Hierarchic competitive equilibria

Without an interiority or strong survival assumption, an equilibrium may not exist in the standard Arrow–Debreu model. We propose a generalized concept of competitive equilibrium, called hierarchic equilibrium. Instead of using standard prices we use hierarchic prices. Existence will be shown without a strong survival assumption and without a non-satiation condition on the preferences. Under standard assumptions this reduces to the Walras equilibrium. Hierarchic equilibria are weakly Pareto optimal and any Pareto optimum can be decentralized without a border condition. We prove the existence of a Pareto optimal hierarchic equilibrium under additional assumptions. Later, we establish a core equivalence result.     

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.     

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.     

Some results on general assumptions about the existence of economic equilibrium

In the present paper we maximally use the possibilities provided by the Nash approach and the Kakutani fixed point theorem for proving the existence of an economic equilibrium. We obtain a general existence theorem which does not require a special form for income distribution functions and producer's objectives, independence of consumers' tastes, ordered preference and zero degree homogeneous price dependence. The role of the non-satiation assumption becomes more clear.     

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.     

Satiation and existence of competitive equilibrium

In this paper, we introduce a new assumption concerning the (non)satiation property of preferences and establish the existence of a competitive equilibrium under it. The assumption is weaker than the standard nonsatiation assumption and “weak nonsatiation” introduced by Allouch and Le Van (2008). In particular, it allows preferences to be satiated only inside the set of individually rational feasible consumptions, while the two nonsatiation assumptions do not. It is also worth noting that just like the two nonsatiation assumptions, our new assumption depends solely on the characteristics of consumers.     

Auctions with resale and bargaining power

We show that when the weak bidder’s bargaining power in the resale market is weakened, the auctioneer’s revenue from the first-price auction with resale is lower. Using the idea of Coase Theorem, we show that when the resale market is a sequential bargaining model with no commitment, the auctioneer’s revenue is substantially reduced, and the ranking is the opposite of Hafalir and Krishna (2009). We establish a version of the Coase Theorem in the context of the auctions with resale. When Coase Theorem holds, we show that the revenue of the auction with resale is lower than the revenue of the same auction without resale. We also provide the existence and uniqueness of equilibrium for our model of auctions with resale.     

No-arbitrage condition and existence of equilibrium with dividends

In this paper we first give an elementary proof of existence of equilibrium with dividends in an economy with possibly satiated consumers. We then introduce a no-arbitrage condition and show that it is equivalent to the existence of equilibrium with dividends.     

Erratum to “Walras and dividends equilibrium with possibly satiated consumers”

The main contribution of the paper is to provide a weaker nonsatiation assumption than the one commonly used in the literature to ensure the existence of competitive equilibrium. Our assumption allows for satiation points in the set of individually feasible consumptions, provided that the consumer has satiation points available to him outside this set. As a result, we show the concept of equilibrium with dividends (see Aumann, R.J., Drèze, J.H., 1986. Values of markets with satiation or fixed prices. Econometrica 54, 1271–1318; Mas-Collel, A., 1992. Equilibrium theory with possibly satiated preferences. In: Majumdar, M. (Ed.), Equilibrium and Dynamics: Proceedings of the Essays in Honour of David Gale. MacMillan, London, pp. 201–213) is pertinent only when the set of satiation points is included in the set of individually feasible consumptions. Our economic motivation stems from the fact that in decentralized markets, increasing the incomes of consumers through dividends, if it is possible, is costly since it involves the intervention of a social planner. Then, we show, in particular, how in securities markets our weak nonsatiation assumption is satisfied by Werner’s (1987) assumption.     

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.     

Stochastic equilibria of an asset pricing model with heterogeneous beliefs and random dividends

We investigate dynamical properties of a heterogeneous agent model with random dividends and further study the relationship between dynamical properties of the random model and those of the corresponding deterministic skeleton, which is obtained by setting the random dividends as their constant mean value. Based on our recent mathematical results, we prove the existence and stability of random fixed points as the perturbation intensity of random dividends is sufficiently small. Furthermore, we prove that the random fixed points converge almost surely to the corresponding fixed points of the deterministic skeleton as the perturbation intensity tends to zero. Moreover, simulations suggest similar behaviors in the case of more complicated attractors. Therefore, the corresponding deterministic skeleton is a good approximation of the random model with sufficiently small random perturbations of dividends. Given that dividends in real markets are generally very low, it is reasonable and significant to some extent to study the effects of heterogeneous agents’ behaviors on price fluctuations by the corresponding deterministic skeleton of the random model.     

