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I would like to know what exactly are the functions of the form $$ \pi(x, y) = P(x+y)x - C(x), $$ i.e., the profit functions arising from symmetric Cournot duopolies with inverse demand functions $P$ and cost functions $C$, where $P$ and $C$ satisfy very mild conditions ($P$ is nonincreasing where its values are positive, and $C$ is nondecreasing, but I want to assume as few analytic properties as possible).

Existence of Cournot equilibria has been studied under very general settings, and I imagined that those studies include answers to my question, but going through some works by Amir, McManus and Vivek didn't lead me to a success.

Has the class of functions of the form above been characterized in the literature for some mild assumptions on $P$ and $C$?

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Acemoglu and Jensen (2013) show that this form of profit functions belongs to the class of "aggregative games". It does not require that the cost function is identical for all firms (just as in Novsheck). In aggregative games, "each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players". Maybe is this reference useful to you:

Acemoglu, Daron , and Martin Kaae Jensen, 2013, Aggregate comparative statics, Games and Economic Behavior, 81, 27-49, https://doi.org/10.1016/j.geb.2013.03.009.

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  • $\begingroup$ This is not a characterization in the sense I intended it to be because not all aggregative games arise from Cournot competitions (right?). $\endgroup$
    – Pteromys
    Commented Sep 2 at 1:55
  • $\begingroup$ @Pteromus: yes, Cournot competition is a special case of aggregative games, and so, all results obtained in their paper apply to Cournot competition. And there is an explicit section about Cournot competition in their paper. $\endgroup$
    – Bertrand
    Commented Sep 2 at 10:42

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