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If a market is given to be a Cuornot duopoly and Q = q1 + q2. What assumptions can you make about q1 and q2 being equal? Is it possible for q1 > q2 and thus for company 1 to have greater market share in an equilibrium?

Edit: I assume they have the same cost and indistinguishable products. I guess my question could be more simply boiled down to: Are they guaranteed to have inverted reaction functions?

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  • $\begingroup$ Hint: Try solving the problem with $MC_1<MC_2$, namely, firm 1 is more efficient / has a lower marginal cost than firm 2. $\endgroup$
    – Herr K.
    Oct 19, 2022 at 2:11
  • $\begingroup$ @HerrK. yes, oops meant to assume, that. I think I need to focus on the reaction function I am just not sure how to prove they have the same reaction function $\endgroup$ Oct 19, 2022 at 2:23
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    $\begingroup$ Hi, what do you mean by an inverted reaction function? $\endgroup$
    – Rick_Morty
    Oct 19, 2022 at 19:03
  • $\begingroup$ @Citrus I mean would one firms reaction function necessarily be the inverse of the other firms reaction function $\endgroup$ Oct 21, 2022 at 23:25

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Even if firms have identical technologies and identical reaction functions, the Cournot equilibrium can be asymmetric, as shown there:

Amir, Rabah & Garcia, Filomena & Knauff, Malgorzata, 2010. "Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis," Journal of Economic Theory, 145, 1968-1986.

Further restrictions on the reaction functions are necessary to guarantee a symmetric equilibrium. A sufficient condition is that they are identical and linear in $y_i$.

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