Weakly rational expectations

Aumann and Drèze (2008) characterised the set of interim expected payoffs that players may have in rational belief systems, in which there is common knowledge of rationality and a common prior. We show here that common knowledge of rationality is not needed: when rationality is satisfied in the support of an action-consistent distribution (a concept introduced by Barelli (2009)), one obtains exactly the same set of rational expectations, despite the fact that in such ‘weakly rational belief systems’ there may not be mutual knowledge of rationality, let alone common knowledge of rationality. In the special case of two-player zero-sum games, the only expected payoff is the minmax value, even under these weak assumptions.     

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L_∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.     

Generalized weak transfer continuity and the Nash equilibrium

This paper introduces the notion of generalized weak transfer continuity and establishes that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in [Carmona, G., 2011. Understanding some recent existence results for discontinuous games. Economic Theory 48, 31–45], [Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16], [Carmona, G., 2009. An existence result for discontinuous games. Journal of Economic Theory 144, 1333–1340], and [Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056].     

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.     

Existence of financial equilibria with restricted participation

We consider a two-date model of a financial exchange economy with finitely many agents having nonordered preferences and portfolio constraints. There is a market for physical commodities at any state today or tomorrow and financial transfers across time and across states are allowed by means of finitely many nominal assets or numéraire assets. We prove a general existence result of equilibria for such a financial exchange economy in which portfolios are defined by linear constraints, extending the framework of linear equality constraints by Balasko et al. (1990), and the existence results in the unconstrained case by Cass (1984, 2006), Werner (1985), Duffie (1987), and Geanakoplos and Polemarchakis (1986). Our main result is a consequence of an auxiliary result, also of interest for itself, in which agents’ portfolio constraints are defined by general closed convex sets and the financial structure is assumed to satisfy a “nonredundancy-type” assumption, weaker than the ones in Radner (1972) and Siconolfi (1989).     

Local public goods as indivisible commodities

A sharper version of Ellickson's (1979) result on the existence of an approximate Tichout equilibrium is shown to be true under much weaker assumptions. It is also shown that there exists a second-best policy regarding the bundle of public goods each local government offers to a resident. The existence result can also be seen as an extension of a result of Anderson, Khan and Rashid (1982) to economies with production. In this context, the main technical difficulty in that the approximate equilibrium allocation does not lie in the attainable and standard techniques of truncation are not adequate.     

Equilibria in vector lattices without ordered preferences or uniform properness

This paper extends a result by Mas-Colell and Richard (Journal of Economic Theory, 1991, 53, 1–11) on the existence of an equilibrium for exchange economies over a vector lattice to cover non-ordered preferences. Moreover, it is shown that the uniformity requirement in the properness assumption made by these authors can be weakened. Furthermore, it is shown that, as a consequence of properness, an equilibrium allocation which is supported by a discontinuous prices can also be supported by continuous prices.     

An equilibrium model of insider trading in continuous time

We study a Bayesian–Nash equilibrium model of insider trading in continuous time. The supply of the risky asset is assumed to be stochastic. This supply can be interpreted as noise from nonrational traders (noise traders). A rational informed investor (the insider) has private information on the growth rate of the dividend flow rewarded by the risky asset. She is risk averse and maximizes her inter-temporal utility rate over an infinite time-horizon. The market is cleared by a risk neutral market maker who sets the price of the risky asset competitively as the conditional present value of future dividends, given the information supplied by the dividend history and the cumulative order flow. Due to the presence of noise traders, the market demand does not fully reveal the insider’s private information, which slowly becomes incorporated in prices. An interesting result of the paper is that a nonstandard linear filtering procedure gives an a priori form for the equilibrium strategy to be postulated. We show the existence of a stationary linear equilibrium where the insider acts strategically by taking advantage of the camouflage provided by the noise which affects the market maker’s estimates on private information. In this equilibrium, we find that the insider’s returns on the stock are uncorrelated over long periods of time. Finally, we show that the instantaneous variance of the price under asymmetric information lies between the instantaneous variance of the price under complete and incomplete information. The converse inequalities hold true for the unconditional variance of the price.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.