Quasi-Competitive Solution

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

Link to original

The market demand of the output of duopolist is a function determined by the total output of both the duopolists. Thus demand function is

$$P=F(q_1=q_2)\text{where}{q}_1\&q_2\text{are output of duopolist}$$

The revenues of the duopolists are given by

$$\begin{array}{rl}{R}_1=p\cdot q_1& =F(q_1+q_2)q_1\\ & =R_1(q_1,q_2)\end{array}$$ $$\begin{array}{rl}{R}_2=p\cdot q_2& =F(q_1+q_2)q_2\\ & =R_2(q_1,q_2)\end{array}$$

Let $c_1(q_1)$ and $c_2(q_2)$ be the cost function:

Then profits are given by

$$\begin{array}{r}{\Pi }_1=R_1-c_1(q_1)\\ {\Pi }_2=R_2-c_2(q_2)\end{array}$$

Here, the basic assumption is that each duopolist maximizes its profit by following the competitive rule $MC=P$

$$\therefore P=F(q_1+q_2)=c_1^{\prime }(q_1)$$ $$P=F(q_1+q_2)=c_2^{\prime }(q_2)$$

By solving these equations, we get $q_1$ and $q_2$.

Substituting them in demand function we get the Price. Upon substitution of quantity in the cost function, we can finally calculate the profits of the firms. This model is called quasi-competitive solution since they are relevant in competition and also monopoly in the behavior of Duopolist.

For example

$P=100-0.5[q_1+q_2]$ be the demand function Let $c_1=5q_1$ ;$c_2=0.5q_2^2$ be the cost function

Find the Max profit for both duopolists

Hint: use $MC=P$ to find $q_1$ and $q_2$ then use $R$ and $c$ to find $\Pi$ for both firms

Solution

By the profit max rule $MC=P$, we get

$$100-0.5q_1-0.5q_2=5⟹q_1+q_2=190$$ $$100-0.5q_1-0.5q_2=q_2⟹q_1+3q_2=200$$

This gives us $q_2=5$ and $q_1=185$

$R_1=p\cdot q_1=100q_1-0.5q_1^2-0.5q_2{q}_1=925$

$R_2=p\cdot q_2=100q_2-0.5q_1{q}_2-0.5q_2^2=25$

So, the max profit is:

${\Pi }_1=925-5\times 185=\boxed{0}$

${\Pi }_2=25-0.5\times 25=\boxed{12.5}$