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

Abstract

We consider a bilateral oligopoly version of the Shapley window model with large traders, represented as atoms, and small traders, represented by an atomless part. For this model, we provide a general existence proof of a Cournot-Nash equilibrium that allows one of the two commodities to be held only by atoms. Then, we show, using a corollary proved by Shitovitz (1973), that a Cournot-Nash allocation is Pareto optimal if and only if it is a Walras allocation.

An introduction to perfect and imperfect competition via bilateral oligopoly

“…It is important to remark that asymmetric oligopoly can be studied by using the partial replica or by considering mixed exchange economies. This alternative approach was introduced by Okuno et al (1980) and further scrutinised by Busetto et al (2008Busetto et al ( , 2011Busetto et al ( , 2017 and Busetto et al (2018Busetto et al ( , 2020.…”

“…This approach based on mixed exchange economies was also extended to noncooperative game theory by Okuno et al (1980), in order to address their critique, and further generalised by Busetto et al (2011) and Busetto et al (2018). Busetto et al (2020) undertake a study of bilateral oligopoly in a mixed exchange economy. In this setting, while all agents have a priori the same strategic position, they found that in equilibrium large agents represented by atoms have market power while small agents on the continuum behave as if they are price-takers.…”

Noncooperative oligopoly equilibrium in markets with hierarchical competition

“…They show that gross substitutes imply uniqueness of equilibrium. More recently, Busetto et al (2020) show the existence of a Cournot-Nash equilibrium for the mixed bilateral oligopoly version of the Shapley window model, i.e. with atoms and an atomless part.…”

Atomic Leontievian Walrasian traders are always Cournotian

“…In this paper, we consider the mixed version of this model introduced by Codognato et al (2015) and further analyzed by Busetto et al (2018b): a mixed exchange economy à la Shitovitz (1973) is studied, where large traders are represented as atoms and small traders are represented by an atomless part; noncooperative exchange is formalized as in the Shapley window model, a strategic market game with complete markets which was first proposed informally by Lloyd S. Shapley and further studied by Sahi and Yao (1989), Codognato and Ghosal (2000), Busetto et In this framework, Codognato et al (2015) showed a theorem establishing that, under the assumptions that all traders' utility functions are continuous, strongly monotone, quasi-concave, and measurable, and atoms' utility functions are also differentiable, 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.…”

“…The strategy selection b * such that b * 12 (t) =8 3 , for eacht ∈ T 0 , b * 21 (2) = b * 21 (3) = 8 3 , is a Cournot-Nash equilibrium and x * (t) = x(t, b * (t), p(b * )), for each t ∈ T . Suppose that b * is not the unique Cournot-Nash equilibrium.…”