<p>I think the missing link for you is not realizing that <span class="math-container">$Q$</span> is actually equal to <span class="math-container">$q_1 + q_2$</span>: in a duopoly, market quantity demanded can only be derived from the two firms in question. With that in mind, we can rewrite the profit functions for both firms in terms of <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span> and then optimize with respect to the <span class="math-container">$q$</span>'s accordingly (since Cournot competition is competition over quantities). We are then going to be left with two <em>best response</em> functions, one that tells you the optimal level of <span class="math-container">$q_1$</span> as a function of <span class="math-container">$q_2$</span> and another that tells you the optimal level of <span class="math-container">$q_2$</span> as a function of <span class="math-container">$q_1$</span>. 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.</p>
<hr />
<p>Firm 1 maximizes profits
<span class="math-container">\begin{align*}
\Pi_1(q_1) = (\alpha - (q_1 + q_2))q_1 - c_1 q_1.
\end{align*}</span></p>
<p>The first-order condition w.r.t. <span class="math-container">$q_1$</span> is
<span class="math-container">\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*}</span></p>
<p>Similarly, Firm 2 maximizes profits
<span class="math-container">\begin{align*}
\Pi_2(q_2) = (\alpha - (q_1 + q_2))q_2 - c_2 q_2.
\end{align*}</span></p>
<p>The first-order condition w.r.t. <span class="math-container">$q_2$</span> is
<span class="math-container">\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*}</span></p>
<p>Solving the system of equations for <span class="math-container">$q_1$</span> and <span class="math-container">$q_2$</span> yields
<span class="math-container">\begin{align*}
q_1^* &amp;= \frac{\alpha - 2c_1 + c_2}{3} \\
q_2^* &amp;= \frac{\alpha - 2c_2 + c_1}{3}, 
\end{align*}</span></p>
<p>which implies that <span class="math-container">$\mathbf{q_1^* &gt; q_2^*}$</span> since <span class="math-container">$c_1 &lt; c_2$</span>.</p>
<p>Note that both firms face the same price <span class="math-container">$P^* = \alpha - (q^*_1 + q^*_2)$</span>. Therefore, we have that
<span class="math-container">\begin{align*}
(P^* - c_1)q_1^* &amp;&gt; (P^* - c_2)q_2^* \\
\implies \mathbf{\Pi^*_1} &amp;&gt; \mathbf{\Pi^*_2}.
\end{align*}</span></p>

CC BY-SA 4.0
2018-10-16T04:05:17.137Z
2022-12-18T07:48:26.770Z
2022-12-18T07:48:26.770Z
Kenneth Rios
<p>There are two firms in a Cournot duopoly that face inverse demand <span class="math-container">$P = \alpha - Q$</span>, but one firm has total costs <span class="math-container">$c_1*q_1$</span> and the other has total costs <span class="math-container">$c_2*q_2$</span> with <span class="math-container">$c_1 &lt; c_2$</span>. I want to show that firm 1 will have greater profits and produce a greater share of market output than firm 2 in equilibrium.</p>

<p>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: <span class="math-container">$Profit_1 = (\alpha - Q)q_1 - c_1*q_1$</span> and <span class="math-container">$Profit_2 = (\alpha - Q)q_2 - c_2 * q_2$</span>, but taking derivatives with respect to the q's doesn't get me anywhere. Would appreciate some help</p>

<microeconomics><game-theory>
Cournot duopoly with differing costs
Megan Hart
# Cournot duopoly with differing costs

(Asked by: username_0 on 2018-10-16)

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

**Tags:** microeconomics, game-theory

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## 1 answers

### Answer by username_1 on 2018-10-16. Score: 4 | Accepted answer

I think the missing link for you is not realizing that $Q$ is actually equal to $q\_1 + q\_2$: in a duopoly, market quantity demanded can only be derived from 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.

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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 w.r.t. $q\_1$ 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 w.r.t. $q\_2$ 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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