Equilibrium in abstract economies without ordered preferences and with a measure space of agents

We report a generalization of Aumann's (1966) existence theorem to economies without ordered preferences and with externalities in consumption. Our work can alternatively be viewed as a generalization of the Shafer–Sonnenschein (1975) theorem to economies with a continuum of agents.     

On the existence of equilibrium in social systems with coordination

Social systems with coordination were introduced by Vind (1983) as a general model of economic institutions. The present paper gives a general existence theorem for equilibria in social systems with coordination extending Vind's result.     

An existence theorem of intertemporal recursive utility in the presence of Lévy jumps

This paper presents an existence theorem for a class of backward stochastic integral equations. The main contribution is a generalization of Duffie and Epstein's [Duffie, D., Epstein, L., 1992. Stochastic differential utility, (Appendix C with Skiadas C.), Econometrica 60, 353–394.] existence theorem of intertemporal recursive utility to allow the information structure to be driven by a Lévy jump process. The existence theorem applies also for a more general class of utility functions, such as recursive utility with habit-formation, and can be used to prove the existence of an equilibrium asset price process as a unique solution to the stochastic Euler equation derived by Ma [Ma, C., 1993b. Valuation of Derivative Securities with Mixed Poisson–Brownian Information and Recursive Utility, McGill University, mimeo.].     

Equilibrium in abstract economies without ordered preferences

We extend a theorem of G. Debreu on the existence of equilibrium in a generalized N-person game (‘A Social Equilibrium Existence Theorem’). Applying standard techniques the result can be used to prove the existence of equilibrium in economies with interdependent preferences, price-dependent preferences and preferences which may be both non-transitive and non-complete. This generalizes a recent theorem of A. Mas-Colell.     

Equilibrium with coordination

The obvious equilibrium concepts in the simplest institutions for transferring ownership of commodities—bilateral exchange—are neither Nash equilibria nor cooperative equilibria. To study such equilibria as special cases of equilibria of a social system it is necessary to introduce coordination. Two or more agents coordinate their actions, if, when they consider an alternative to a state, they take as given—for agents with whom they coordinate—the alternative state. If there is no coordination we obtain Nash equilibrium as a special case. If there is complete coordination we obtain optimality as a special case. The main result is an existence theorem for a social system with coordination. This theorem is then applied to prove existence of exchange equilibria in an economy with bilateral exchange.     

Nonlinear generalization of the Frobenius theorem: A symmetric approach

A further generalization of the Samuelson-Solow's results for the Frobenius theorem is presented. We ask the existence of a semi-positive vector x with a non-negative scalar λ for an equation F(x;λ)=0, where F is a vector function from Rⁿ into itself with the parameter value λ. The nature of our extension as well as economic interpretation of our assumptions is also discussed in the paper.     

Pure and randomized equilibria in the stochastic von Neumann–Gale model

The paper examines the problem of the existence of equilibrium for the stochastic analogue of the von Neumann–Gale model of economic growth. The mathematical framework of the model is a theory of set-valued random dynamical systems defined by positive stochastic operators with certain properties of convexity and homogeneity. Existence theorems for equilibria in such systems may be regarded as generalizations of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The known results of this kind are obtained under rather restrictive assumptions. We show that these assumptions can be substantially relaxed if one allows for randomization. The main result of the paper is an existence theorem for randomized equilibria. Some special cases (models defined by positive matrices) are considered in which the existence of pure equilibria can be established.     

Existence of maximal elements and equilibria in linear topological spaces

We present some mathematical theorems which are used to generalize previous results on the existence of maximal elements and of equilibrium. Our main theorem in this paper is a new existence proof for an equilibrium in an abstract economy, which is closely related to a previous result of Borglin–Keiding, and Shafer–Sonneschein, but allows for an infinite number of commodities and a countably infinite number of agents.     

On disequilibrium adjustment processes

Here, we study a class of disequilibrium processes defined by differential inequalities, where prices are adjusted according to the agent's short-term demands, in both pure exchange and monetary settings. We establish convergence theorem and accessibility theorem without convexity hypotheses on the utility functions.     

Anonymous sequential games with aggregate uncertainty

In this paper we extend a result of Jovanovic and Rosenthal (Anonymous sequential games, Journal of Mathematical Economics 17, 1988) on the existence of equilibrium in Anonymous Sequential Games. Jovanovic and Rosenthal prove existence in the case where the aggregate distribution of agents' characteristics evolves nonstochastically - the case of ‘no aggregate uncertainty’. Here we show how aggregate uncertainty can be introduced into the model (so the aggregate distribution evolves stochastically) and extend the Jovanovic-Rosenthal existence theorem to this case.     

Duality and maximum principle in multi-period convex programming

Starting from works in Rockafellar (1967), a duality theorem, existence theorem, reciprocity principle and a max–min principle are obtained in a class of dynamic programming which has a possibility of wide economic applications. These theorems serve to summarize the theory of the efficient price vectors developed in activity analysis and to provide a linkage between that theory and the maximum principle in the optimal control theory.     

