Quasi-equilibria in Banach spaces with lower semi-continuous preferences

Summary. This paper establishes a general result on the existence of non-trivial quasi-equilibria for economies with infinitely many commodities, in which consumers preferences are assumed to be lower semi-continuous, but not necessarily to have open lower sections nor to be open valued. Such preference relations include the lexicographic orderings. In addition, production technologies are not required to satisfy free disposal condition.Received: 25 May 2004, Revised: Received: 25 May 2004; revised version: September: 14, 2004, JEL Classification Numbers: D51, D61, D62.I am grateful to Professors Sho-Ichiro Kusumoto, Atsushi Kajii, Yoshihiko Otani, Zaifu Yang and one anonymous referee for their helpful commente and suggestions.     

A two-stage core with applications to asset market and differential information economies

Summary. We introduce a new core concept, called the two-stage core, which is appropriate for economies with sequential trade. We prove a general existence theorem and present two applications of the two-stage core: (i) In asset markets economies where we extend our existence proof to the case of consumption sets with no lower bound, in order to capture the case of arbitrary short sales of assets. Further, we show that the two-stage core is non empty in the Hart (1975) example where a rational expectations equilibrium fails to exist. (ii) In differential information economies where we provide sufficient conditions for the incentive compatibility of trades. Namely, that no coalition of agents can misreport the true state and provide improvements to all its members, even by redistributing the benefits from misreporting. Received: December 20, 1995; revised version: December 6, 1996     

Sequential decisions with several agents

Summary. We consider k agents who have different subjective probabilities and are utility maximizers. A planner, who knows the beliefs of the agents, maximizes the social expected utility, which is increasing and symmetric in the utilities of the agents. She does that by optimally stopping the flow of information released to the agents. The explicit form of the optimal stopping time is given. Received: October 23, 1995; revised version: June 25, 1997     

Multiple unit auctions with strategic price-quantity decisions

Summary. I study a multiple unit auction where symmetric risk-neutral bidders choose prices and quantities endogenously. In the model, bidders (a) may place non-linear valuations on the auctioned units, and (b) bid for several units at the same price (“lumpy” bids). I characterize quantity-symmetric and strictly monotone-increasing price equilibria for discriminatory and competitive auctions, and show that (i) if quantity strategy profiles are equal across auctions revenue- equivalence holds, (ii) expected revenue is higher if bidders bid for the entire supply rather than for shares of it, and (iii) equilibrium allocations may fail to be Pareto-optimal. Received: April 14, 1995; revised version: September 3, 1997     

The least core, kernel and bargaining sets of large games

Summary. We study the least core, the kernel and bargaining sets of coalitional games with a countable set of players. We show that the least core of a continuous superadditive game with a countable set of players is a non-empty (norm-compact) subset of the space of all countably additive measures. Then we show that in such games the intersection of the prekernel and the least core is non-empty. Finally, we show that the Aumann-Maschler and the Mas-Colell bargaining sets contain the set of all countably additive payoff measures in the prekernel. Received: June 6, 1996; revised version: March 1, 1997     

Strategy-proofness versus efficiency for small domains of preferences over public goods

Summary. It has long been known that when agents have von Neumann-Morgenstern preferences over lotteries, there is an incompatibility between strategy-proofness and efficiency (Gibbard, [9]; Hylland, [12]) – a solution satisfying those properties must be dictatorial. We strengthen this result by showing that it follows from the same incompatibility on a series of much smaller domains of preferences. Specifically, we first show the incompatibility to hold on our smallest domain, in which two agents are restricted to have linear preferences over one private good and one public good produced from the private good (Kolm triangle economies). This result then implies the same incompatibility on increasingly larger domains of preferences, ending finally with the class of von Neumann-Morgenstern preferences over lotteries. Received: February 6, 1997; revised version: January 29, 1998     

Utility representation of continuous preferences

Summary.  By generalising the Arrow-Hahn approach, we give new conditions under which a preference relation on a metric space can be represented by a continuous utility function. Received: December 11, 1995; revised version July 29, 1996     

Incomplete markets, continuum of states and default

Summary. We show that when bankruptcy, subject to penalties, is allowed, it is possible to prove the existence of equilibrium in a model with a continuum of states without imposing any assumptions on ex-post endowments.Received: November 20, 1995; revised version: September 16, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Quasiconcavity and the kernel of a separable utility

Summary. In this paper I prove that a quasiconcave separable utility function defined on an atomless space is concave. Received: August 11, 1997; revised version: October 2, 1997     

A note on the “size” of determinacy and indeterminacy

Summary. It is shown in this note that in an incomplete markets economy with uncountably many states of the world there may be uncountably many isolated equilibria as well as uncountably many non-isolated equilibria. Moreover, both subsets can be simultaneously of second category. Therefore, none of the subsets can be considered negligible with respect to the other, neither from a cardinality point of view nor from a topological one. Unfortunately, this fact prevents from claiming that these economies may have “typically” determinate equilibria – even though uncountably many of them – as would have been desirable for comparative statics exercises. Received: May 19, 1995; revised version: March 24, 1997     

