Oligopoly in Markets with a Continuum of Traders

Atomic Cournotian traders may be Walrasian

“…We provide four examples which show that this characterization holds non-vacuously. When our condition fails to hold, we also confirm, through some examples, the result obtained by Okuno et al (1980): small traders always have a negligible influence on prices, while the large traders keep their strategic power even when their behavior turns out to be Walrasian in the cooperative framework considered by Gabszewicz and Mertens (1971) and Shitovitz (1973). …”

“…The condition which requires that the atoms are not "too" big, introduced by Gabszewicz and Mertens (1971), is not necessary for the equivalence between the core and the set of Walras allocations, as shown by Theorem 3, but it is sufficient for this equivalence, by Theorem 2; moreover, it is neither necessary nor sufficient for a nonempty intersection between the sets of Walras and Cournot-Nash allocations as shown, respectively, by Examples 7 and 4. The condition which requires that there are only atoms of the same type, introduced by Shitovitz (1973), is not necessary for the equivalence between the core and the set of Walras allocations, as shown by Theorem 2, but it is sufficient for this equivalence, by Theorem 3; moreover, it is neither necessary nor sufficient for a nonempty intersection between the sets of Walras and Cournot-Nash allocations as shown, respectively, by Examples 6 and 5. Theorem 4 states that the condition which characterizes the nonempty intersection of the sets of Walras and Cournot-Nash allocations requires that each atom demands a null amount of one commodity.…”

“…Gabszewicz and Mertens (1971) and Shitovitz (1973) showed that an equivalence between the core and the set of Walras allocations may hold even when the space of traders contains atoms. In order to state their two main theorems, we need to introduce some further notation and definitions.…”

“…Some years later, Gabszewicz and Mertens (1971) and Shitovitz (1973) introduced the notion of a mixed exchange economy, i.e., an exchange economy with large traders, represented as atoms, and small traders, represented by an atomless part, in order to analyze oligopoly in a general equilibrium framework. Gabszewicz and Mertens (1971) showed that, if atoms are not "too" big, the core still coincides with the set of Walras allocations whereas Shitovitz (1973), in his Theorem B, proved that this proving a theorem which states that demanding a null amount of one of the two commodities by all the atoms is a necessary and sufficient condition for a CournotNash allocation to be a Walras allocation. We also provide four examples which show that this characterization theorem is non-vacuous.…”

A price-setting game with a nonatomic fringe

“…Mixed measure theoretic models have been considered, for instance, by Gabszewicz and Mertens (1971), Shitovitz (1973), and Okuno, Postlewaite, and Roberts (1980) in a general equilibrium framework. Sadanand and Sadanand (1996) used in their analysis a partial equilibrium model containing a dominant firm and a nonatomic competitive fringe in order to investigate the timing of quantity-setting oligopoly games.…”