Cournot Model

A duopoly is a market situation where there are two sellers, for each set of actions and reaction of the duopolist, there corresponds a solution. Thus there is no unique solution for duopoly equilibrium. The solution depends on the actions of one duopolist and the counter of another duopolist.^duopoly-definition

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In this model, the basic assumption is that each duopolist will maximize his profit assuming that his rival’s actions are invariant under his own actions. Thus, duopolist $I$ maximizes ${\Pi }_1$ w.r.t $q_1$ treating $q_2$ as a parameter.

Similarly, duopolist $II$ will maximize ${\Pi }_2$ w.r.t $q_2$ treating $q_1$ as a parameter.

We maximize ${\Pi }_1=R_1-C_1$ w.r.t $q_1$, and thus $\frac{\partial {\Pi }_1}{\partial q_1}=\frac{\partial R_1}{\partial q_1}-\frac{\partial C_1}{\partial q_1}=0$.

Upon solving this, we will get solution for $q_1$ in terms of $q_2$ as $q_1=\Phi (q_2)$.

Thus, the function gives $I$’s output if $II$’s output is known. This will tell how $I$ will react for each action of $II$. Thus, it is called reaction function for $I$ . Similarly, for $II$ , ${\Pi }_2=R_2-C_2$ we get $q_2$.

So, $\frac{\partial {\Pi }_2}{\partial q_2}=\frac{\partial R_2}{\partial q_2}-\frac{\partial C_2}{\partial q_2}=0$

$⟹\frac{\partial R_2}{\partial q_2}=\frac{\partial C_2}{\partial q_2}$ solving which we get the reaction function of $II$ as $q_2=\Phi (q_1)$ which gives output for $II$ for given output of $I$.

If $q_1$ and $q_2$ can be formed such that both the reaction functions are satisfied there there will be Static Equilibrium otherwise there will be disequilibrium.

Example

$$P=100-0.5(q_1+q_2),c_1=5q,c_2=0.5q_2^2$$

We will find the reaction functions for both the firms.

$$\begin{array}{rl}{\Pi }_1& =100q_1-0.5q_1^2-0.5q_1{q}_2-5q_1\\ \frac{\partial {\Pi }_1}{\partial q_1}& =100-q_1-0.5q_2-5\\ & \\ \therefore q_1& =95-0.5q_2\end{array}\text{\hspace{1em}(1)}$$

which is the reaction function of firm $I$.

Similarly,

$$\begin{array}{rl}{\Pi }_2& =100q_2-0.5q_1{q}_2-0.5q_2^2-0.5q_2^2\\ \frac{\partial {\Pi }_2}{\partial q_2}& =100-0.5q_1-2q_2\\ & \\ \therefore q_2& =50-0.25q_1\end{array}\text{\hspace{1em}(2)}$$

which is the reaction function for firm $II$.

Solving (1) and (2), we get $q_1=80$, $q_2=30$, ${\Pi }_1=3200$ and ${\Pi }_2=900$.

q1 = 80
q2 = 30
pi1 = 100*q1 - 0.5*q1^2  - 0.5*q1*q2 - 5*q1
pi2 = 100*q2 - 0.5*q1*q2  - q2^2

Thus there is Coward Equilibrium.