EXAMPLE

Weiss (1981) established core equivalence and the existence of competitive equilibria in finitely additive exchange economies. To underline the relevance of finitely additive economies we present in this note an example with a close connection to finite exchange economies.

Excess price volatility and financial innovation

In a three-period finite exchange economy with incomplete financial markets and retrading, we study the effects of the degree of incompleteness and of changes in the financial structure on asset price volatility. In what are essentially no aggregate risk economies, asset price volatility is a sunspot-like phenomenon. If markets are completed by financial innovation, asset price volatility reduction is generic. With aggregate risk, changes in the financial structure affect asset price volatility through a pecuniary externality. Financial innovation which decreases equilibrium price volatility can be crafted under conditions of sufficient market incompleteness. Numerical examples illustrate the role of risk aversion for volatility changes and show that, with or without aggregate risk, reducing the degree of incompleteness per se is not necessarily associated with a volatility reduction.

Scalarization techniques or relationship between a social welfare function and a Pareto optimality concept

This paper discusses some scalarization techniques and one application of the multi-objective optimization problem into a mathematical economics. A function of social welfare and a concept of Pareto optimality in a finite pure exchange economy are considered. Here it is proved that a set of Pareto optimality allocations is path-connected and uncountable, if the utility functions of economical agents are monotone, concave and strictly quasi-concave. In the end, it is proved that there exist non-negative weights and Pareto optimality allocations such that the social welfare function has maximum not proving continuity of this function.

Structure of the Pareto optimality set with fixed total resources and consumption sets

In present paper we study a Pareto optimality set in finite pure exchange economies with fixed total resources and consumption sets. It is proved that a Pareto optimality set is nonempty, path-connected and compact, if the utility functions of the economical agents are continuous, monotone, concave and strictly quasi-concave.

An elementary core equivalence theorem¶in a countable economy

We consider an infinite exchange economy with countably many traders, which can be regarded as a natural extension of finite exchange economies to an infinite one. In our countable economy the core defined in the traditional manner would be empty. To avoid this unwanted situation we have to strengthen the notion of “improves upon”. We will achieve this based on the idea that forming coalitions involve costs.

Financial Innovation and Price Volatility

In a three-period finite competitive exchange economy with incomplete financial markets and retrading, we show the generic existence of financial innovation which decreases equilibrium price volatility (as well as innovation which incresases it). The existence is obtained under conditions of sufficient market incompleteness. The financial innnovation may consist of an asset which is only traded at time zero, or retreded, and with payoffs only at the terminal date. The existence is shown to be robust in the asset payoff space.

Acceptable and Walrasian allocations

The paper provides an axiomatic approach to mechanisms which determine the individuals' `contributions' to the total welfare of society, for every finite pure exchange economy and for every allocation of the total resources. An efficient allocation is acceptable if it induces individual contributions that are not lower than the respective contributions with the initial allocation. We characterize the mechanism that satisfies our axioms, and in addition guarantees the existence of an acceptable allocation for every economy. We then show that the set of acceptable allocations (with respect to this mechanism) coincides with the set of Walrasian allocations with respect to the initial allocation.

Nonconvergence of the Mas-Colell and Zhou Bargaining Sets

In a nontransferable utility exchange economy with a continuum of agents, the Mas-Collel bargaining set coincides with the set of Walrasian equilibria. In this paper, we show that the Mas-Colell bargaining set, as well as a smaller bargaining set due to Zhou, may fail to converge to competitive outcomes in large finite NTU exchange economies.

A note on the finiteness of the set of equilibria in an exchange economy with constrained endowments

Debreu, G. (1970), proved that for smooth finite exchange economies, the set of endowment assignments which result in a finite number of price equilibria is “large” in the sense that it has null closed complement. We show that if the individual endowment space of each agent does not include all the commodities, then this conclusion need not hold. In particular, we consider the case where each agent's endowment space includes only one good. We construct a class of examples in which every economy in an open subset of the constrained endowment space gives rise to a continuum of equilibria.

Interpretation of Aubin's fuzzy coalitions and their extension: Relaxation of finite exchange economics

In this article a new interpretation of Aubin's fuzzy coalitions is given. This interpretation is based on associating each finite pure exchange economy with some large nonatomic economy. Then some connections between fuzzy core of finite economy and the core of associated nonatomic economy are established. From these results it is derived that for the case of convex preferences an allocation belonging to the fuzzy core is an equilibrium allocation. The assertion is not true in nonconvex case. We proceed by using the idea of J.P. Aubin — amplifying the class of coalitions in such a manner that an allocation unblocked by them is an equilibrium allocation also in the nonconvex case.

