Production equilibria in vector lattices

The general purpose of this paper is to prove quasiequilibrium existence theorems for production economies with general consumption sets in an infinite dimensional commodity space, without assuming any monotonicity of preferences or free-disposal in production. The commodity space is a vector lattice commodity space whose topological dual is a sublattice of its order dual. We formulate two kinds of properness concepts for agents' preferences and production sets, which reduce to more classical ones when the commodity space is locally convex and the consumption sets coincide with the positive cone. Assuming properness allows for extension theorems of quasiequilibrium prices obtained for the economy restricted to some order ideal of the commodity space. As an application, the existence of quasiequilibrium in the whole economy is proved without any assumption of monotonicity of preferences or free-disposal in production. Received: March 9, 1999; revised version: September 4, 2000     

Equilibrium in a reinsurance market with short sale constraints

Summary. This paper deals with the existence of equilibrium in a dynamic reinsurance market with short sale constraints, driven by a marked point process, as studied in Bernis and Jouini (2001). We use the set of reinsurance treaties as consumption set, which is the positive orthant of some Banach lattice that can be identified to a space of martingales, . The properness of preferences is a key assumption for us to prove the existence of an equilibrium. We provide a sufficient condition for the preferences to be proper in term of loading factor of the reinsurance premium. Received: June 15, 2000; revised version: May 17, 2001     

Equilibria in infinite-dimensional production economies with convexified coalitions

Summary. This paper deals with a private ownership production economy assuming that the commodity space is infinite-dimensional. It is first showed that the fuzzy core allocations, a concept that goes back to J.-P. Aubin, are in a one-to-one correspondence with certain core allocations of a continuum economy suitably defined. This result is obtained under convexity of preferences and production sets and separability of the commodity space. In the case of nonconvex preferences and production sets, the set of fuzzy coalitions can be enlarged in order to obtain that every allocation of the core accordingly defined is supported by a non zero price. The proof of the equivalence result when the positive cone of the commodity space has the empty interior, is obtained under assumptions of properness for preferences relations and production sets. Received: July 9, 1998; revised version: December 6, 1999     

Stochastic equilibria with incomplete financial markets

We demonstrate the existence of equilibria with incomplete financial markets for stochastic economies whose information structure is given by an event tree, restricting attention to purely financial securities, those paying in units of account (e.g., “dollars”). Financial markets may be incomplete: some consumption streams may be impossible to obtain by any trading strategy. Securities may be individually precluded from trade at arbitrary states and dates. Sufficient conditions for the existence of stochastic equilibria are: continuous, convex, strictly monotonic preferences and strictly positive aggregate endowments. These conditions are weakened. A corollary states that any regime of security prices precluding arbitrage can be embedded in an equilibrium.     

Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies

In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofré (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spacesThis work was partially supported by Nucleo Complex Engineering System. The successive versions of the paper were partly prepared during visits of Alejandro Jofré to CERMSEM and of Monique Florenzano and Pascal Gourdel to the Centro de Modelamiento Matematico. The hospitality of both institutions and the support of the french Coopération régionale Cone Sud are gratefully aknowledged. The authors thank Ali Khan for stimulating exchange of ideas and literature, Roko Aliprantis, Jean-Marc Bonnisseau, Alain Chateauneuf, Roger Guesnerie, Filipe Martins Da Rocha, Moncef Meddeb, B. Mordukovich, Lionel Thibault and Rabee Tourky for valuable discussions     

Equilibria in incomplete assets economies with infinite dimensional spot markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a finite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an “exceptional” set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium—in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point. The paper establishes that even with our very general commodity spaces there exists an equilibrium for a “very” dense set of assets structures. Our approach is in the main convex analytic and the results do not require that preferences be smooth or complete or transitive. The concepts and techniques studied in this paper have important finite as well as infinite dimensional applications. This paper has benefited from the comments of Martine Quinzii, Wayne Shafer, Manuel Santos and Yeneng Sun. The research of C. D. Aliprantis is supported by the NSF Grants SES-0128039, DMS-0437210, and ACI-0325846. The research of R. Tourky is funded by the Australian Research Council Grant A00103450.     

Optimal growth with many consumers

A method is described for constructing all Pareto-optimal allocations for a dynamic economy with many heterogeneous consumers, under certainty, in which both the technology and consumer preferences are recursive but preferences need not be additively separable over time. Optimal (perfect foresight competitive equilibrium) allocations are obtained through the study of a dynamic program. For an economy with one consumption good, sufficient conditions are given for the existence of a unique interior stationary distribution of consumption and wealth. For a two-person exchange economy, sufficient conditions are given for the global asymptotic stability of the unique interior stationary point.     

Decentralizing Edgeworth equilibria in economies with many commodities

Summary. This paper proves core-equivalence theorems for exchange economies without ordered preferences, defined on locally convex Riesz commodity spaces such that the price space is a lattice. Properness assumptions are borrowed from some recent equilibrium existence results. Received: January 15, 1998; revised version: August 19, 1998     

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     

Strategy-Proof Probabilistic Mechanisms in Economies with Pure Public Goods

Public good economies where agents are endowed with strictly convex continuous single-peaked preferences on a convex subset of Euclidean space are considered. Such an economy arises for instance in the classical problem of allocating a given budget to finance the provision of several public goods where the agents have monotonically increasing strictly convex continuous preferences. A probabilistic mechanism assigns a probability distribution over the feasible alternatives to any profile of reported preferences. The main result of the paper establishes that any strategy-proof (in the sense of A. Gibbard, Econometrica45 (1977), 665-681) and unanimous mechanism must be a random dictatorship. Journal of Economic Literature Classification Numbers: D70, D71, H40, C60.     