Inconsequential arbitrage

We introduce the concept of inconsequential arbitrage and, in the context of a model allowing short-sales and half-lines in indifference surfaces, prove that inconsequential arbitrage is sufficient for existence of equilibrium. Moreover, with a slightly stronger condition of nonsatiation than that required for existence of equilibrium and with a mild uniformity condition on arbitrage opportunities, we show that inconsequential arbitrage, the existence of a Pareto optimal allocation, and compactness of the set of utility possibilities are equivalent. Thus, when all equilibria are Pareto optimal — for example, when local nonsatiation holds — inconsequential arbitrage is necessary and sufficient for existence of an equilibrium. By further strengthening our nonsatiation condition, we obtain a second welfare theorem for exchange economies allowing short sales.Finally, we compare inconsequential arbitrage to the conditions limiting arbitrage of Hart [Hart, O.D., 1974. J. Econ. Theory 9, 293–311], Werner [Werner, J., 1987. Econometrica 55, abs1403–1418], Dana et al. [Dana, R.A., Le Van, C., Magnien, F., 1999. J. Econ. Theory 87, 169–193] and Allouch [Allouch, N., 1999. Equilibrium and no market arbitrage. CERMSEM, Universite de Paris I]. For example, we show that the condition of Hart (translated to a general equilibrium setting) and the condition of werner are equivalent. We then show that the Hart/Werner conditions imply inconsequential arbitrage. To highlight the extent to which we extend Hart and Werner, we construct an example of an exchange economy in which inconsequential arbitrage holds (and is necessary and sufficient for existence), while the Hart/Werner conditions do not hold.     

Bayesian incentive compatible beliefs

The problem of incentives for correct revelation is studied as a game with incomplete information where players have individual beliefs concerning other's types. General conditions on the beliefs are given which are shown to be sufficient for the existence of a Pareto-efficient mechanism for which truth-telling is a Bayesian equilibrium.     

On the structure of the equilibrium manifold

An elementary proof is given of Y. Balasko's theorem that the equilibrium manifold is diffeomorphic to R^lm. Our proof includes the case (l,m)=(2,2), which the original proof does not cover.     

Tiebout's tale in spatial economies: Entrepreneurship, self-selection, and efficiency

This paper establishes the existence and efficiency of equilibrium in a local public goods economy with spatial structures by formalizing Hamilton's [Hamilton, B.W., 1975. Zoning and property taxation in a system of local governments Urban Studies 12, 205–211] elaboration of Tiebout's [Tiebout, C., 1956. A pure theory of local public expenditures. Journal of Political Economy 64, 416–424] tale. We use a well-known equilibrium concept from Rothschild and Stiglitz [Rothschild, M., Stiglitz, J.E., 1976. Equilibrium in competitive insurance markets: an essay on the economics of imperfect information. Quarterly Journal of Economics 40, 629–649] in a market with asymmetric information, and show that Hamilton's zoning policy plays an essential role in proving the existence and efficiency of equilibrium. We use an idealized large economy following Ellickson, Grodal, Scotchmer and Zame [Ellickson, B., Grodal, B., Scotchmer, S., Zame, W.R., 1999. Clubs and the market, Econometrica 67, 1185–1217] and Allouch, Conley and Wooders [Allouch, N., Conley, J.P., Wooders, M.H., The Tiebout Hypothesis: On the Existence of Pareto Efficient Competitive Equilibria, (2004), mimeograph]. Our theorem is directly applicable to the existence and efficiency of a discrete spatial approximation of mono- or multi-centric city equilibria in an urban economy with commuting time costs, even if we allow the existence of multiple qualities of (collective) residences, when externalities due to traffic congestion are not present.     

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.     

On cores and indivisibility

An economic model of trading in commodities that are inherently indivisible, like houses, is investigated from a game-theoretic point of view. The concepts of balanced game and core are developed, and a general theorem of Scarf's is applied to prove that the market in question has a nonempty core, that is, at least one outcome that no subset of traders can improve upon. A number of examples are discussed, and the final section reviews a series of other models involving indivisible commodities, with references to the literature.     

Existence of equilibria in atomless production economies with price dependent preferences

The existence of a competitive equilibrium for an economy with a measure space of agents is proved by reducing the economy to a three-person game and then applying Debreu's lemma (1952). In addition to the relative simplicity and shortness of the proof, we generalize known results [Aumann (1966), Hildenbrand (1970), Schmeidler (1969)] by allowing both price dependent preferences and satiation.     

Arbitrage, equilibrium, and gains from trade: A counterexample

We present a counterexample to a theorem due to Chichilnisky (Bulletin of the American Mathematical Society, 1993, 29, 189–207; American Economic Review, 1994, 84, 427–434). Chichilnisky's theorem states that her condition of limited arbitrage is necessary and sufficient for the existence of an equilibrium in an economy with unbounded short sales. Our counterexample shows that the condition defined by Chichilnisky is not sufficient for existence of equilibrium. We also discuss difficulties in Chichilnisky (Economic Theory, 1995, 5, 79–107).     

An existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy

We establish an existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy. Instead of the traditional approach which depends on Kakutani’s fixed point theorem, we employ the theories of aggregative games and best reply potential games. We show that, if there exists a representative consumer, under some conditions on preferences and production technologies, the profit maximization game is a (pseudo) best reply potential game. Hence, the existence of the equilibria is proved independently of the well known convex-valued assumption on the best responses. Although our assumptions result in the additive separability on a utility function of a representative consumer, the existence of increasing returns and indivisible productions can be allowed. In our model, it is shown that the game played by firms exhibits strategic substitutes whether the products of firms are substitutes or complements, and this plays an important role for the existence of the equilibria.