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There are two firms in a Cournot duopoly that face inverse demand $P = \alpha - Q$, but one firm has total costs $c_1*q_1$ and the other has total costs $c_2*q_2$ with $c_1 < c_2$. I want to show that firm 1 will have greater profits and produce a greater share of market output than firm 2 in equilibrium.

Intuitively that makes sense to me because if one firm is more cost efficient they should be able to produce more and capture more of the market, right? I'm just having trouble showing this mathematically. I've gotten to the point of writing down profits for each: $Profit_1 = (\alpha - Q)q_1 - c_1*q_1$ and $Profit_2 = (\alpha - Q)q_2 - c_2 * q_2$, but taking derivatives with respect to the q's doesn't get me anywhere. Would appreciate some help

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I think the missing link for you is not realizing that $Q$ is actually equal to $q_1 + q_2$: in a duopoly, quantity demanded can only be derived for the two firms in question. With that in mind, we can rewrite the profit functions for both firms in terms of $q_1$ and $q_2$ and then optimize with respect to the $q$'s accordingly (since Cournot competition is competition over quantities). We are then going to be left with two best response functions, one that tells you the optimal level of $q_1$ as a function of $q_2$ and another that tells you the optimal level of $q_2$ as a function of $q_1$. Then it just becomes an algebra problem to find the Nash equilibrium and the relationship of quantities and profits between the firms will become clear.


Firm 1 maximizes profits \begin{align*} \Pi_1(q_1) = (\alpha - (q_1 + q_2))q_1 - c_1 q_1. \end{align*}

The first-order condition is \begin{gather*} \frac{d \Pi_1(q_1)}{dq_1} = 0 \\ \implies \alpha - 2q_1 + q_2 = c_1 \\ \implies q_1(q_2) = \frac{\alpha - c_1}{2} + \frac{q_2}{2}. \end{gather*}

Similarly, Firm 2 maximizes profits \begin{align*} \Pi_2(q_2) = (\alpha - (q_1 + q_2))q_2 - c_2 q_2. \end{align*}

The first-order condition is \begin{gather*} \frac{d \Pi_2(q_2)}{dq_2} = 0 \\ \implies \alpha - 2q_2 + q_1 = c_2 \\ \implies q_2(q_1) = \frac{\alpha - c_2}{2} + \frac{q_1}{2}. \end{gather*}

Solving the system of equations for $q_1$ and $q_2$ yields \begin{align*} q_1^* &= \frac{\alpha - 2c_1 + c_2}{3} \\ q_2^* &= \frac{\alpha - 2c_2 + c_1}{3}, \end{align*}

which implies that $\mathbf{q_1^* > q_2^*}$ since $c_1 < c_2$.

Note that both firms face the same price $P^* = \alpha - (q^*_1 + q^*_2)$. Therefore, we have that \begin{align*} (P^* - c_1)q_1^* &> (P^* - c_2)q_2^* \\ \implies \mathbf{\Pi^*_1} &> \mathbf{\Pi^*_2}. \end{align*}

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