Question

Two firms produce luxury sheepskin auto seat covers: Western Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by

C(q) = 30q + 1.5q2

The market demand for these seat covers is represented by the inverse demand equation

P = 300 - 3Q

where Q = q1 + q2, total output.

1. If each firm acts to maximize its profits, taking its rival’s output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm?
2. It occurs to the managers of WW and BBBS that they could do a lot better by colluding. If the two firms collude, what will be the profit-maximizing choice of output? The industry price? The output and the profit for each firm in this case?
3. The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. To aid in making the decision, the manager of WW constructs a payoff matrix like the one below. Fill in each box with the profit of WW and the profit of BBBS. Given this payoff matrix, what output strategy is each firm likely to pursue

   PROFIT PAYOFF MAXTRIX (WW PROFIT, BBBS PROFIT)	BBBS
   PRODUCECOURNOT q	PRODUCE CARTEL q
   WW	PRODUCE COURNOT q
   PRODUCE CARTEL q

d. Suppose WW can set its output level before BBBS does. How much will WW choose to produce in this case? How much will BBBS produce? What is the market price, and what is the profit for each firm? Is WW better off by choosing its output first? Explain why or why not.

Short answer

1. Each firm will produce 22.5 units. The market output will be 45 units at $165. The profit of each firm will be $2,278.13.
2. The output choice will be half of the monopoly output. The industry price will be $192. Each firm will produce 18 units, and the profit of each firm will be $2,430.
3. The payoff profit of WW and the profit of BBBS will be:

   PROFIT PAYOFF MAXTRIX (WW PROFIT, BBBS PROFIT)		BBBS
   		PRODUCE COURNOT q	PRODUCE CARTEL q
   WW	PRODUCE COURNOT q	2278, 2278	2582, 2187
   	PRODUCE CARTEL q	2187, 2582	2430, 2430

The output strategy will be Cournot Nash Equilibrium.

d. WW will produce 25.7 units. BBBS will produce 21.4 units. The market price will be $158.70; the profit of WW will be $2,316.86, and BBBS will be $2,067.24. Yes, WW will be better off by choosing its output first. WW is better off because it acts as a leader in the market; thus, it takes the decision independently.

Step-by-step solution

Explanation for part (a)

In Cournot oligopolists, the firms decide the output simultaneously. Thus, the reaction curve is of both the firm is calculated below:

The reaction curve of firm 1 is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1={\mathrm{TR}}_1-{\mathrm{TC}}_1 \\ =\left(300-3{\mathrm{q}}_1-3{\mathrm{q}}_2\right){\mathrm{q}}_1-30{\mathrm{q}}_1-1.5{\mathrm{q}}_1^2 \\ =300{\mathrm{q}}_1-3{\mathrm{q}}_1^2-3{\mathrm{q}}_1{\mathrm{q}}_2-30{\mathrm{q}}_1-1.5{\mathrm{q}}_1^2 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dq}}_1}=300-6{\mathrm{q}}_1-3{\mathrm{q}}_2-30-3{\mathrm{q}}_1=0 \\ 270-9{\mathrm{q}}_1-3{\mathrm{q}}_2=0 \\ 9{\mathrm{q}}_1=270-3{\mathrm{q}}_2 \\ {\mathrm{q}}_1=\frac{270-3{\mathrm{q}}_2}{9}=30-\frac{1}{3}{\mathrm{q}}_2 \end{gathered}$$

The reaction curve of firm 2 will be the same as firm 1 as the cost function is identical. Thus, the reaction curve of firm 2 will be$q_2=30-\frac{1}{3}{q}_1$.

From both the reaction curve the output of each firm will be:

$$\begin{gathered} q_1=30-\frac{1}{3}\left(30-\frac{1}{3}{q}_1\right) \\ {q}_1=30-10+\frac{1}{9}{q}_1 \\ 9q_1-q_1=180 \\ 8q_1=180 \\ {q}_1=22.5 \\ {q}_2=30-\frac{1}{3}\times 22.5 \\ =30-7.5 \\ =22.5 \end{gathered}$$

The market output, market price, and profit of each firm will be:

$$\begin{gathered} Q=22.5+22.5 \\ =45 \\ P=300-3\left(45\right) \\ =300-135 \\ =\$165 \\ {\mathrm{\pi }}_1=\left(165\times 22.5\right)-30\left(22.5\right)-1.5{\left(22.5\right)}^2 \\ =3712.5-675-759.375 \\ =\$2278.13 \\ {\mathrm{\pi }}_2=\left(165\times 22.5\right)-30\left(22.5\right)-1.5{\left(22.5\right)}^2 \\ =3712.5-675-759.375 \\ =\$2278.13 \end{gathered}$$

The market output will be 45 units, the market price will be $165, and the profit for each firm will be $2,278.13.

Explanation for part (b)

If two firms collude, the output decision will be taken as a monopoly and split among the firms; thus, each firm will produce half the monopoly output. The colluded firms act as a monopoly because, in the industry, it represented as one single firm. The monopolist optimum level will be where the marginal revenue is equal to marginal cost. Thus, the optimum level will be:

$$\begin{gathered} P=300-3Q \\ TR=300Q-3Q^2 \\ MR=300-6Q \\ C=30\frac{Q}{2}+1.5{\left(\frac{Q}{2}\right)}^2 \\ TC=2C \\ =30Q+3{\left(\frac{Q}{2}\right)}^2 \\ MC=30+\frac{3}{2}Q \\ MR=MC \\ 300-6Q=30+\frac{3}{2}Q \\ \frac{15}{2}Q=270 \\ Q=\frac{270\times 2}{15} \\ =36 \\ P=300-3\left(36\right) \\ =300-108 \\ =\$192 \end{gathered}$$

Each firm will produce 18 units (=36/2).

