Collusion 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 duopolist collide with each other and try tom maximize the total profit instead of individual profit, then, they will distribute the profit among themselves according to the output which they produce.

Thus, this will maximize $\Pi ={\Pi }_1+{\Pi }_2$

$$=R_1-C_1+R_2-C_2$$

w.r.t $q_1$ and $q_2$ for which F.O.C are

$$\frac{\partial \Pi }{\partial q_1}=0\frac{\partial \Pi }{\partial q_2}=0$$

Solving them, we get individual output $q_1$ and $q_2$ and from which we get individual profits ${\Pi }_1$ and ${\Pi }_2$.

$$\begin{array}{rl}{\Pi }_1& =100q_1-0.5q_1(q_1+q_2)-5q_1\\ {\Pi }_2& =100q_2-0.5q_2(q_1+q_2)-0.5q_2^2\\ \Pi ={\Pi }_1+{\Pi }_2& =100(q_1+q_2)-0.5(q_1+q_2{)}^2-5q_1-0.5q_2^2\end{array}$$

$\therefore$ F.O.C

$$\frac{\partial \Pi }{\partial q_1}=0\frac{\partial \Pi }{\partial q_2}=0$$ $$\begin{array}{rl}& 100-(q_1+q_2)-5\\ ⟹& q_1+q_2=95\\ & \\ & 100-(q_1+q_2)-q_2=0\\ ⟹& q_1+2q_2=100\end{array}\text{\hspace{1em}(1)(2)}$$

From (1) and (2), we get $q_2=5$ and $\therefore q_1=90$

Now, we calculate the profits:

$$\begin{array}{rlr}P& =100-0.5\times 95& =52.5\\ {\Pi }_1& =52.5\times 90-5\times 90& =4275\\ {\Pi }_2& =52.5\times 5-0.5\times 25& =250\end{array}$$

P = 100 - 0.5(95)
q1 = 90
q2 = 5
P1 = P*q1 - 5q1
P2 = P*q2 - 0.5q2^2