91Ó°ÊÓ

The Nash equilibrium will be obtained at the interaction of both the firm’s reaction curves.

The reaction curve of firm 1 is calculated below:

$$\begin{gathered} {\mathrm{Ï€}}_1={\mathrm{P}}_1{\mathrm{Q}}_1-{\mathrm{C}}_1 \\ =20{\mathrm{P}}_1-{\mathrm{P}}_1^2-{\mathrm{P}}_1{\mathrm{P}}_2-0 \\ \frac{{\mathrm{»åÏ€}}_1}{{\mathrm{dP}}_1}=20-2{\mathrm{P}}_1+{\mathrm{P}}_2=0 \\ 20-2{\mathrm{P}}_1+{\mathrm{P}}_2=0 \\ 2{\mathrm{P}}_1=20+{\mathrm{P}}_2 \\ {\mathrm{P}}_1=10+0.5{\mathrm{P}}_2 \end{gathered}$$

The reaction curve of firm 2 is calculated below:

$$\begin{gathered} {\mathrm{Ï€}}_2={\mathrm{P}}_2{\mathrm{Q}}_2-{\mathrm{C}}_2 \\ =20{\mathrm{P}}_2-{\mathrm{P}}_2^2+{\mathrm{P}}_1{\mathrm{P}}_2-0 \\ \frac{{\mathrm{»åÏ€}}_2}{{\mathrm{dP}}_2}=20-2{\mathrm{P}}_2+{\mathrm{P}}_1=0 \\ 20-2{\mathrm{P}}_2+{\mathrm{P}}_1=0 \\ 2{\mathrm{P}}_2=20+{\mathrm{P}}_1 \\ {\mathrm{P}}_2=10+0.5{\mathrm{P}}_1 \end{gathered}$$

From both the reaction curve,

$$\begin{gathered} {\mathrm{P}}_1=10+0.5\left(10+0.5{\mathrm{P}}_1\right) \\ =10+5+0.25{\mathrm{P}}_1 \\ 0.75{\mathrm{P}}_1=15 \\ {\mathrm{P}}_1=\$20 \\ {\mathrm{P}}_2=10+0.5\left(20\right) \\ =10+10 \\ =\$20 \end{gathered}$$

Each firm will charge $20.

The output of the firm is calculated below:

$$\begin{gathered} Q_1=20-20+20=20 \\ {Q}_2=20+20-20=20 \end{gathered}$$

Each firm will produce 20 units.

The profit of each firm is calculated below:

$$\begin{gathered} {\mathrm{π}}_1=\left(20×20\right)-0=\$400 \\ {\mathrm{π}}_2=\left(20×20\right)-0=\$400 \end{gathered}$$

Each firm profit will be $400.

Firm 1 is the leading firm and sets the price; thus, it takes the reaction curve of firm 2 into account while selecting the price. Therefore, firm 1 price will be:

$$\begin{gathered} {\mathrm{Ï€}}_1=20{\mathrm{P}}_1-{\mathrm{P}}_1^2+{\mathrm{P}}_1\left(10+0.5{\mathrm{P}}_1\right)-0 \\ =30{\mathrm{P}}_1-0.5{\mathrm{P}}_1^2 \\ \frac{{\mathrm{»åÏ€}}_1}{{\mathrm{dP}}_1}=30-{\mathrm{P}}_1=0 \\ {\mathrm{P}}_1=\$30 \end{gathered}$$

Firm 2 price will be:

$P_2=10+0.5\left(30\right)=\$25$

The price charged by firm 1 will be $30, and by firm 2 will be $25.

The quantity of both firms is calculated below:

$$\begin{gathered} {\text{Q}}_{\text{1}}\text{= 20 - 30 + 25} \\ \text{= 15} \\ {\text{Q}}_{\text{2}}\text{= 20 + 30 - 25} \\ \text{= 25} \end{gathered}$$

Firm 1 will produce 15 units, and firm 2 will be 25 units.

The profit of both the firm will be:

$$\begin{gathered} {\mathrm{π}}_1=\left(30×15\right)-0=\$450 \\ {\mathrm{π}}_2=\left(25×25\right)-0=\$625 \end{gathered}$$

The profit of firm 1 will be $450, and firm 2 will be $625.

The nash equilibrium profit will be $400. The profit of the leader firm will be $450, and for the follower firm, it will be $625. Thus, option (iii) will be preferred as it gives higher profit.