Continuum Of Traders Research Articles

Equivalence of Competitive Equilibria, Fuzzy Cores, and Fuzzy Bargaining Sets in Finite Production Economies

For an exchange economy with a continuum of traders and a finite-dimensional commodity space under some standard assumptions, Aumann showed that the core and the set of competitive allocations (two most important solutions) coincide and Mas-Colell proved that the bargaining set and the set of competitive allocations coincide. However, in the case of exchange economies with a finite number of traders, it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we establish the equivalence of the fuzzy core, the fuzzy bargaining set (or Aubin bargaining set), and the set of competitive allocations in a finite coalition production economy with an infinite-dimensional commodity space under some standard assumptions. We first derive a continuous equivalence theorem and then discretize it to obtain the desired equivalence in finite economies.

Market Selection in Large Economies: A Matter of Luck

Within a general equilibrium model with a continuum of traders, we investigate the Market Selection Hypothesis (MSH) - markets select traders with more accurate beliefs. We find, contrary to known results for economies with (only) finitely many traders, that risk attitudes affect survival and that markets might select against traders with accurate beliefs. Remarkably, even in these cases, asymptotic equilibrium prices reflect accurate beliefs. Thus, unlike known violations of MSH, we corroborate Freedman’s conjecture that market selection forces induce rational expectations.

Market selection in large economies: A matter of luck

In a general equilibrium model with a continuum of traders and bounded aggregate endowment, I investigate the market selection hypothesis that markets favor traders with accurate beliefs. Contrary to known results for economies with (only) finitely many traders, I find that risk attitudes affect traders' survival and that markets can favor “lucky” traders with incorrect beliefs over “skilled” traders with accurate beliefs. My model allows for a clear distinction between luck and skills, and it shows that market selection forces induce efficient prices even when accurate traders do not survive in the long run.

Noncooperative oligopoly in markets with a continuum of traders and a strongly connected set of commodities

We show the existence of a Cournot–Nash equilibrium for a mixed version of the Shapley window model, where large traders are represented as atoms and small traders are represented by an atomless part. Previous existence theorems for the Shapley window model, provided by Sahi and Yao (1989) in the case of economies with a finite number of traders and by Busetto et al. (2011) in the case of mixed exchange economies, are essentially based on the assumption that there are at least two atoms with strictly positive endowments and indifference curves contained in the strict interior of the commodity space. Our result does not require this restriction. It relies on the characteristics of the atomless part of the economy and exploits the fact that traders belonging to the atomless part have an endogenous “Walrasian” behavior.

Existence of competitive equilibrium in coalition production economies with a continuum of agents

Aumann (Econometrica 34: 1–17, 1966) established the existence of competitive equilibria in exchange economies with a continuum of traders. This result has been extended to special production economies under the condition that the production sets are countably additive or superadditive in the literature. In this paper, we extend Aumann’s result to coalition production economies with a continuum of traders in which each coalition can have a different production set and the production sets need not to be additive or superadditive. Our new existence theorem for competitive equilibrium implies the classical result by Hildenbrand (Econometrica 38: 608–623, 1970), and implies those by Sondermann (J Econ Theory 8: 259–291, 1974) and Greenberg et al. (J Math Econ 6: 31–41, 1979) under the assumption that the production sets are convex.

Equivalence of the Aubin bargaining set and the set of competitive equilibria in a finite coalition production economy

Mas-Colell (Mas-Colell, 1989) proved that the bargaining set and the set of competitive allocations coincide in an exchange economy with a continuum of traders under some standard assumptions. In the case of pure exchange economies with a finite number of traders it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we show that the Aubin bargaining set (or fuzzy bargaining set) and the set of competitive allocations coincide in a finite coalition production economy under some standard assumptions. We also show that the (Mas-Colell) bargaining set shrinks to the set of competitive allocations in a finite coalition production economy E under some standard conditions when E is replicated. As a consequence, the existence of competitive equilibrium in a finite coalition production economy implies the nonemptiness of Aubin bargaining sets.

Coincidence of the Mas-Colell bargaining set and the set of competitive equilibria in a continuum coalition production economy

Mas-Colell (J Math Econ 18:129–139, 1989) proved that the bargaining set and the set of competitive allocations coincide in an exchange economy with a continuum of traders under some standard assumptions. We extend this result to continuum coalition production economies and prove that the bargaining set and the set of competitive allocations coincide in a coalition production economy with a continuum of traders under some standard assumptions. As a consequence, we obtain a coincidence theorem for the core and the set of competitive allocations in a coalition production economy which extends the well-known coincidence theorem by Aumann (Econometrica 32:39–50, 1964).

