Existence Of Equilibrium For Economy Research Articles

Vector Equilibrium Problems—A Unified Approach and Applications

We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some “interiority” properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given.

A new approach to the existence and regularity of linear equilibrium in a noisy rational expectations economy

This paper gives a new approach to show the existence and regularity of linear equilibrium established by Lou ⓡ al. (2019) for a noisy rational expectations economy. Different from the existing method which essentially requires to find a fixed point of a system of nonlinear algebraic equations, the new approach is operated directly on an alternative form of market-clearing conditions. One main advantage of the new approach is that besides homogeneous-valuation economies, it can also handle the existence of equilibrium in economies with heterogeneous valuations where the existing method for dealing with homogeneous-valuation economies fails to work.

Tropical convexity in Riesz spaces

Various properties of tropical convex sets of the positive cone of a Riesz space are established. Under suitable, but standard assumptions, we show that they have fixed points properties, for single valued and multivalued maps, analogous to those associated to the usual convex sets in topological vector spaces. Selection theorems and extension theorems similar to those of Michael and Dugundji also hold and, in the appropriate setting, tropical convex sets are absolute extensors for the class of metric spaces and absolute retracts. We also give some applications to standard problems of mathematical economics: Nash equilibria, existence of elements for non transitive preference relations and existence of equilibria for abstract economies in Riesz spaces under tropical convexity assumptions. This work can be conceived as a contribution to infinite dimensional idempotent analysis.

Existence of General Competitive Equilibria: A Variational Approach

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

Existence of Equilibrium in Economies with Indivisible Commodities and Savings

We consider a two-period exchange economy without uncertainty. Every commodity can be consumed only in integer amounts and there exists one financial (nominal) asset whose trading unit is divisible. Holding the asset does not directly affect agents' preferences. We prove the existence of an equilibrium if there exists a continuum of agents and if the distribution of agents' initial holdings of the divisible asset is dispersed in the sense that the set of agents who initially have the same amount of the asset is negligible relative to the size of the entire economy. Since agents cannot run out of their wealths because of the indivisibilities, our equilibrium concept permits the presence of some commodities in excess supply. This causes the inefficiency of equilibrium allocation. Also, our economy exhibits the real indeterminacy of equilibrium allocation. Equilibrium allocations depend on the price level of the second-date commodities.

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L ∞ ( M , M , μ ) , the space of all μ -essentially bounded M -measurable functions on M . As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough. Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.

Equilibria of noncompact abstract economies in general almost convex strategy spaces

We first extend the concept of almost convex condition and establish a fixed point theorem for correspondences with convex values only on an almost convex subset for their ranges. This generalizes both results of Himmelberg [C.J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207] and the result of Jafari and Sehgal [F. Jafari, V.M. Sehgal, An extension to a theorem of Himmelberg, J. Math. Anal. Appl. 327 (2007) 298–301]. Furthermore, applying it, we have existence theorems for equilibria of noncompact abstract economies in general almost convex strategy spaces. Our theorems generalize the corresponding results of Zhou [Jianxin Zhou, On the existence of equilibrium for abstract economies, J. Math. Anal. Appl. 193 (1995) 839–858], Tan and Wu [K.-K. Tan, Z. Wu, A note on abstract economies with upper semicontinuous correspondence, Appl. Math. Lett. 11 (5) (1998) 21–22] in several ways. In particular, we answer the question raised by Zhou in the reference cited above in the affirmative with weaker hypotheses.

Anonymous price taking equilibrium in Tiebout economies with a continuum of agents: Existence and characterization

We introduce a model of a local public goods economy with a continuum of agents and jurisdictions with finite but unbounded populations, where the set of possible projects for each jurisdiction/club is unrestricted in size. Under boundedness of per capita payoffs, which simply ensures that equal treatment payoffs are bounded above, we apply results of Kaneko and Wooders (1986) to obtain nonemptiness of the core of the economy. We then demonstrate, under the stronger condition of strict small group effectiveness, that the equal treatment core coincides with the set of price-taking equilibrium outcomes with anonymous prices—that is, prices for public goods depend only on observable characteristics of agents. Existence of equilibrium follows from nonemptiness of the core and equivalence of the core to the set of equilibrium outcomes. Our approach provides a new technique for showing existence of equilibrium in economies with a continuum of agents.

An answer to a question by Herings et al.