Equilibrium interest rate and liquidity premium with transaction costs

Summary. In this article we study the effects of transaction costs on asset prices. We assume an overlapping generations economy with two riskless assets. The first asset is liquid while the second asset carries proportional transaction costs. We show that agents buy the liquid asset for short-term investment and the illiquid asset for long-term investment. When transaction costs increase, the price of the liquid asset increases. The price of the illiquid asset decreases if the asset is in small supply, but may increase if the supply is large. These results have implications for the effects of transaction taxes and commission deregulation. Received: December 5, 1997; revised version: March 19, 1998     

A comparison between the core and the monopoly solutions in a mixed exchange economy

Summary. Using a mixed market model for analyzing imperfectly competitive economies, we maximize the oligopolists' Welfare Function, given individual rationality and feasibility constraints. We prove that solutions belong to the core for a large class of economies. This class includes, in particular, every monopoly having a single type of small traders. Note that all such solutions yield the large trader, utility-wise, strictly more than at any monopoly solution, where the monopolist plays strategically, and the ocean of small traders act as being as price-takers. Received: March 4, 1996; revised version July 7, 1996     

A unified treatment of finite and infinite economies: limited arbitrage is necessary and sufficient for the existence of equilibrium and the core

Summary. A single condition, limited arbitrage, is shown to be necessary and sufficient for the existence of a competitive equilibrium and the core in economies with any number of markets, finite or infinite, with or without short sales. This extends earlier results of Chichilnisky [8] for finite economies. This unification of finite and infinite economies is achieved by proving that in Hilbert spaces limited arbitrage is necessary and sufficient for the compactness of the Pareto frontier. Limited arbitrage has also been shown to be necessary and sufficient for a resolution of the social choice paradox [9], [10], [12], [13], [14]. Received: August 4, 1995; revised version: April 11, 1997     

Recursive utility and preferences for information

Summary. This paper presents an axiomatic foundation for recursive utility that captures the role of the timing of resolution of uncertainty without relying on exogenously specified objective beliefs. Two main representation results are proved. In the first one, future utility enters the recursion through the type of general aggregators considered in Skiadas (1997a), and as a result the formulation is purely ordinal and free of any probabilities. In the second representation these aggregators are conditional expectations relative to subjective beliefs. A new recursive representation incorporating disappointment aversion is also suggested. The main methodological innovation of the paper derives from the fact that the basic objects of choice are taken to be pairs of state-contingent consumption plans and information filtrations, rather than the temporal (objective) lotteries of the existing literature. It is shown that this approach has the additional benefit of being directly applicable to the continuous-time version of recursive utility developed by Duffie and Epstein (1992). Received: February 18, 1997; revised version: July 18, 1997     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Manifolds of golden rule and balanced steady state equilibria

Summary. We show that the set of balanced steady state (resp. golden rule) equilibria, parameterized by endowments, of stationary overlapping-generations economies are smooth manifolds diffeomorphic to Euclidean spaces. These properties extend similar properties of the Walrasian equilibria and enable one to apply the natural projection approach to the study of these equilibria. Received: October 30, 1995; revised version: October 10, 1996     

Bundling decisions for selling multiple objects

Summary. Auctioneers often face the decision of whether to bundle two or more different objects before selling them. Under a Vickrey auction (or any other revenue equivalent auction form) there is a unique critical number for each pair of objects such that when the number of bidders is fewer than that critical number the seller strictly prefers a bundled sale and when there are more bidders the seller prefers unbundled sales. This property holds even when the valuations for the objects are correlated for a given bidder. The results have been proved using a mathematical technique of quantiles that can be extremely useful for similar analysis. Received: November 5, 1996; revised version: January 21, 1998     

Characterizing C2 aggregate excess demand functions

Summary. We integrate and sharpen two characterizations of aggregate excess demand functions: we obtain Mas-Colell's (1977) equilibrium invariance, and strengthen Geanakoplos' (1984) weakly concave utility functions to strictly concave ones. Our proofs modify and extend Geanakoplos' utility-construction. We note two applications: a sharper characterization of equilibrium price sets (cf. Mas-Colell (1977)), and a basis for the studies of computable economies (Richter and Wong (1996)). Received: July 29, 1996; revised version: February 20, 1997     

General allocation problems with single-peaked preferences: path-independence and related topics

We face the problem of allocationg a fixed amount of a perfectly divisible good among a group of agents with single-peaked preferences. We survey the three different cases studied in the literature: the pure distribution case, the redistribution case, and the gerneral case. The so called general case provide with a natural framework to analyze the idea of path-independence. In this framework, we explore the existence of rules fulfilling this property. Our first result is negative: a strong version of this property cannot be fulfilled together with efficiency. Nonetheless, some restricted versions of the path-independence property are compatible with interesting properties, in particular no manipulability, and no envy. We then identify two solutions satisfying this sort of property: the equal distance rule, and a new extension of the uniform rule.     

Double implementation in economies with production technologies unknown to the designer

Summary. This paper considers double implementation of Walrasian allocations and Lindahl allocations in Nash and strong Nash equilibria for both private and public goods economies when preferences, initial endowments, production technologies, and coalition patterns are all unknown to the designer. It will be noted that the mechanisms presented here are feasible and continuous. In addition, unlike most mechanisms proposed in the literature, our mechanism works not only for three or more agents, but also for two-agent economies, and thus it is a unified mechanism which is irrespective of the number of agents. Received: March 12, 1998; revised version: March 12, 1998