Optimality Criteria of Economic Growth as Examples of Optimal Resource Allocation Over Time

This is my PhD dissertation defended in 1966 at the University of Minnesota and it was never published. It has been cited and used in other researchers work so I now make it available. The problem of optimal resource allocation over time is formulated as an optimal control problem. This is where the existence of optimal programs is investigated (and sufficient conditions are provided) for the first time in economics. When earlier authors spoke of existence, they spoke of convergence of the integral defining the optimality criterion. Optimal programs were characterized in terms of market equilibria at each point in time as well as imputed prices for stocks of capital.One of the propositions in the paper establishes the principle of centralization of planning and decentralization of implementation (this closely relates to polycenterism advocated by European socialist parties in the late 1950s. If one uses a measure space of traders instead of the basic time interval and assume pure exchange then the existence theorem of this paper can greatly simplify the proof of existence of equilibrium in a or a much more flexible model of exchange with continuum of traders.

Three Models of Noncooperative Oligopoly in Markets with a Continuum of Traders

Trois modèles d’oligopole en équilibre général avec un continu d’agents Cet article propose une reconstruction de la théorie de l’oligopole en équilibre général en univers non coopératif. Trois types de modèles en échange pur sont étudiés : le modèle de Cournot-Walras analysé dans Codognato et Gabszewicz (1991) ; le modèle de jeu de marché à la Cournot-Nash proposé initialement par Shapley ; et le modèle de Cournot-Walras étudié par Busetto et al. (2008). Nous établissons la relation qui existe entre les équilibres de ces trois modèles avec l’équilibre concurrentiel walrassien. Ensuite, nous analysons les relations entre les trois équilibres stratégiques associés à ces trois modèles. Classification JEL : C72, D51.

Three Models of Noncooperative Oligopoly in Markets with a Continuum of Traders

SummaryIn this paper, we reconstruct the main developments of the theory of noncooperative oligopoly in general equilibrium, by focusing on the analysis of three prototypical models for pure exchange economies: the model of Cournot-Walras equilibrium of Codognato and Gabszewicz (1991); the model of Cournot-Nash equilibrium originally proposed by Lloyd S. Shapley and known as window model; the model of Cournot-Walras equilibrium of Busetto et al. (2008). We establish, in a systematic way, the relationship between the three notions of equilibrium proposed in these models and the notion of Walras equilibrium. Then, we investigate the relationships among those three notions of equilibrium.

Asymmetric Information in Bilateral Trade and in Markets: An Inversion Result

I consider a simple bilateral trading game between a seller and a buyer who have private valuations for an indivisible good. The seller makes a price offer which the buyer can either accept or reject. If the seller can observe the valuation of the buyer (if information is symmetric), then bilateral trade is trivially efficient. If the seller cannot observe the valuation (if information is asymmetric), then bilateral trade is inefficient. This bilateral trading game between a single buyer and a single seller is embedded into a matching market with a continuum of traders. I consider steady-state equilibria in stationary strategies. I show that, on the overall market level, the relation between the informational regime and efficiency is inverted when frictions are small. In particular, if frictions are small and if information is asymmetric, the trading outcome in the market is close to being efficient. If information is symmetric, however, the trading outcome can be very inefficient -- even if frictions vanish.

Stackelberg-Walras and Cournot-Walras Equilibria in Mixed Markets: A Comparison

In this note, we compare two strategic general equilibrium concepts: the Stackelberg-Walras equilibrium and the Cournot-Walras equilibrium. We thus consider a market exchange economy embodying atoms and a continuum of traders. It is shown that, when the preferences of the small traders are represented by Cobb-Douglas utility functions, the Stackel-berg-Walras and the Cournot-Walras equilibria can coincide only if 1) the endowments and preferences of atoms are identical and 2) the elasticity of the followers’ best response functions are equal to zero in equilibrium.

Stackelberg-Cournot and Cournot Equilibria in a Mixed Markets Exchange Economy

In this note, we compare two strategic general equilibrium concepts: the Stackelberg-Cournot equilibrium and the Cournot equilibrium. We thus consider a market exchange economy including atoms and a continuum of traders, who behave strategically. We show that, when the preferences of the small traders are represented by Cobb-Douglas utility functions and the atoms have the same utility functions and endowments, the Stackelberg-Cournot and the Cournot equilibrium equilibria coincide if and only if the followers’ best responses functions have a zero slope at the SCE.