We answer an open question by Herings et al. [J.J. Herings, G. van der Laan, D. Talman, Z. Yang, A fixed point theorem for discontinuous functions, Operations Research Letters 36 (1) (2008) 89–93], by proving that their fixed point theorem for discontinuous functions works for mappings defined on convex compact subset of R n , and not only polytopes. This rests on a fixed point result of Toussaint [S. Toussaint, On the existence of equilibria in economies with infinitely many commodities and without ordered preferences, Journal of Economic Theory 33 (1) (1984) 98–115].

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514–540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039–1053].

Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and [formula omitted]-convexity

B -convexity was introduced in [W. Briec, C. Horvath, B -convexity, Optimization 53 (2004) 103–127]. Separation and Hahn–Banach like theorems can be found in [G. Adilov, A.M. Rubinov, B -convex sets and functions, Numer. Funct. Anal. Optim. 27 (2006) 237–257] and [W. Briec, C.D. Horvath, A. Rubinov, Separation in B -convexity, Pacific J. Optim. 1 (2005) 13–30]. We show here that all the basic results related to fixed point theorems are available in B -convexity. Ky Fan inequality, existence of Nash equilibria and existence of equilibria for abstract economies are established in the framework of B -convexity. Monotone analysis, or analysis on Maslov semimodules [V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and Its Applications, Math. Appl., vol. 401, Kluwer Academic, 1997; V.P. Litvinov, V.P. Maslov, G.B. Shpitz, Idempotent functional analysis: An algebraic approach, Math. Notes 69 (2001) 696–729; V.P. Maslov, S.N. Samborski (Eds.), Idempotent Analysis, Advances in Soviet Mathematics, Amer. Math. Soc., Providence, RI, 1992], is the natural framework for these results. From this point of view Max-Plus convexity and B -convexity are isomorphic Maslov semimodules structures over isomorphic semirings. Therefore all the results of this paper hold in the context of Max-Plus convexity.

More on equilibria in competitive markets with externalities and a continuum of agents

Equilibrium existence results are presented for competitive markets with externalities in price and consumption and a measure space of agents. These unify and extend [Balder, E.J., 2003. Existence of competitive equilibria in economies with a measure space of consumers and consumption externalities. Preprint, in press, electronically available at http://www.math.uu.nl/publications/preprints/1294.ps.gz; Balder, E.J., 2005. More about equilibrium distributions for competitive markets with externalities. Working paper, Department of Economics, University of Illinois.] and generalize the main existence results by Aumann [Aumann, R.J., 1964. Markets with a continuum of traders. Econometrica 32, 39–50], Schmeidler [Schmeidler, D., 1969. Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37, 578–585; Schmeidler, D., 1973. Equilibrium points of nonatomic games. Journal of Statististical Physics 7, 295–300.], Greenberg et al. [Greenberg, J., Shitovitz, B., Wieczorek, A., 1979. Existence of equilibria in atomless production economies with price dependent preferences. Journal of Mathematical Economics 6, 31–41.], Yamazaki [Yamazaki, A., 1978. An equilibrium existence theorem without convexity assumptions. Econometrica 46, 541–555.], Noguchi [Noguchi, M., 2005. Interdependent preferences with a continuum of agents. Working paper, Meijo University, 2001. Journal of Mathematical Economics 41, 665–686.], Cornet [Cornet, B., Topuzu, M., 2005. Existence of equilibria for economies with externalities and a measure space of consumers. Economic Theory 26, 397–421.], and Noguchi and Zame [Noguchi, M. and Zame, W.R., 2004. Equilibrium distributions with externalities. Preprint.].

Bewley's limiting approach to infinite dimensional economies with l.s.c. preferences

We examine the existence of quasi-equilibria in economies with a general topological commodity space and l.s.c. preferences by the limiting technique of Bewley [Bewley, T., 1972. Existence of equilibria in economies with infinitely many commodities. Journal of Economic Theory 4, 514–540]. However, there are two major difficulties in using Bewley's technique for such economies. To overcome these difficulties, we propose a class of new preferences, called bipolar preferences, and impose a strong l.s.c. assumption upon the economies. As a result, an existence theorem of quasi-equilibria in economies with a general topological commodity space and l.s.c. preferences is established.

