Question

Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, C(q) = 40q. Assume that the demand curve for the industry is given by P = 100 - Q and that each firm expects the other to behave as a Cournot competitor.

1. Calculate the Cournot-Nash equilibrium for each firm, assuming that each chooses the output level that maximizes its profits when taking its rival’s output as given. What are the profits of each firm?
2. What would be the equilibrium quantity if Texas Air had constant marginal and average costs of \(25 and American had constant marginal and average costs of \)40?
3. Assuming that both firms have the original cost function, C(q) = 40q, how much should Texas Air be willing to invest to lower its marginal cost from 40 to 25, assuming that American will not follow suit? How much should American be willing to spend to reduce its marginal cost to 25, assuming that Texas Air will have marginal costs of 25 regardless of American’s actions?

Short answer

1. Each firm produces 20 units at $60. The profit of each firm will be $400.
2. The equilibrium quantity will be 45 units.
3. Texas Air will be willing to spend $500 to reduce the marginal cost to $25. Americans will be willing to spend $400 to reduce the marginal cost to $25.

Step-by-step solution

Explanation for part (a)

Texas is considered firm 1, and America is considered to be firm 2.

The Cournot-Nash equilibrium is calculated below:

The firm 1 reaction curve will be;

$$\begin{gathered} P=100-Q_1-Q_2 \\ C=40Q \\ {\mathrm{\pi }}_1={\mathrm{TR}}_1-{\mathrm{C}}_1 \\ =100{\mathrm{Q}}_1-{\mathrm{Q}}_1^2-{\mathrm{Q}}_1{\mathrm{Q}}_2-40{\mathrm{Q}}_1 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dQ}}_1}=100-2{\mathrm{Q}}_1-{\mathrm{Q}}_2-40=0 \\ 60-2{\mathrm{Q}}_1-{\mathrm{Q}}_2=0 \\ {\mathrm{Q}}_1=\frac{60-{\mathrm{Q}}_2}{2} \end{gathered}$$

The firm 2 reaction curve will be:

The marginal cost of both firms is the same; thus, the reaction curve of firm 2 will be the same as firm 1. Thus, the firm 2 reaction curve will be ${\mathrm{Q}}_2=\frac{60-{\mathrm{Q}}_1}{2}$.

The equilibrium output is calculated below:

role="math" localid="1644240743037" $$\begin{gathered} Q_1=\frac{60-\left(\frac{60-{\mathrm{Q}}_1}{2}\right)}{2} \\ 4Q_1=60+Q_1 \\ 3Q_1=60 \\ {Q}_1=20 \\ {Q}_2=\frac{60-20}{2}=20 \\ {Q}_T=Q_1+Q_2 \\ =20+20 \\ =40 \\ P=100-40 \\ =\$60 \end{gathered}$$

The equilibrium output will be 40 units at $60.

The profit of each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(60\times 20\right)-\left(40\times 20\right) \\ =1200-800 \\ =\$400 \\ {\mathrm{\pi }}_2=\left(60\times 20\right)-\left(40\times 20\right) \\ =1200-800 \\ =\$400 \end{gathered}$$

The profit of each firm will be $400.

Explanation for part (b)

The reaction curve of firm 1 is calculated below:

$$\begin{gathered} P=100-Q_1-Q_2 \\ C=25Q \\ {\mathrm{\pi }}_1={\mathrm{TR}}_1-{\mathrm{C}}_1 \\ =100{\mathrm{Q}}_1-{\mathrm{Q}}_1^2-{\mathrm{Q}}_1{\mathrm{Q}}_2-25{\mathrm{Q}}_1 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dQ}}_1}=100-2{\mathrm{Q}}_1-{\mathrm{Q}}_2-25=0 \\ 75-2{\mathrm{Q}}_1-{\mathrm{Q}}_2=0 \\ {\mathrm{Q}}_1=\frac{75-{\mathrm{Q}}_2}{2} \end{gathered}$$

The firm 2 reaction curve is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_2={\mathrm{TR}}_2-{\mathrm{C}}_2 \\ =100{\mathrm{Q}}_2-{\mathrm{Q}}_2^2-{\mathrm{Q}}_1{\mathrm{Q}}_2-40{\mathrm{Q}}_2 \\ \frac{{\mathrm{d\pi }}_2}{{\mathrm{dQ}}_2}=100-2{\mathrm{Q}}_2-{\mathrm{Q}}_1-40=0 \\ 60-2{\mathrm{Q}}_2-{\mathrm{Q}}_1=0 \\ {\mathrm{Q}}_2=\frac{60-{\mathrm{Q}}_1}{2} \end{gathered}$$

The equilibrium output is calculated below:

$$\begin{gathered} Q_1=\frac{75-\left(\frac{60-{\mathrm{Q}}_1}{2}\right)}{2} \\ 4Q_1=150-60+Q_1 \\ 3Q_1=90 \\ {Q}_1=30 \\ {Q}_2=\frac{60-30}{2}=15 \\ {Q}_T=Q_1+Q_2 \\ =30+15 \\ =45 \end{gathered}$$

The equilibrium output will be 45 units.

Explanation for part (c)

The profit of Texas when marginal cost is $25. The profit will be

When marginal cost is $40, then profit is $400, and when marginal cost is $25, the profit will be $900; thus, the difference will be $500. Hence, Texas should be willing to invest $500 to lower the marginal cost from $40 to $25.

The profit for Americans before the investment is calculated below:

The profit will be $225.

The profit after the investment is calculated below:

The reaction curve of firm1 and firm 2 will be:

The output of firm 1, firm 2, and the price will be:

When marginal cost is $40, then profit is $225, and when marginal cost is $25,the profit will be $625; thus, the difference will be $400. Hence, Americans should invest $400 to lower the marginal cost from $40 to $25.