Evolution paths on the equilibrium manifold

In a pure exchange smooth economy with fixed total resources, we define the length between two regular equilibria belonging to the equilibrium manifold as the number of intersection points of the evolution path connecting them with the set of critical equilibria. We show that there exists a minimal path according to this definition of length.     

General consumption constraints and regular economies

In the spirit of Smale’s work, we consider pure exchange economies with general consumption sets. In this paper, the consumption set of each household is described in terms of a function called possibility function. The main innovation comes from the dependency of each possibility function with respect to the individual endowments. We prove that, generically in the space of endowments and possibility functions, economies are regular. A regular economy has a finite number of equilibria, which locally depend on endowments and possibility functions in a continuous manner.     

On the structure of the set of fixed price equilibria

Working in the framework suggested by Drèze, this paper studies the number of fixed price equilibria and their continuity with respect to the price system. In an exchange economy, the concept of a rationing scheme is introduced, which specifies how shortages are shared among agents. For given utility functions and a given rationing scheme, under standard assumptions, an existence theorem is recalled, and it is shown that the graph of the equilibrium correspondence, when prices and initial endowments vary, is a piecewise continuously differentiable manifold. Moreover, generically, the number of equilibria for an economy, at given prices, is finite and the set of equilibria varies continuously with the price system and the initial endowments.     

Equilibria without the survival assumption

It is well known that an equilibrium in the Arrow–Debreu model may fail to exist if a very restrictive condition called the survival assumption is not satisfied. We study two approaches that allow for the relaxation of this condition. Danilov and Sotskov [Danilov, V.I., Sotskov, A.I., 1990. A generalized economic equilibrium. Journal of Mathematical Economics 19, 341–356], and Florig [Florig, M., 2001. Hierarchic competitive equilibria. Journal of Mathematical Economics 35, 515–546] developed a concept of a generalized equilibrium based on a notion of hierarchic prices. Marakulin [Marakulin, V., 1988. An equilibrium with nonstandard prices and its properties in mathematical models of economy. Discussion Paper No. 18. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 51 pp. (in Russian); Marakulin, V., 1990. Equilibrium with nonstandard prices in exchange economies. In: Quandt, R., Triska, D. (Eds.), Optimal Decisions in Market and Planned Economies. Westview Press, London, pp. 268–282] proposed a concept of an equilibrium with non-standard prices. In this paper, we establish the equivalence between non-standard and hierarchic equilibria. Furthermore, we show that for any specified system of dividends the set of such equilibria is generically finite. As a consequence, we have generic finiteness of Mas-Colell’s equilibria with slack, uniform dividend equilibria, and other special cases of our concept.     

Smooth preferences and the regularity of equilibria

To study equilibria we describe an economy by its distribution of consumers' preferences and endowments. All preferences are smooth and weakly convex. Demand of an economy need not be single valued, but there is an open dense set of economies for which demand is a C¹-function in a neighborhood of the equilibrium prices. We call an economy regular if its excess demand is transversal to zero. A regular economy has locally unique equilibria. It is shown that regular economies form an open dense set on which the equilibrium price correspondence varies continuously and the number of equilibria is locally constant.     

Utility perturbations and the equilibrium price set

For pure exchange economies in which agents are described by a compact smooth manifold of smooth strictly monotonic and strictly concave utilities, it is shown that, at least generically, the equilibrium price set is a smooth manifold of the same dimension. Given any smooth selection from the equilibrium price manifold and any sufficiently close smooth function, the function is a selection from the equilibrium price correspondence for some manifold of economies close to the original one. In particular, the set of equilibria corresponding to any open neighborhood of an economy contains an open subset of the price simplex.     

Public good provision and the Smith Process

An existing public good provision mechanism known as the Smith Process (SP) is extended to allow for non-zero fixed cost, non-constant marginal cost and imperfectly divisible output. Two versions of SP are considered: unrestricted (USP) and restricted with a unanimity rule (RSPU). USP implements efficient choice provided the gap between marginal and average cost is sufficiently low. RSPU relaxes the conditions for efficient implementation but increases the set of equilibria involving inefficient choice. Furthermore, if weakly dominated strategies are eliminated, then non-provision is no longer an equilibrium under RSPU but continues to be one under USP.     

A continuous location problem with different norms

This paper considers a set of problems of fixing the number and location of facilities to cover a given set of demand points, such that each demand point is served within a range of known distance standards, where each facility can be located on the entire plane and the distance can be either measured in rectilinear, Euclidean or Tchebycheff units. It is shown that an optimum solution can always be found on a small finite set of points. Thus the optimum itself can be found by either complete enumeration of these points or by zero one integer linear programming techniques. Furthermore, it is shown that the least and the largest number of facilities are required for the rectilinear and the Tchebycheff cases respectively.     

