Question

Suppose the market for tennis shoes has one dominant firm and five fringe firms. The market demand is Q = 400 - 2 P. The dominant firm has a constant marginal cost of 20. The fringe firms each have a marginal cost of MC = 20 + 5q.

a. Verify that the total supply curve for the five fringe firms is Q_f = P - 20.

b. Find the dominant firm’s demand curve.

c. Find the profit-maximizing quantity produced and the price charged by the dominant firm, and the quantity produced and the price charged by each of the fringe firms.

d. Suppose there are 10 fringe firms instead of five. How does this change your results?

e. Suppose there continue to be five fringe firms but that each manages to reduce its marginal cost to MC = 20 + 2q. How does this change your results?

Short answer

a. Yes, the total supply curve for the five fringe firms is Q_f = P - 20.

b. The dominant firm’s demand curve will be Q_D = 420 – 3P.

c. The profit-maximizing quantity of the dominant firm will be 180 units at $80. Each fringe will produce 12 units at $80.

d. The profit-maximizing quantity of the dominant firm will be 180 units at $65. Each fringe will produce 9 units at $65.

e. The profit-maximizing quantity of the dominant firm will be 180 units at $60. Each fringe will produce 20 units at $80.

Step-by-step solution

Explanation for part (a)

The supply curve of fringe is calculated below:

$$\begin{gathered} \text{MC = 20 + 5q} \\ \text{P = MC} \\ \text{P = 20 + 5q} \\ \text{5q = P - 20} \\ \text{q =}\frac{\text{P}}{\text{5}}\text{- 4} \end{gathered}$$

The total supply curve of 5 fringes is calculated below:

$$\begin{gathered} \text{Q = 5q} \\ \text{Q = 5}\left(\frac{\text{P}}{\text{5}}\text{- 4}\right) \\ {\text{Q}}_{\text{f}}\text{= P - 20} \end{gathered}$$

Hence, proved.

Explanation for part (b)

The dominant demand curve is the total market demand minus the total supply of fringe.

The dominant firm demand curve is calculated below:

$$\begin{gathered} {\text{Q}}_{\text{D}}\text{= 400 - 2P -}\left(\text{P - 20}\right) \\ {\text{Q}}_{\text{D}}\text{= 400 - 2P - P + 20} \\ {\text{Q}}_{\text{D}}\text{= 420 - 3P} \end{gathered}$$

The dominant firm demand curve will be Q_D = 420 – 3P.

Explanation for part (c)

The marginal revenue is equal to the marginal cost at the optimum level of the dominant firm. The marginal revenue curve of the dominant firm will be:

$$\begin{gathered} {\text{Q}}_{\text{D}}\text{= 420 - 3P} \\ \text{P = 140 -}\frac{\text{1}}{\text{3}}\text{Q} \\ \text{TR = 140Q -}\frac{\text{1}}{\text{3}}{\text{Q}}^{\text{2}} \\ \text{MR = 140 -}\frac{\text{2}}{\text{3}}\text{Q} \end{gathered}$$

At the optimal level,

$$\begin{gathered} MR=MC \\ 140-\frac{2}{3}Q=20 \\ \frac{2}{3}Q=120 \\ Q=60\times 3 \\ Q=180 \\ P=140-\frac{1}{3}\times 180 \\ =140-60 \\ =\$80 \end{gathered}$$

The dominant firm will produce 180 units at $80.

The output for fringe at $80 will be:

$$\begin{gathered} Q=P-20 \\ =80-20 \\ =60 \\ q=\frac{60}{5}=12 \end{gathered}$$

Each fringe will supply 12 units at $80.

Explanation for part (d)

The new total supply curve of fringe will be:

$$\begin{gathered} Q=10q \\ =10\left(\frac{p}{5}-4\right) \\ {Q}_f=2P-40 \end{gathered}$$

The dominant firm new demand curve is calculated below:

$$\begin{gathered} Q_D=400-2P-\left(2P-40\right) \\ {Q}_D=400-2P-2P+40 \\ =440-4P \end{gathered}$$

The new marginal revenue curve of the dominant firm will be:

$$\begin{gathered} Q_D=440-4P \\ P=110-\frac{1}{4}Q \\ TR=110Q-\frac{1}{4}{Q}^2 \\ MR=110-\frac{1}{2}Q \end{gathered}$$

At the optimal level,

$$\begin{gathered} MR=MC \\ 110-\frac{1}{2}Q=20 \\ \frac{1}{2}Q=90 \\ Q=180 \\ P=110-\frac{1}{4}\times 180 \\ =110-45 \\ =\$65 \end{gathered}$$

The dominant firm will produce 180 units at $65.

The output for fringe at $65 will be:

$$\begin{gathered} Q=2P-40 \\ =2\left(65\right)-40 \\ =130-40 \\ =90 \\ q=\frac{90}{10}=9 \end{gathered}$$

Each fringe will supply 9 units at $65.

Explanation for part (e)

The supply curve of fringe is calculated below:

$$\begin{gathered} MC=20+2q \\ P=MC \\ P=20+2q \\ 2q=P-20 \\ q=\frac{P}{2}-10 \end{gathered}$$

The total supply curve of 5 fringes is calculated below:

$$\begin{gathered} Q=5q \\ Q=5\left(\frac{P}{2}-10\right) \\ Q=2.5P-50 \end{gathered}$$

The dominant firm demand curve is calculated below:

$$\begin{gathered} Q_D=400-2P-\left(\frac{5}{2}P-50\right) \\ {Q}_D=400-2P-\frac{5}{2}P+50 \\ {Q}_D=450-4.5P \end{gathered}$$

The marginal revenue curve of the dominant firm will be:

$$\begin{gathered} Q_D=450-4.5P \\ P=100-\frac{Q}{4.5} \\ TR=100Q-\frac{Q^2}{4.5} \\ MR=100-\frac{2}{4.5}Q \end{gathered}$$

At the optimal level,

$$\begin{gathered} MR=MC \\ 100-\frac{2}{4.5}Q=20 \\ \frac{2}{4.5}Q=80 \\ Q=\frac{80\times 4.5}{2} \\ =40\times 4.5=180 \\ P=100-\frac{180}{4.5} \\ =100-40 \\ =\$60 \end{gathered}$$

The dominant firm will produce 180 units at $60.

The output for fringe at $80 will be:

$$\begin{gathered} Q=2.5P-50 \\ =2.5\left(60\right)-50 \\ =150-50 \\ =100 \\ q=\frac{100}{5}=20 \end{gathered}$$

Each fringe will supply 20 units at $60.