Fixed endpoint optimal control

Summary. Properness has been introduced in the expected utility framework and it recently has been transfered to mean-variance utility functions. Here, we show that properness implies the slope of the mean-standard deviation indifference curve being convex in the standard deviation. This indifference curve property allows us to characterize the comparative static effects of changing the background risk and dependency structure in a simple portfolio choice model.Received: 8 December 2003, Revised: 16 February 2004, JEL Classification Numbers: D81.     

Convergence of the Aumann–Davis–Maschler and Geanakoplos bargaining sets

Summary. Geanakoplos [17] defined a notion of bargaining set, and proved that his bargaining set is approximately competitive in large finite transferable utility (TU) exchange economies with smooth preferences. Shapley and Shubik [26] showed that the Aumann–Davis–Maschler bargaining set is approximately competitive in replica sequences of TU exchange economies with smooth preferences. We extend Geanakoplos result to nontransferable utility (NTU) exchange economies without smooth preferences, and we extend the Shapley and Shubik result to non-replica sequences of NTU exchange economies with smooth preferences.Received: November 11, 1996This revised version was published online in February 2005 with corrections to the cover date.     

An algebraic theory of portfolio allocation

Summary. Using group and majorization theory, we explore what can be established about allocation of funds among assets when asymmetries in the returns vector are carefully controlled. The key insight is that preferences over allocations can be partially ordered via majorized convex hulls that have been generated by a permutation group. Group transitivity suffices to ensure complete portfolio diversification. Point-wise stabilizer subgroups admit sectoral separability in fund allocations. We also bound the admissible allocation vector by a set of linear constraints the coefficients of which are determined by group operations on location and scale asymmetries in the rate of returns vector. For a distribution that is symmetric under a reflection group, the linear constraints may be further strengthened whenever there exists an hyperplane that separates convex sets. Received: May 15, 2001; revised version: March 20, 2002 RID="*" ID="*" Journal paper No. J-19797 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 3463, and supported by Hatch Act and State of Iowa funds. Correspondence to: D. A. Hennessy     

Dynamic and stochastic structures in tourism demand modeling

In this paper we consider a model for international tourism demand. The point of departure of the analysis is a utility function that is both dynamic and stochastic. In the model the stochastic component is interpreted as random changes in preferences for goods and services, while the dynamic component can be seen as either habit formation or as interdependent preferences. The resulting demand functions are estimated as a multivariate state space model, where the stochastic components enter the model as stochastic seasonal and trend components. An application is constructed for different segments of the Swedish tourism market. The results indicate the importance of including both dynamic and stochastic components in the utility function, and the importance of using disaggregate data to enable investigation of each market segment.I am grateful to two anonymous referees for useful comments. Much of the research was done while I was a visiting scholar at University of California Berkeley. The hospitality of the RIPM division is gratefully acknowledged. The research was supported by grants from the Wallander Foundation.First version received: January 2003/Final version received: February 2004     

General equilibrium and resource economics

Summary. For a number of reasons a large class of general equilibrium models from the field of resource economics does not allow for an equilibrium analysis along the lines of the theory of infinite dimensional commodity spaces. The reasons concern the choice of the commodity space and the applicability of properness assumptions with respect to preferences and the technology. This paper illustrates the difficulties and shows for a prototype model how the problems can successfully be tackled by the use of a limit argument on equilibria in the truncated economies. Received: May 2, 1996; revised version: May 13, 1998     

Boiteux's solution to the shifting-peak problem and the equilibrium price density in continuous time

Summary. Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production functions are Mackey continuous. A general form for production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. This gives a framework for settling Boiteux's conjecture on the shifting-peak problem. To make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges. The general equilibrium results are supplemented by results for prices supporting individual consumer or producer optima. Received: February 16, 2000; revised version: July 7, 2001     

How Proper Is Sequential Equilibrium?

A strategy profile of a normal form game is proper if and only if it is quasi-perfect in every extensive form (with that normal form). Thus, properness requires optimality along a sequence of supporting trembles, while sequentiality only requires optimality in the limit. A decision-theoretic implementation of sequential rationality, strategic independence respecting equilibrium (SIRE), is defined and compared to proper equilibrium, using lexicographic probability systems. Finally, we give tremble-based characterizations, which do not involve structural features of the game, of the rankings of strategies that underlie proper equilibrium and SIRE.Journal of Economic LiteratureClassification Numbers: C70, C72.     

On the computation of stable sets and strictly perfect equilibria

Summary. In this paper a procedure is described that computes for a given bimatrix game all stable sets in the sense of Kohlberg and Mertens (1986). Further the procedure is refined to find the strictly perfect equilibria (if any) of such a game. Recieved: December 29, 1999; revised version: February 17, 2000     

Economies with price-dependent preferences

With the exception of existence, not much is known of economies where individual preferences may depend on prices. We show that the main properties of competitive equilibria that fall under the general heading of qualitative comparative statics remain true when preferences are price-dependent, provided that the total resources are allowed to vary in the parameter space. The equilibrium set is a smooth manifold diffeomorphic to a Euclidean space; the natural projection is a smooth proper map, with topological degree one; equilibria always exist, and are locally unique for an open and dense set of economies.     

Ordinal Efficiency and the Polyhedral Separating Hyperplane Theorem

We study problems in which each of finitely many agents must be allocated a single object, based on the agents' rankings of pure outcomes. A random allocation is ordinally efficient if it is not ordinally dominated in the sense of there being another random assignment that gives each agent a first order stochastically dominant distribution of objects. We show that any ordinally efficient random assignment maximizes the sum of expected utilities for some vector of vNM utility functions that are consistent with the given ordinal preferences. One method of proof uses a new version of the separating hyperplane theorem for polyhedra. Journal of Economic Literature Classification Numbers: C78, D61.