No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.     

Arbitrage and equilibrium in unbounded exchange economies with satiation

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (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 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.     

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.     

The geometry of arbitrage and the existence of competitive equilibrium

We present the basic geometry of arbitrage, and use this basic geometry to shed new light on the relationships between various no-arbitrage conditions found in the literature. For example, under very mild conditions, we show that the no-arbitrage conditions of Hart [Journal of Economic Theory 9 (1974) 293] and Werner [Econometrica 55 (1987) 1403] are equivalent and imply the compactness of the set of utility possibilities. Moreover, we show that if agents’ sets of useless net trades are linearly independent, then the Hart–Werner conditions are equivalent to the stronger condition of no-unbounded-arbitrage due to Page [Journal of Economic theory 41 (1987) 392]—and, in turn, all are equivalent to compactness of the set of rational allocations. We also consider the problem of existence of equilibrium. We show, for example, that under a uniformity condition on preferences weaker than Werner’s uniformity condition, the Hart–Werner no-arbitrage conditions are sufficient for existence. With an additional condition of weak no-half-lines—a condition weaker than Werner’s no-half-lines condition—we show that the Hart–Werner conditions are both necessary and sufficient for existence.     

Equilibrium and arbitrage in incomplete asset markets with fixed prices

At arbitrary prices of commodities and assets, fix-price equilibria exist under weak assumptions: endowments need not satisfy an interiority condition, utility functions need only satisfy a very weak monotonicity requirement, and the asset return matrix allows for redundant assets. Prices of assets may permit arbitrage. At equilibrium, though restricted through endogenously determined trading constraints, arbitrage possibilities may persist; in an example, an individual holds an arbitrage portfolio.     

Core equivalence and welfare properties without divisible goods

We study an economy where all goods entering preferences or production processes are indivisible. Fiat money not entering consumers’ preferences is an additional perfectly divisible parameter. We establish a First and Second Welfare Theorem and a core equivalence result for the rationing equilibrium concept introduced in Florig and Rivera (2005a). The rationing equilibrium can be considered as a natural extension of the Walrasian notion when all goods are indivisible at the individual level but perfectly divisible at the level of the entire economy.     

Inter-temporal preference for flexibility and risky choice

We derive an inter-temporal theory of choice, in the spirit of Kreps and Porteus [Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200], where decision makers have incomplete preferences. This can be used to model indecisiveness as well as unforeseen contingencies. The key to our approach is a time consistency condition and therefore the normative connection between ex-ante and ex-post choice. The time consistency condition enables a representation that is a straight forward extension of recursive utility with the exception that it features an inter-temporal ‘utility for flexibility’.     

Unbounded exchange economies with satiation: How far can we go?

We unify and generalize the existence results in Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55 (6), 1403–1418], Dana et al. [Dana, R.-A., Le Van, C., Magnien, F., 1999. On the different notions of arbitrage and existence of equilibrium. Journal of Economic Theory 87 (1), 169–193], Allouch et al. [Allouch, N., Le Van, C., Page Jr., F.H., 2006. Arbitrage and equilibrium in unbounded exchange economies with satiation. Journal of Mathematical Economics 42 (6), 661–674], Allouch and Le Van [Allouch, N., Le Van, C., 2008. Erratum to “Walras and dividends equilibrium with possibly satiated consumers”. Journal of Mathematical Economics 45 (3–4), 320–328]. We also show that, in terms of weakening the set of assumptions, we cannot go too far.     

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.     

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).     

Arbitrage and equilibrium in economies with infinitely many commodities

An ‘arbitrage opportunity’ for a class of agents is a commodity bundle that will increase the utility of any of the agents and that has non-positive price. The non-existence of ‘arbitrage opportunities’ is necessary and sufficient for the existence of an economic equilibrium. A bundle is ‘priced by arbitrage’ if there is a unique price that it can command without causing an ‘arbitrage opportunity’ to exist. For economies that have infinitely many commodities, appropriate notions of ‘arbitrage opportunities’ and ‘bundles priced by arbitrage’ depend on the continuity of agents’ preferences. This paper develops these notions, thereby providing a foundation for recent work in financial theory concerning arbitrage in continuous-time models of securities markets.     

Patience in some non-additive models

Our aim is to give an axiomatization of preferences over infinite consumption streams. At first we adopt the additive case, and give a characterization of preferences which satisfy patience [Marinacci, M., 1998. An axiomatic approach to complete patience and time invariance. Journal of Economic Theory 83, 105–144] or equivalently what Diamond [Diamond, P.A., 1965. The evaluation of infinite utility streams. Econometrica 33, 170–177] named equal treatment of all generations and then, focus on stationary additive preferences. It appears that this class of functionals contains the discounting functionals axiomatized in Koopmans [Koopmans, T.C., 1972. In: McGuire, C.B., Radner, R. (Eds.), Representations of Preference Orderings Over Time. Decision and Organization, North-Holland, Amsterdam] and what is known as Banach-Mazur limit functionals. These results are extended to non-additives preferences where similar results are generalized and naive patience receives a positive treatement through the liminf criterion.     

