Question

Demand for light bulbs can be characterized by Q = 100 - P, where Q is in millions of boxes of lights sold and P is the price per box. There are two producers of lights, Everglow and Dimlit. They have identical cost functions: C_i = 10Q_i +1/2Q_i²(i = E, D) Q = QE + QD

1. Unable to recognize the potential for collusion, the two firms act as short-run perfect competitors. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?
2. Top management in both firms is replaced. Each new manager independently recognizes the oligopolistic nature of the light bulb industry and plays Cournot. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?
3. Suppose the Everglow manager guesses correctly that Dimlit is playing Cournot, so Everglow plays Stackelberg. What are the equilibrium values of QE, QD, and P? What are each firm’s profits?
4. If the managers of the two companies collude, what are the equilibrium values of QE, QD, and P? What are each firm’s profits?

Short answer

1. Each firm will produce 30 units, and the price will be $40. The profit of each firm will be $450 million.
2. Each firm will produce 22.5 units, and the price will be $55. The profit of each firm will be $759.4 million.
3. Everglow will produce 25.7 units, Dimlit will produce 21.4 units, and the price will be $52.90. The profit for Everglow will be $772.3 million, and for Dimlit will be $689.1 million.
4. Each firm will produce 18 units, and the price will be $64. The profit of each firm will be $810.

Step-by-step solution

Explanation for part (a)

In a perfect competition market, the firm operates where the price is equal to marginal cost. Let firm 1 be Everglow and firm 2 be Dimlit.

Firm 1 will be at the optimum where the price is equal to marginal cost.

$$\begin{gathered} {\text{P = 100 -Q}}_E{\text{-Q}}_D \\ {\text{C}}_E{\text{= 10Q}}_E\text{+}\frac{\text{1}}{\text{2}}{\text{Q}}_E^{\text{2}} \\ {\text{MC}}_E{\text{= 10 +Q}}_E \\ \text{P = MC} \\ {\text{100 -Q}}_E{\text{-Q}}_D{\text{= 10 +Q}}_E \end{gathered}$$

Firm 2’s optimum level will be:

$$\begin{gathered} {\text{C}}_D{\text{= 10Q}}_D\text{+}\frac{\text{1}}{\text{2}}{\text{Q}}_D^{\text{2}} \\ {\text{MC}}_D{\text{= 10 +Q}}_D \\ \text{P = MC} \\ {\text{100 -Q}}_E{\text{-Q}}_D{\text{= 10 +Q}}_D \\ {\text{2Q}}_D{\text{= 90 -Q}}_E \\ {\text{Q}}_D\text{=}\frac{{\text{90 -Q}}_E}{\text{2}} \end{gathered}$$

Putting the value of Q_D in the optimum level equation of firm 1 to get the value of Q_E.

$$\begin{gathered} {\text{100 -Q}}_E\text{-}\frac{{\text{90 -Q}}_E}{\text{2}}{\text{= 10 +Q}}_E \\ {\text{100 -Q}}_E{\text{- 45 + 0.5Q}}_E{\text{= 10 +Q}}_E \\ {\text{1.5Q}}_E\text{= 45} \\ {\text{Q}}_E\text{= 30} \\ {\text{Q}}_D\text{=}\frac{\text{90 - 30}}{\text{2}} \\ \text{=}\frac{\text{60}}{\text{2}} \\ \text{= 30} \end{gathered}$$

The equilibrium quantity and price will be:

$$\begin{gathered} {\text{Q}}_{\text{T}}\text{= 30 + 30} \\ \text{= 60} \\ \text{P = 100 - 60} \\ \text{= \$ 40} \end{gathered}$$

The equilibrium quantity will be 60 units at $40.

The profit for each firm will be the same as the cost function is identical.

The profit for each firm will be:

$$\begin{gathered} \mathrm{\pi }=\left(40\times 30\right)-\left(10\times 30\right)-0.5\times {\left(30\right)}^2 \\ =1200-300-450 \\ =\$450\mathrm{million} \end{gathered}$$

Thus, Everglow and Dimlit both earn a profit of $450 million.

