<p>Acemoglu and Jensen (2013) show that this form of profit functions belongs to the class of &quot;aggregative games&quot;. It does not require that the cost function is identical for all firms (just as in Novsheck). In aggregative games, &quot;each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players&quot;. Maybe is this reference useful to you:</p>
<blockquote>
<p>Acemoglu, Daron , and Martin Kaae Jensen, 2013, Aggregate comparative statics, <em>Games and Economic Behavior</em>, 81, 27-49,
<a href="https://doi.org/10.1016/j.geb.2013.03.009" rel="nofollow noreferrer">https://doi.org/10.1016/j.geb.2013.03.009</a>.</p>
</blockquote>

CC BY-SA 4.0
2024-09-02T01:55:27.690Z
This is not a characterization in the sense I intended it to be because not all aggregative games arise from Cournot competitions (right?).
Pteromys
CC BY-SA 4.0
2024-09-02T10:42:04.370Z
@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.
Bertrand
CC BY-SA 4.0
2024-08-30T11:50:06.530Z
2024-08-30T11:50:06.530Z
Bertrand
<p>I would like to know what exactly are the functions of the form
<span class="math-container">$$
\pi(x, y) = P(x+y)x - C(x),
$$</span>
i.e., the profit functions arising from symmetric Cournot duopolies with inverse demand functions <span class="math-container">$P$</span> and cost functions <span class="math-container">$C$</span>, where <span class="math-container">$P$</span> and <span class="math-container">$C$</span> satisfy very mild conditions (<span class="math-container">$P$</span> is nonincreasing where its values are positive, and <span class="math-container">$C$</span> is nondecreasing, but I want to assume as few analytic properties as possible).</p>
<p>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.</p>
<p>Has the class of functions of the form above been characterized in the literature for some mild assumptions on <span class="math-container">$P$</span> and <span class="math-container">$C$</span>?</p>

<microeconomics><game-theory><reference-request><theory><oligopoly>
Characterization of profit functions arising from Cournot competitions
Pteromys
# Characterization of profit functions arising from Cournot competitions

(Asked by: username_0 on 2024-08-30)

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 $C$?

**Tags:** microeconomics, game-theory, reference-request, theory, oligopoly

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## 1 answers

### Answer by username_1 on 2024-08-30. Score: 1

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>.

> **Comments:**
> **username_0**: This is not a characterization in the sense I intended it to be because not all aggregative games arise from Cournot competitions (right?).  
> **username_1**: @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.  

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