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 (Econometrica 41:467–501, 1973), that a Cournot–Nash allocation is Pareto optimal if and only if it is a Walras allocation.
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Notes
The symbol 0 denotes the origin of \(R^2_+\) as well as the real number zero: no confusion will result.
\(\text{ card }(A)\) denotes the cardinality of a set A.
In this paper, differentiability means continuous differentiability and is to be understood to include the case of infinite partial derivatives along the boundary of the consumption set (for a discussion of this case, see, for instance, Kreps (2012), p. 58).
In order to save in notation, with some abuse we denote by \(\mathbf{x}\) both the function \(\mathbf{x}(t)\) and the function \(\mathbf{x}(t,\mathbf{b}(t),p(\mathbf{b}))\).
\(\Vert \cdot \Vert \) denotes the Euclidean norm and \(e^j\) denotes the vector whose \(j\hbox {th}\) coordinate is 1 and whose other coordinate vanishes.
For a discussion of this literature, see Giraud (2003), p. 359 and p. 365.
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We would like to thank two anonymous referees for their comments and suggestions. Francesca Busetto and Giulio Codognato gratefully acknowledge financial support from PRID2018-2-DIES005.
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Busetto, F., Codognato, G., Ghosal, S. et al. Existence and optimality of Cournot–Nash equilibria in a bilateral oligopoly with atoms and an atomless part. Int J Game Theory 49, 933–951 (2020). https://doi.org/10.1007/s00182-020-00719-z
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DOI: https://doi.org/10.1007/s00182-020-00719-z