Existence of equilibria for economies with externalities and a measure space of consumers

This paper considers an exchange economy with a measure space of agents and consumption externalities, which take into account two possible external effects on consumers’ preferences: dependence upon prices and dependence upon other agents’ consumption. We first consider a model with a general externality mapping and we then treat the particular case of reference coalition externalities, in which the preferences of each agent a are influenced by prices and by the global or the mean consumption of the agents in finitely many (exogenously given) reference coalitions associated with agent a. Our paper provides existence results of equilibria in both models when consumers have transitive preferences. It extends in exchange economies the standard results by Aumann [2], Schmeidler [16], Hildenbrand [12], and previous results by Greenberg et al. [11] for price dependent preferences, Schmeidler [17] for fixed reference coalitions and Noguchi [15] for a more particular concept of reference coalitions. We also mention related results obtained independently by Balder [4].

Existence of Equilibria in Economies with Bads

We consider an exchange economy in which there are infinitely many consumers and some commodities are bads, that is, cause disutility to consumers. We give an example of such an economy for which there is no competitive equilibrium or its variants (quasi- or pseudo-equilibrium), and an example of the failure of the so-called uniform integrability condition of equilibrium allocations of increasingly populous finite economies.

A note on equilibria for abstract economies

In this paper, we use new fixed-point results in Fréchet spaces to generalize previous results on the existence of equilibria for abstract economies.

The Riesz–Kantorovich formula and general equilibrium theory

Let L be an ordered topological vector space with topological dual L′ and order dual L ~. Also, let f and g be two order-bounded linear functionals on L for which the supremum f∨ g exists in L. We say that f∨ g satisfies the Riesz–Kantorovich formula if for any 0≤ ω∈ L we have f∨g(ω)= sup 0≤x≤ω [f(x)+g(ω−x)]. This is always the case when L is a vector lattice and more generally when L has the Riesz Decomposition Property and its cone is generating. The formula has appeared as the crucial step in many recent proofs of the existence of equilibrium in economies with infinite dimensional commodity spaces. It has also been interpreted by the authors in terms of the revenue function of a discriminatory price auction for commodity bundles and has been used to extend the existence of equilibrium results in models beyond the vector lattice settings. This paper addresses the following open mathematical question: We show that if f and g are continuous, then f∨ g must satisfy the Riesz–Kantorovich formula when L has an order unit and has weakly compact order intervals. If in addition L is locally convex, f∨ g exists in L ~ for any pair of continuous linear functionals f and g if and only if L has the Riesz Decomposition Property. In particular, if L ~ separates points in L and order intervals are σ( L, L ~)-compact, then the order dual L ~ is a vector lattice if and only if L has the Riesz Decomposition Property — that is, if and only if commodity bundles are perfectly divisible.

Existence of equilibrium for economies with infinitely many commodities and infinitely countable consumers

This paper examines the existence of equilibria for double infinite eonomies. S.F. Richard and S. Srivastava have established the existence of equilibria for economies with infinitely countable consumers when commodity space isL ∞ (M, M, μ). However, most Banach Lattices as commodity spaces haven’t interior points in their positive cones, so their result can’t be applied to many cases. We here consider a general Banach Lattice as commodity space and introduce a concept of equiproperness on preferences. Under the assumption the existence of equilibrium for economy is established. Finally, we examine the existence of equilibria for production economies.

Existence of equilibria in economies with increasing returns and infinitely many commodities

In this paper, we prove the existence of equilibria in a model with infinitely many commodities and where production sets exhibit increasing returns to scale or more general types of non-convexities. We distinguish two cases. In the first, producers follow loss free pricing rules like the average cost or the profit maximizing pricing rule. The second case is devoted to bounded loss pricing rules. In each case, we give an existence result under assumptions which extend those considered in the finite dimensional case. In particular, they are satisfied by a firm with a convex production set which maximizes its profits. We also give a new sufficient condition to have an economically meaningful equilibrium price.

Existence of Equilibria in Economies with Externalities and Nonconvexities

We consider a general equilibrium model of an economy in which the production possibilities, the consumption sets and the preferences of the consumers are represented by set-valued mappings which depend on the environment to take into account the possibility of external effect. In order to encompass all kinds of nonconvexities, we do not put any convexity assumption either on the graph of the set-valued mapping which describes the technological possibilities or on the production set for a given environment. The firms are instructed to set their prices according to general pricing rules which may depend on the production plans of other producers and on consumption plans. We report an existence result of general equilibria. As in the model without external effects, the key hypotheses are bounded loss and survival assumptions. Nevertheless, we also assume that the set-valued mappings which describe the fundamentals of the economy are lower semi-continuous and have a closed graph. Our framework is sufficiently large to generalize previous works on the existence of competitive equilibria with externalities when the firms have convex production sets and on the existence of equilibria with general pricing rule without externality.