Question

Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q1 + Q2. The firms’ cost functions are C1(Q1) = 20 + 10 Q1 and C2(Q2) = 10 + 12 Q2.

1. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?
2. What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves and show the equilibrium.
3. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but a takeover is not?

Short answer

a. The joint profit-maximizing level of output is 4 units.

Firm 1 will produce 4 units, and firm 2 will produce 0 units. If only Firm 1 operates, its profit will be $60, and if only Firm 2 operates, its profit will be $62.2.

Firm 2 would have earned a higher profit than firm 1.

b. Firm 1 output will be 2.8 units, and firm 2 output will be 2.4 units if they do not cooperate. The profit for firm 1 will be $19.20, and for firm 2 will be $18.80.

The reaction curve of both the firm is shown below:

c. Firm 1 will be willing to pay $40.80 to firm 2.

Step-by-step solution

Explanation for part (a)

The joint profit-maximizing level of the firm will be at the level where marginal cost is equal to marginal revenue. The marginal cost of firm 1 will be lower; thus, it will be considered for the output decision. The output and price will be:

$$\begin{gathered} \text{P = 50 - 5Q} \\ {\text{TR = 50Q - 5Q}}^{\text{2}} \\ \text{MR = 50 - 10Q} \\ {\text{C}}_{\text{1}}{\text{= 20 + 10Q}}_{\text{1}} \\ \text{MC = 10} \\ \text{MR = MC} \\ \text{50 - 10Q = 10} \\ \text{10Q = 40} \\ \text{Q = 4} \\ \text{P = 50 - 5}\left(\text{4}\right) \\ \text{= 50 - 20} \\ \text{= \$ 30} \end{gathered}$$

The market output will be 4 units at $30.

As the marginal cost of firm 1 is 10 and the marginal cost of firm 2 is 12; thus, firm 1’s marginal cost is lower than firm 2, then all the output will be produced by firm1, and firm 2 produces 0 units.

The profit of both the firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(30\times 4\right)-20-10\left(4\right) \\ =120-20-40 \\ =\$60 \\ {\mathrm{\pi }}_1=\left(30\times 0\right)-10-12\left(4\right) \\ =\$0 \end{gathered}$$

Firm 1’s profit will be $60, and firm 2’s profit will be $0.

If there is only firm 1, then the profit for firm 1 will be $60, and for firm 2 will be $0.

If only firm 2 will be there, then the profit of firm 2 is calculated below:

$$\begin{gathered} MR=50-10Q_2 \\ C=10+12Q_2 \\ MC=12 \\ MR=MC \\ 50-10Q_2=12 \\ 10Q_2=38 \\ {Q}_2=3.8 \\ P=50-5\left(3.8\right) \\ =50-19 \\ =\$31 \\ {\mathrm{\pi }}_2=\left(31\times 3.8\right)-10-12\left(3.8\right) \\ =117.8-10-45.6 \\ =\$62.2 \end{gathered}$$

Firm 2’s profit will be $62.2. Thus, if each firm operates solely, firm 2 profit will be higher than firm 1.

Explanation for part (b)

If the firms operate in the Cournot model, then the reaction curve for each firm will be:

Firm 1’s reaction curve will be:

Firm 2’s reaction curve will be:

From both the firm reaction curve:

The market price and profit of both the firms are calculated below:

The market price will be $24. The profit of firm 1 will be $19.20, and for firm 2 will be 18.80.

The reaction curve of both the firm is shown below:

The equilibrium output from firm 1 will be 2.8 units, and for firm 2 will be 2.4 units.

Explanation for part (c)

The payment by firm 1 to firm 2 will be the difference of the profit when firm 2 is not present and when the firm is present in the market. When firm 2 is not present, the profit of firm 1 will be $60, and when firm 2 is present in the market, then firm 1 will be $19.20. Thus, firm 1 will be willing to pay $40.80 (=60 – 19.20).