<p>Let the profit functions of the two firms be 
$$
\begin{aligned}
\pi_1(q_1,q_2)&amp;=p(q_1,q_2)q_1-C_1(q_1)\\
\pi_2(q_1,q_2)&amp;=p(q_1,q_2)q_2-C_2(q_2)
\end{aligned}
$$
where $p(q_1,q_2)$ is the inverse demand depending on the total output of the two firms $q_1+q_2$, and $C_i$ is the total cost function of firm $i\in\{1,2\}$. Thus $C_i'$ is the marginal cost (MC) of firm $i$, and we can impose the condition that $C_1'\ne C_2'$ on the cost functions so that the firms have different MCs. </p>

<p>Now we can solve for the Nash equilibrium (NE). Profit maximization would yield the following first-order conditions for the two firms:
$$
\begin{aligned}
p'_1(q_1^*,q_2)q_1^*+p(q_1^*,q_2)-C_1'(q_1^*)&amp;=0\qquad\qquad(1)\\
p'_2(q_1,q_2^*)q_2^*+p(q_1,q_2^*)-C_2'(q_2^*)&amp;=0\qquad\qquad(2)
\end{aligned}
$$
These are the usual MR=MC condition (<em>for each firm</em>). Solving for $q_1^*(q_2)$ in $(1)$, we get the <strong>best response of Firm 1</strong> as a function of Firm 2's output $q_2$. Likewise, from $(2)$ we get <strong>Firm 2's best response</strong> to Firm 1's output, $q_2^*(q_1)$. </p>

<p>Recall the a NE in a two-player game is <strong>a pair of <em>mutual best responses</em></strong>. Therefore the NE of this two-firm Cournot game with different MCs is the pair $(q_1^*(q_2^*),q_2^*(q_1^*))$, where each firm is best responding to the other firm's best response (to the former firm). Practically, this means to plug $q_2^*$ into $q_1^*(q_2)$ and solve for $q_1^*$, and then plug $q_1^*$ back into $q_2^*(q_1)$ and solve for $q_2^*$. </p>

CC BY-SA 3.0
2016-05-24T17:20:34.697Z
2016-05-24T17:20:34.697Z
Herr K.
<p>When MC function is different for both the firms, how will MR = MC work ?</p>

<nash-equilibrium>
how to find Cournot equilibrium for 2 firms having different MC?
ADubey
# how to find Cournot equilibrium for 2 firms having different MC?

(Asked by: username_0 on 2016-05-24)

When MC function is different for both the firms, how will MR = MC work ?

**Tags:** nash-equilibrium

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## 1 answers

### Answer by username_1 on 2016-05-24. Score: 0 | Accepted answer

Let the profit functions of the two firms be
$$
\begin{aligned}
\pi\_1(q\_1,q\_2)&=p(q\_1,q\_2)q\_1-C\_1(q\_1)\\
\pi\_2(q\_1,q\_2)&=p(q\_1,q\_2)q\_2-C\_2(q\_2)
\end{aligned}
$$
where $p(q\_1,q\_2)$ is the inverse demand depending on the total output of the two firms $q\_1+q\_2$, and $C\_i$ is the total cost function of firm $i\in\{1,2\}$. Thus $C\_i'$ is the marginal cost (MC) of firm $i$, and we can impose the condition that $C\_1'\ne C\_2'$ on the cost functions so that the firms have different MCs.

Now we can solve for the Nash equilibrium (NE). Profit maximization would yield the following first-order conditions for the two firms:
$$
\begin{aligned}
p'\_1(q\_1^\*,q\_2)q\_1^\*+p(q\_1^\*,q\_2)-C\_1'(q\_1^\*)&=0\qquad\qquad(1)\\
p'\_2(q\_1,q\_2^\*)q\_2^\*+p(q\_1,q\_2^\*)-C\_2'(q\_2^\*)&=0\qquad\qquad(2)
\end{aligned}
$$
These are the usual MR=MC condition (*for each firm*). Solving for $q\_1^\*(q\_2)$ in $(1)$, we get the **best response of Firm 1** as a function of Firm 2's output $q\_2$. Likewise, from $(2)$ we get **Firm 2's best response** to Firm 1's output, $q\_2^\*(q\_1)$.

Recall the a NE in a two-player game is **a pair of *mutual best responses***. Therefore the NE of this two-firm Cournot game with different MCs is the pair $(q\_1^\*(q\_2^\*),q\_2^\*(q\_1^\*))$, where each firm is best responding to the other firm's best response (to the former firm). Practically, this means to plug $q\_2^\*$ into $q\_1^\*(q\_2)$ and solve for $q\_1^\*$, and then plug $q\_1^\*$ back into $q\_2^\*(q\_1)$ and solve for $q\_2^\*$.


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