No arbitrage condition and existence of equilibrium in inflnite or flnite dimension with expected risk averse utilities (original) (raw)

Equilibrium theory with a measure space of possibly satiated consumers

Journal of Mathematical Economics, 2003

Walras equilibria may not exist when consumers' preferences are possibly satiated and to overcome this difficulty, several extended notions of equilibria have been proposed, which all reduce to Walras equilibria under nonsatiation and free disposal. This includes the notions of equilibria with slack (also called dividend equilibria) as in Müller (1980), Makarov (1981), and , monetary equilibria as in , or weak equilibria as in , which are all defined when there are finitely many consumers. This includes also the notion of free disposal equilibrium, when markets clear in a weak sense, allowing free disposal. Our paper considers an economy with a measure space of consumers and provides a general existence result of equilibria for the various existing notions. This result extends in particular the result by Hildenbrand (1970) on the existence of Walras equilibria.

Equilibrium Theory beyond Arbitrage

2003

The notion of arbitrage is predominantly used as a conceptual framework of finance and economics for studying equilibrium as well as pricing relations of asset markets. This may be no longer useful, however, once the transitivity preferences is dropped because equilibrium can exist in an economy which admits arbitrage opportunities. Thus, the no arbitrage conditions are no longer necessary for the existence of equilibrium with non-transitive preferences. We propose a new condition which is necessary and sufficient for the existence of equilibrium of the economy where preferences need not be transitive and the consumption set of each agent need not be bounded from below.

On the Existence of Equilibrium with Incomplete Markets and Non-Monotonic Preferences

Brazilian Review of Econometrics, 2008

We provide a shorter proof than Geanakoplos and Polemarchakis (1986) of the existence of equilibrium in an incomplete financial market economy with numeraire assets, under the weak assumption that asset returns are non-negative. Furthermore, we relax the strict monotonicity assumption on preferences and as an application we prove the existence of equilibrium when agents may disagree on zero probability events but do not plan to go bankrupt in any state.

Arbitrage and equilibrium in unbounded exchange economies with satiation

Journal of Mathematical Economics, 2006

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (Econometrica, 1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner's existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. First, we show that in unbounded exchange economies, even if some agents' preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades -that is, as long as agents' preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to * The authors are grateful to an anonymous referee for many helpful comments. transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation -a condition considerably weaker than local or global nonsatiation -that makes possible this transformation.

On the Different Notions of Arbitrage and Existence of Equilibrium

Journal of Economic Theory, 1999

In this paper we first give an existence of equilibrium theorem with unbounded below consumption sets, by the demand approach, assuming only that the individually rational utility set is compact. vVe then classify different notions of absence of arbitrage and give conditions under which they are equivalent to the existence of an equilibrium. SUR LES DIFFERENTES NOTIONS D'ARBITRAGE ET L'EXISTENCE DE L'EQUILIBRE Résumé•-Dans ce papier, d'abord nous donnons un théorème d'existence d'un équilibre avec des ensembles de consommation non bornés inférieurement par l'approche de la demande en suµµosant seulement que l'ensemble des utilités individuellement rationnelles est compact. Ensuite. nous classons les différentes notions d'absence d'arbitrage et donnons des conditions qui les rendent équivalentes à l'existence d'un équilibre.

From Utility Maximization to Arbitrage Pricing, and Back

Social Science Research Network, 1996

In an incomplete market there exists a multiplicity of valuation operators. A legitimate question has been raised regarding which valuation operator should be used in such a market. To make this choice without relying on an equilibrium model and maintaining the spirit of arbitrage models in that risk attitudes do not influence the selection criteria, several authors (e.g., Rubinstein, 1994, Stutzer, 1996 and, in a slightly different context, Hansen and Jagannathan, 1997) suggest ad hoc criteria. This paper develops a unifying framework for choosing a valuation operator from among the multiplicity. It is shown that under mild regularity conditions every utility function induces a selection criterion and, conversely, every selection criterion is induced by a certain utility function. Utilizing our framework we show that the selection criteria in the above papers have strong links to utility maximization. In fact, the chosen valuation operator is the marginal rate of substitution of an agent maximizing his or her utility of profit emanating from selecting a portfolio subject to a zero budget constraint. This utility maximization problem is a "relaxation" of the problem of maximizing arbitrage profit as utilized in the definition of the no-arbitrage condition. It is the "relaxation" in the sense that the maximization of arbitrage profit now is solved with reference to risk, utilizing a certain utility function. Rather than require that the value of the self-financing portfolio (sought for in maximizing arbitrage profit) be nonnegative in every state of nature, the portfolio is allowed to have negative values in some states of nature while the utility of its payoff is maximized. The induced utility function must exhibit "strong enough" risk aversion to prevent the problem of maximizing utility from being unbounded. Consequently, the selection criteria in the above papers not only have a link to risk preferences, but further assume, implicitly though perhaps unintentionally, a certain degree of risk aversion.