Extension of a theorem of Smale on equilibria for pure exchange economies

The methods used by Smale in his paper Global Analysis and Economics IIa, seem to be heavily dependent on the hypothesis that utility functions have no critical points. In this paper we prove that the no critical point hypothesis can be dropped and the class of regular economies, for which the equilibria are locally continuous functions of the parameters of the economy, can be extended.     

Myopia and monetary equilibria

In an infinitely lived, representative individual economy, important properties of competitive equilibria, such as determinacy and the non-existence of monetary equilibria, are not robust to the introduction of myopia. An individual is myopic if, at each date, he plans consumption only for that date and few periods that immediately follow; that is, his planning horizon, n, is finite. Equilibria with myopia can display real indeterminacy and allow for monetary as well as non-monetary steady states; thus, they share some of the features of equilibria in economies of overlapping generation. The equilibrium price dynamics (but not the consumption dynamics) of an exchange economy with extreme myopia, n = 1, are identical to the dynamics of an overlapping generation economy with two-period lives.     

The graph of Lindahl correspondence as the unique von Neumann-Morgenstern abstract stable set

In this paper, we consider a continuum economy with a finite number of types of agent, and several private and public goods. The main result of the paper is that the graph of the equal-treatment Lindahl equilibria mapping is the unique abstract stable set with respect to the dominance relation in economies with crowding effects introduced by Vasil'ev et al. in 1995. The external stability of this mapping implies that, for any equal-treatment allocation x in , that is not a Lindahl equilibrium, there exists a subeconomy of such that one of its equal-treatment Lindahl allocations blocks x. This result is a counterpart of the theorem of Mas-Colell for Aumann's atomless market with private goods.     

Sequential regularity in smooth production economies

In general equilibrium models where agents trade sequentially in multiple periods, the equilibria that endogenously occur in later periods can be robustly indeterminate if production sets are not everywhere differentiable (as. for example, with linear activities). The present paper proves that if technology is smooth, then equilibria are sequentially regular. That is. the equilibria of the endogenously generated economies occurring in later periods that confirm the unanimous expectations formed by agents in earlier periods are regular (and thus, for example, isolated from other equilibria). The paper also proves and utilizes the standard result that the overall intertemporal equilibria are regular.     

A characterization of equilibria in the Groves–Ledyard mechanism

In this paper, we characterize all interior and boundary equilibria of the Groves–Ledyard mechanism for a large class of economies and determine their stability properties. We show that the mechanism admits three types of equilibria: a symmetric, efficient, stable interior equilibrium, a large set of asymmetric, efficient, unstable, interior equilibria, and a large set of asymmetric, inefficient, stable boundary equilibria. We further show that asymmetric equilibria fail to exist for large values of the punishment parameter or if the message space is bounded sufficiently. The boundary equilibria previously had not been located nor had the instability of the asymmetric equilibria been known. Interestingly, the stability of the symmetric equilibrium rests on two dynamics that individually produce instability.     

Private ownership economies with a finite number of equilibria

Interest in economies with a finite number of equilibria has been first drawn over by Debreu (1970). The present paper shows that such economies form a subset of dense interior of the set of private ownership economies and proves some continuity properties of the Walras correspondence. It generalizes similar results obtained for exchange economies by Dierker and Dierker (1972). Demonstrations also rest heavily on transversality theory but use somewhat stronger theorems.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.     

Equilibria and core of large economies

In this paper we apply results on regular economies to study equilibria and core in a non-differentiable framework. We show that the distributions of agents' characteristics of regular economies form a dense subset of all distributions of agents' characteristics. Therefore ‘most’ economies have equilibria which are contained in finitely many ?-balls. And the core of ‘most’ sufficiently large economies is contained in finitely many ?-balls centered at equilibrium allocations of these economies.     

Cost sharing: efficiency and implementation

We study environments where a production process is jointly shared by a finite group of agents. The social decision involves the determination of input contribution and output distribution. We define a competitive solution when there is decreasing-returns-to-scale which leads to a Pareto optimal outcome. Since there is a finite number of agents, the competitive solution is prone to manipulation. We construct a mechanism for which the set of Nash equilibria coincides with the set of competitive solution outcomes. We define a marginal-cost-pricing equilibrium (MCPE) solution for environments with increasing returns to scale. These solutions are Pareto optimal under certain conditions. We construct another mechanism that realizes the MCPE.     

Large but finite games with asymmetric information

Carmona considered an increasing sequence of finite games in each of which players are characterized by payoff functions that are restricted to vary within a uniformly equicontinuous set and choose their strategies from a common compact metric strategy set. Then Carmona proved that each finite game in an upper tail of such a sequence admits an approximate Nash equilibrium in pure strategies.