Atomic Cournotian traders may be Walrasian

Abstract

In a bilateral oligopoly, with large traders, represented as atoms, and small traders, represented by an atomless part, when is there a non-empty intersection between the sets of Walras and Cournot-Nash allocations? Using a two-commodity version of the Shapley window model, we show that a necessary and sufficient condition for a Cournot-Nash allocation to be a Walras allocation is that all atoms demand a null amount of one of the two commodities. 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).

Existence and optimality of Cournot–Nash equilibria in a bilateral oligopoly with atoms and an atomless part

“…Suppose thatx 1 (t) = 0 orx 2 (t) = 0, for each t ∈ T 1 . Then, the pair (p,x) is a Walras equilibrium, by Theorem 4 in Codognato et al (2015). But then, x is Pareto optimal, by Theorem 2.…”

“…This result establishes a relationship among the Cournotian tradition of oligopoly, the Walrasian tradition of perfect competition, and the Paretian analysis of optimality. Some examples computed by Codognato et al (2015) provide evidence that this characterization holds non-vacuously.…”

“…Then, the pair (p,x) is a Walras equilibrium, by Theorem 2. But then,x 1 (t) = 0 orx 2 (t) = 0, for each t ∈ T 1 , by Theorem 4 in Codognato et al (2015). Suppose thatx 1 (t) = 0 orx 2 (t) = 0, for each t ∈ T 1 .…”

“…Borrowing from Codognato et al (2015), we introduce now the two-commodity version of the Shapley window model. A strategy correspondence is a correspondence B :…”

“…The next proposition provides a characterization of Pareto optimal Cournot-Nash allocations. To prove it, we use a result obtained in Codognato et al (2015) which provides a necessary and sufficient condition for a Cournot-Nash allocation to be a Walras allocation. This characterization result requires the following assumption.…”