The profit for each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_1=\left(192\times 18\right)-30\left(18\right)-1.5{\left(18\right)}^2 \\ =3456-540-486 \\ =\$2430 \\ {\mathrm{\pi }}_1=\left(192\times 18\right)-30\left(18\right)-1.5{\left(18\right)}^2 \\ =3456-540-486 \\ =\$2430 \end{gathered}$$

The profit for each firm will be $2,430.

Explanation for part (c)

If both firms operate in the Cournot oligopoly, each firm's profit will be $2,278. If both the firm collude, then the profit of each firm will be $2,430.

The profit when WW operate in Cournot oligopoly and BBBS operate in Colluded oligopoly will be:

$$\begin{gathered} \mathrm{Q}=22.5+18 \\ =40.5 \\ \mathrm{P}=300-3\left(40.5\right) \\ =300-121.5 \\ =\$178.5 \\ {\mathrm{\pi }}_{WW}=\left(178.5\times 22.5\right)-30\left(22.5\right)-1.5{\left(22.5\right)}^2 \\ =4016.25-675-759.375 \\ =\$2581.88 \\ {\mathrm{\pi }}_{BBS}=\left(178.5\times 18\right)-30\left(18\right)-1.5{\left(18\right)}^2 \\ =3213-540-486 \\ =\$2187 \end{gathered}$$

The profit when WW operate in Colluded oligopoly and BBBS operate in Cournot oligopoly will be:

$$\begin{gathered} {\mathrm{\pi }}_{WW}=\left(178.5\times 18\right)-30\left(18\right)-1.5{\left(18\right)}^2 \\ =3213-540-486 \\ {\mathrm{\pi }}_{BBS}=\$2187=\left(178.5\times 22.5\right)-30\left(22.5\right)-1.5{\left(22.5\right)}^2 \\ =4016.25-675-759.375 \\ =\$2581.88 \end{gathered}$$

The payoff profit of WW and the profit of BBBS will be:

PROFIT PAYOFF MAXTRIX (WW PROFIT, BBBS PROFIT)		BBBS
		PRODUCE COURNOT q	PRODUCE CARTEL q
WW	PRODUCE COURNOT q	2278, 2278	2582, 2187
	PRODUCE CARTEL q	2187, 2582	2430, 2430

If BBBS operates in Cournot Oligopoly, WW is better off operating in Cournot Oligopoly, and if BBBS operates in Cartel Oligopoly, WW is better off operating Cournot Oligopoly. If WW operates in Cournot Oligopoly, BBBS is better off operating in Cournot Oligopoly, and if WW operates in Cartel Oligopoly, BBS is better off operating Cournot Oligopoly. Thus, Cournot Nash Equilibrium for the industry.

Explanation for part (d)

WW (firm 1) uses the Stackelberg Oligopoly; thus, it decides the output first. WW uses the reaction curve of BBS (firm 2) to calculate its output. The firm 2 reaction curve will be $q_2=30-\frac{1}{3}{q}_1$.

The output of both the firms will be:

$$\begin{gathered} {\mathrm{\pi }}_1=300-3{\mathrm{q}}_1^2-3{\mathrm{q}}_1\left(30-\frac{1}{3}{\mathrm{q}}_1\right)-30{\mathrm{q}}_1-1.5{\mathrm{q}}_1^2 \\ =300{\mathrm{q}}_1-3{\mathrm{q}}_1^2-90{\mathrm{q}}_1+{\mathrm{q}}_1^2-30{\mathrm{q}}_1-1.5{\mathrm{q}}_1^2 \\ \frac{{\mathrm{d\pi }}_1}{{\mathrm{dq}}_1}=300-6{\mathrm{q}}_1-90+2q_1-30-3{\mathrm{q}}_1=0 \\ 180-7{\mathrm{q}}_1=0 \\ 7{\mathrm{q}}_1=180 \\ {\mathrm{q}}_1=\frac{180}{7}=25.7 \\ {\mathrm{q}}_2=30-\frac{1}{3}\times 25.7 \\ =30-8.57=21.4 \end{gathered}$$

The output of firm 1 will be 25.7 units, and for firm 2 will be 21.4 units.

The market price and profit for each firm will be:

$$\begin{gathered} \mathrm{P}=300-3\left(25.7+21.4\right) \\ =300-141.3 \\ =\$158.7 \\ {\mathrm{\pi }}_1=\left(158.7\times 25.7\right)-30\left(25.7\right)-1.5{\left(25.7\right)}^2 \\ =4078.59-771-990.735 \\ =\$2316.86 \\ {\mathrm{\pi }}_2=\left(158.7\times 21.4\right)-30\left(21.4\right)-1.5{\left(21.4\right)}^2 \\ =3396.18-642-686.94 \\ =\$2067.24 \end{gathered}$$

The market price will be $158.70; the profit of WW will be $2,316.86, and BBBS will be $2,067.24.

WW is better off as it chooses the output first; thus, it takes the decision independently. Choosing the output first takes the major share of the market and has higher profit than BBBS.