Representation of non-transferable utility games by coalition production economies

We prove that every compactly generated non-transferable utility (NTU) game can be generated by a coalition production economy. The set of Walrasian payoff vectors for our induced coalition production economy coincides with the inner core of the balanced cover of the original game. This equivalence depends heavily on our representation. We exemplify that this equivalence need not hold in other representations. We also give a sufficient condition for the existence of a Walrasian equilibrium for our induced coalition production economy.     

Slack in incomplete markets with nominal assets: A symmetric proof

In this paper, we provide an equilibrium analysis in the framework of incomplete markets where some agents’ preferences are possibly satiated at some state of the nature. We will consider nominal assets with exogenously fixed asset prices. We extend the notion of equilibrium with slack – introduced by Drèze and Müller [Drèze, J., Müller, H., 1980. Optimality properties of rationing schemes. Journal of Economic Theory 23, 150–159] in a fixed price setting – to the GEI framework.     

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.     

Can we identify Walrasian allocations?

Abstract. We consider a discrete time, pure exchange infinite horizon economy with consumers and consumption goods per period. Within the framework of decentralized mechanisms, we show that for any given consumption trade at any period of time, say at time one, the consumers will need in general an infinite dimensional (informational) space to identify such a trade as an intertemporal Walrasian one. However, we show a set of environments where the Walrasian trades at each period of time can be achieved as the equilibrium trades of a sequence of decentralized competitive mechanisms, using only both current prices and quantities to coordinate decisions. Received: 1 December 1999 / Accepted: 31 October 2000     

CONTINUOUS PREFERENCE ORDERINGS REPRESENTABLE BY UTILITY FUNCTIONS

This paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. We start with a historical perspective on the notions of utility and preferences, continue by defining the mathematical concepts employed in this literature, and then list several key contributions to the topic of representability. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.     

Perturbed utility and general equilibrium analysis

We study general equilibrium theory of complete markets in an otherwise standard economy with each household having an additive perturbed utility function. Since this function represents a type of stochastic choice theory, the equilibrium of the corresponding economy is defined to be a price vector that makes its mean expected demand equal its mean endowment. We begin with a study of the economic meaning of this notion, by showing that at any given price vector, there always exists an economy with deterministic utilities whose mean demand is just the mean expected demand of our economy with additive perturbed utilities. We then show the existence of equilibrium, its Pareto inefficiency, and the upper hemi-continuity of the equilibrium set correspondence. Specializing to the case of regular economies, we finally demonstrate that almost every economy is regular and the equilibrium set correspondence in this regular case is continuous and locally constant.     

Nonparametric Bayesian modelling of monotone preferences for discrete choice experiments

Discrete choice experiments are widely used to learn about the distribution of individual preferences for product attributes. Such experiments are often designed and conducted deliberately for the purpose of designing new products. There is a long-standing literature on nonparametric and Bayesian modelling of preferences for the study of consumer choice when there is a market for each product, but this work does not apply when such markets fail to exist as is the case with most product attributes. This paper takes up the common case in which attributes can be quantified and preferences over these attributes are monotone. It shows that monotonicity is the only shape constraint appropriate for a utility function in these circumstances. The paper models components of utility using a Dirichlet prior distribution and demonstrates that all monotone nondecreasing utility functions are supported by the prior. It develops a Markov chain Monte Carlo algorithm for posterior simulation that is reliable and practical given the number of attributes, choices and sample sizes characteristic of discrete choice experiments. The paper uses the algorithm to demonstrate the flexibility of the model in capturing heterogeneous preferences and applies it to a discrete choice experiment that elicits preferences for different auto insurance policies.     

Unique induced preference representations

Machina [Machina, M.J., 1984. Temporal risk and the nature of induced preferences. Journal of Economic Theory 33, 199–231] considers an individual who has to choose from a set of alternative temporal uncertain prospects, and must take an action before the uncertainty is resolved, seeking to maximize the expected value of an (action determined) von Neumann-Morgenstern utility index. It is natural to ask if the set of underlying von Neumann-Morgenstern utility indices can be uniquely recovered solely on the basis of the thus induced (ordinal) preferences over temporal prospects. Machina’s conclusion is that “ordinal preferences alone will not suffice.” However, we show that it is possible to recover the action–utility set inducing the preferences uniquely if we restrict attention to action–utility sets for which no two actions induce the same preference relation on the space of temporal prospects, no action is redundant, and no action leads to a risk free outcome.