how to find Cournot equilibrium for 2 firms having different MC?

Question

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

Answer 1

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 $q_2^*$.