An Elementary Approach to Approximate Equilibria with Infinitely Many Commodities

We prove the existence of approximate equilibria in a finite exchange economy with a countably infinite number of commodities and nonconvex preferences, when every trader has an excess demand set that is finitely spannable, i.e., that could be covered by a union of its convex subsets in finitely many steps. We show that the bound on the norm of the per capita aggregate excess demand is reciprocally related to the square root of the population. Extensions are also made to the case where countably many commodities are indivisible. The proofs are elementary. Journal of Economic Literature Classification Numbers: D50, C62, D52.

A strategic market game of a finite exchange economy with a mutual bank

We introduce a strategic market game for an exchange economy not having enough commodity money. We show the existence of a non-cooperative equilibrium for any finite replication economy with a mutual bank, we then show that efficient trade can be achieved in the limiting economy by expanding the money supply through the use of fractional reserves, where the commodity money is demonetized and used for reserves. The means of exchange becomes bank credit backed, in part, by ‘gold’. However, efficiency cannot be achieved in general as a non-cooperative equilibrium of a finite player game.

Existence and fairness of value allocation without convex preferences

Shafer ( Econometrica, 48 (1980), 467–476) proved that in a finite exchange economy value allocations exist, provided that each agent has convex, complete, transitive, compact and monotone preferences. However, if preferences are not convex, then value allocations may not exist. To remedy this difficulty we enlarge the set of value allocations by introducing the concept of approximate value allocations, and show that in a finite exchange economy approximate value allocations exist, even if preferences are not convex, or compact, or monotone. This value existence result can be used to provide a very general value existence theorem for a sequence of finite economies. Further, we show that value allocations may discriminate in favor of or against a coalition of agents.

Pareto-optimal Nash equilibria are competitive in a repeated economy

Consider a finite exchange economy first as a static, 1 period, economy and then as a repeated economy over T periods when the utility of each agent is the mean utility over T. A family of strategic games is defined via a set of six general properties the most distinct of which is the ability of agents to move commodities forward in time. Now consider Pareto optimal allocations in the T period economy which are also Nash equilibria in this family of strategic games. We prove that as T becomes large this set converges to the set of competitive utility allocations in the one period economy. The key idea is that a repetition of the economy when agents can move commodities forward in the time acts as a convexification of the set of individually feasible outcomes for player i holding all other strategies fixed.

Approximate equilibria in markets with indivisible commodities

This paper presents results on the existence of approximate equilibria in large but finite exchange economies in which all the traders have non-convex consumption sets. As such, the paper gives a natural extension, to the finite economy, of Mas-Colell's results on existence with indivisible commodities and a continuum of agents. It also presents an approximate equilibrium theorem without any assumption of compactness or bounded non-convexity of preferences.

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.

An admissible set occurring in various bargaining situations

Given a set of alternatives S and a binary relation M on S the admissible set of the pair ( S, M) is defined to be the set of maximal elements with respect to the transitive closure of M. It is shown that existing solutions in game theory and mathematical economics are special cases of this concept (they are admissible sets of a natural S and M). These include the core of an n-person cooperative game, Nash equilibria of a noncooperative game, and the max-min solution of a two-person zero sum game. The competitive equilibrium prices of a finite exchange economy are always contained in its admissible set. Special general properties of the admissible set are discussed. These include existence, stability, and a stochastic dynamic process which leads to outcomes in the admissible set with high probability.

Existence of Approximate Equilibria and Cores

IT IS WELL KNOWN that for a finite exchange economy, where preferences are not assumed to be convex, there may be no price or even the core may be empty. For this reason it was proposed to enlarge the set of price and the core by introducing the concepts of equilibrium and core. The existence of approximate for exchange economies, where preferences are not assumed to be convex, has been investigated by R. Starr [6]. He showed that there exists a quasi-equilibrium, provided the number of participants is large enough and there is a bound on the degree of non-convexity [6, p. 30, Assumption D]. In this note we shall show the existence of equilibria (a stronger concept than the one considered by Starr [6, p. 31]) for large economies where the preferences are neither assumed to be convex nor complete. To obtain our result we shall assume that the preferences and the endowments of all participating agents belong to a compact set. In [5] Shapley and Shubik proved that, for a large replica of a given economy with transferable utility, the e-core is nonempty. We shall generalize this result to large economies without transferable utility by using the concept of ?-core as introduced by Kannai [2]. The nonemptiness of the e-core follows easily from the existence of approximate and a relationship between the set of approximate and e-core. The existence of c-core for large economies (with a fixed number of types) can also be deduced from Kannai's Theorem C' [2] in its stronger form (Theorem C in [3]).