Question

Apply reaction (or best-response) functions to identify optimal decisions and likely competitor responses in oligopoly settings.

Short answer

The optimum response of the Firm 1 is to produce 20 units whereas the competitior firm will respond by producing 18 units.

Step-by-step solution

Definition of Oligopoly

It refers to the market structure where there are only a few firms in the market and are comparatively larger than the market itself.

Let us understand the optimal decision and competitor responses through Cournot Duopoly Model.

First, consider the market demand and market cost conditions for each firm.

The inverse demand function is given as,

$\mathrm{P}\text{=}200\text{-}3({\mathrm{Q}}_1+{\mathrm{Q}}_2)$

The cost functions are which is identical for all firms is,

For firm 1:

$\mathrm{C}1\left({\mathrm{Q}}_1\right)=26{\mathrm{Q}}_1$

For firm 2:

${\mathrm{C}}_2\left({\mathrm{Q}}_2\right)=\text{32}{\mathrm{Q}}_2$

Cournot Model

The profit function for is given by

$\begin{array}{c}{\mathrm{\pi }}_1=(\mathrm{P}\times {\mathrm{Q}}_1)-{\mathrm{C}}_1\left({\mathrm{Q}}_1\right)\\ =\left[200\text{-}3({\mathrm{Q}}_1+{\mathrm{Q}}_2)\right]{\mathrm{Q}}_1-26{\mathrm{Q}}_1\\ =200{\mathrm{Q}}_1-3{\mathrm{Q}}_1^2-3{\mathrm{Q}}_1{\mathrm{Q}}_2-26\mathrm{Q}{}_1\\ {\mathrm{\pi }}_1=174{\mathrm{Q}}_1-3{\mathrm{Q}}_1^2-3{\mathrm{Q}}_1{\mathrm{Q}}_2\end{array}$

$\begin{array}{c}{\mathrm{\pi }}_2=(\mathrm{P}\times {\mathrm{Q}}_2)-{\mathrm{C}}_2\left({\mathrm{Q}}_2\right)\\ =\left[200\text{-}3({\mathrm{Q}}_1+{\mathrm{Q}}_2)\right]{\mathrm{Q}}_2-32{\mathrm{Q}}_2\\ =200{\mathrm{Q}}_2-3{\mathrm{Q}}_1{\mathrm{Q}}_2-3{\mathrm{Q}}_2^2-32\mathrm{Q}{}_2\\ {\mathrm{\pi }}_2=168{\mathrm{Q}}_2-3{\mathrm{Q}}_1{\mathrm{Q}}_2-3{\mathrm{Q}}_2^2\end{array}$

Maximizing Profits:

$\begin{array}{c}\frac{\partial {\pi }_1}{\partial Q_1}=174-6Q_1-3Q_2\\ \frac{\partial {\pi }_1}{\partial Q_1}=0\\ 174-6Q_1-3Q_2=0\\ 174-3Q_2=6Q_1\\ 29-\frac{1}{2}{Q}_2=Q_1\end{array}$

$\begin{array}{c}\frac{\partial {\pi }_2}{\partial Q_2}=168-3Q_1-6Q_2\\ \frac{\partial {\pi }_2}{\partial Q_2}=0\\ 168-3Q_1-6Q_2=0\\ 168-3Q_1=6Q_2\\ 28-\frac{1}{2}{Q}_1=Q_2\end{array}$

The reaction functions are,

$\begin{array}{c}{Q}_1=r_1\left(Q_2\right)\\ =29-\frac{1}{2}{Q}_2\\ Q_2=r_2\left(Q_1\right)\\ =28-\frac{1}{2}{Q}_1\end{array}$

Computing both the reaction functions,

$\begin{array}{c}{Q}_1=29-\frac{1}{2}\left(28-\frac{1}{2}{Q}_1\right)\\ Q_1=29-14+\frac{1}{4}{Q}_1\\ Q_1=15+\frac{1}{4}{Q}_1\\ Q_1-\frac{1}{4}{Q}_1=15\\ 3Q_1=60\\ Q_1=20\end{array}$

$\begin{array}{c}{\mathrm{Q}}_2={\mathrm{r}}_2\left({\mathrm{Q}}_1\right)\\ =28-\frac{1}{2}{\mathrm{Q}}_1\\ =28-\frac{1}{2}\times 20\\ =28-10\\ =18\end{array}$

Therefore the profits are:

$\begin{array}{c}{\mathrm{\pi }}_1=174{\mathrm{Q}}_1-3{\mathrm{Q}}_1^2-3{\mathrm{Q}}_1{\mathrm{Q}}_2\\ =174\left(20\right)-3{\left(20\right)}^2-3(20\times 18)\\ =3480-1200-1080\\ =3480-2280\\ =1200\end{array}$

$\begin{array}{c}{\mathrm{\pi }}_2=168{\mathrm{Q}}_2-3{\mathrm{Q}}_1{\mathrm{Q}}_2-3{\mathrm{Q}}_2^2\\ =168\left(18\right)-3(18\times 20)-3{\left(18\right)}^2\\ =3024-1080-972\\ =3024-2052\\ =972\end{array}$

Hence, it can be concluded that in Cournot duopoly model Firm 1 has achieved $1200 of profit which is comparatively more than Firm 2’s profit of $972, where Firm 1’s optimum output is 20 and Firm2’s optimum output is 18.