A Search-Based Pricing Model of Credit Default Swaps

We characterize the equilibrium spreads of Credit Default Swaps traded in an over-the-counter market. The OTC market involves a continuum of traders indexed by their beliefs about risk. Traders encounter standard search frictions. Once agents are matched, they adjust their preferred spread according to their subjective risk measures of both the reference entity and their counterparty. In the absence of search frictions, agents do not consider counterparty risk in determining the optimal spread. Under the assumption that buyers do not own the underlying debt, gainful trade occurs only when beliefs about risk differ. When buyers own the debt, the spread grows and gainful trade can occur even when beliefs about risk coincide. The limiting spread is the risk-neutral spread up to an affine transformation.

Noncooperative oligopoly in markets with a continuum of traders

We show the existence of a pure strategy Cournot–Nash equilibrium for a model of noncooperative exchange where large traders, represented as atoms, and small traders, represented by an atomless part, are allowed to buy and sell all commodities.

Existence of Competitive Equilibrium of Large Security-Spot Market with Incomplete Asset Structure

With the concept of security-spot market or security market with measurable space of agents, which was first given by Zhang (1998, 2000), we continue to discuss the existence of equilibrium in such large economies. Under common conditions, we have claimed the existence of equilibrium in incomplete security-spot market with a continuum of traders. Because of the development from complete market to incomplete market, we have not only generalized Aumann’s results in 1966, but also achieved a breakthrough on researching methods.

More on equilibria in competitive markets with externalities and a continuum of agents

Equilibrium existence results are presented for competitive markets with externalities in price and consumption and a measure space of agents. These unify and extend [Balder, E.J., 2003. Existence of competitive equilibria in economies with a measure space of consumers and consumption externalities. Preprint, in press, electronically available at http://www.math.uu.nl/publications/preprints/1294.ps.gz; Balder, E.J., 2005. More about equilibrium distributions for competitive markets with externalities. Working paper, Department of Economics, University of Illinois.] and generalize the main existence results by Aumann [Aumann, R.J., 1964. Markets with a continuum of traders. Econometrica 32, 39–50], Schmeidler [Schmeidler, D., 1969. Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37, 578–585; Schmeidler, D., 1973. Equilibrium points of nonatomic games. Journal of Statististical Physics 7, 295–300.], Greenberg et al. [Greenberg, J., Shitovitz, B., Wieczorek, A., 1979. Existence of equilibria in atomless production economies with price dependent preferences. Journal of Mathematical Economics 6, 31–41.], Yamazaki [Yamazaki, A., 1978. An equilibrium existence theorem without convexity assumptions. Econometrica 46, 541–555.], Noguchi [Noguchi, M., 2005. Interdependent preferences with a continuum of agents. Working paper, Meijo University, 2001. Journal of Mathematical Economics 41, 665–686.], Cornet [Cornet, B., Topuzu, M., 2005. Existence of equilibria for economies with externalities and a measure space of consumers. Economic Theory 26, 397–421.], and Noguchi and Zame [Noguchi, M. and Zame, W.R., 2004. Equilibrium distributions with externalities. Preprint.].

No-arbitrage condition and existence of equilibrium in asset markets with a continuum of traders

In the present paper, we prove that a no-arbitrage condotion (a la Werner) is necessary and sufficient for the existence of an equilibrium with a continuum of traders and a finite of assets. As in Aumann (1966), Hildenbrand (1974) and Schmeidler (1969), preferences are not assume to be convex. We do not use Fatou's Lemma and do not assume that the consumption sets are compact

A Variational Problem Related to a Continuous-Time Allocation Process for a Continuum of Traders

We provide an existence theorem to continuous-time allocation process for a continuum of traders, in the absence of a concavity condition on the utility function, and under the presence of some economic parameters.

The bargaining set of a large economy with differential information

We study the Mas-Collel bargaining set of an exchange economy with differential information and a continuum of traders. We established the equivalence of the private bargaining set and the set of Radner competitive equilibrium allocations. As for the weak fine bargaining set, we show that it contains the set of competitive equilibrium allocations of an associated symmetric information economy in which each trader has the joint information» of all the traders in the original economy, but unlike the weak fine core and the set of fine value allocations, it may also contain allocations which are not competitive in the associated economy.