Explanation for part (b)

When both the firm operates in the Cournot model, then the reaction curves of each firm is calculated below.

Everglow’s reaction curve will be:

$$\begin{gathered} {\mathrm{\pi }}_{\mathrm{E}}=100{\mathrm{Q}}_{\mathrm{E}}-{\mathrm{Q}}_{\mathrm{E}}^2-{\mathrm{Q}}_{\mathrm{E}}{\mathrm{Q}}_{\mathrm{D}}-10{\mathrm{Q}}_{\mathrm{E}}-\frac{1}{2}{\mathrm{Q}}_{\mathrm{E}}^2 \\ \frac{{\mathrm{d\pi }}_{\mathrm{E}}}{{\mathrm{dQ}}_{\mathrm{E}}}=100-2{\mathrm{Q}}_{\mathrm{E}}-{\mathrm{Q}}_{\mathrm{D}}-10-{\mathrm{Q}}_{\mathrm{E}}=0 \\ 3{\mathrm{Q}}_{\mathrm{E}}=90-{\mathrm{Q}}_{\mathrm{D}} \\ {\mathrm{Q}}_{\mathrm{E}}=\frac{90-{\mathrm{Q}}_{\mathrm{D}}}{3} \end{gathered}$$

The reaction curve for Dimlit will be the same as Everglow as their cost function is identical. The reaction curve of Dimlit will be $Q_D=\frac{90-Q_E}{3}$.

From the reaction curve,

$$\begin{gathered} Q_E=\frac{90-\left(\frac{90-{\mathrm{Q}}_{\mathrm{D}}}{3}\right)}{3} \\ 9Q_E=240-90+Q_E \\ 8Q_E=180 \\ {Q}_E=22.5 \\ {Q}_D=22.5 \end{gathered}$$

The equilibrium quantity and price will be:

$$\begin{gathered} Q_T=22.5+22.5 \\ =45 \\ P=100-45 \\ =\$55 \end{gathered}$$

The profit for both the firms will be the same as the cost function is identical. The profit of both the firms will be:

$$\begin{gathered} {\mathrm{\pi }}_{\mathrm{E}}=\left(55\times 22.5\right)-10\left(22.5\right)-0.5{\left(22.5\right)}^2 \\ =1237.5-225-253.125 \\ =\$759.4\mathrm{million} \\ {\mathrm{\pi }}_{\mathrm{D}}=\$759.4\mathrm{million} \end{gathered}$$

The profit of both the firm will be $759.4 million.

Explanation for part (c)

Everglow is a leader firm; thus, set the quantity first, knowing the reaction curve of Dimlit. Therefore, Everglow’soutput will be:

$$\begin{gathered} {\mathrm{\pi }}_{\mathrm{E}}=100{\mathrm{Q}}_{\mathrm{E}}-{\mathrm{Q}}_{\mathrm{E}}^2-{\mathrm{Q}}_{\mathrm{E}}\left(\frac{90-{\mathrm{Q}}_{\mathrm{E}}}{3}\right)-10{\mathrm{Q}}_{\mathrm{E}}-\frac{1}{2}{\mathrm{Q}}_{\mathrm{E}}^2 \\ =100{\mathrm{Q}}_{\mathrm{E}}-{\mathrm{Q}}_{\mathrm{E}}^2-30{\mathrm{Q}}_{\mathrm{E}}+\frac{{\mathrm{Q}}_{\mathrm{E}}^2}{3}-10{\mathrm{Q}}_{\mathrm{E}}-\frac{1}{2}{\mathrm{Q}}_{\mathrm{E}}^2 \\ \frac{{\mathrm{d\pi }}_{\mathrm{E}}}{{\mathrm{dQ}}_{\mathrm{E}}}=100-2{\mathrm{Q}}_{\mathrm{E}}-30+\frac{2{\mathrm{Q}}_{\mathrm{E}}}{3}-10-{\mathrm{Q}}_{\mathrm{E}}=0 \\ 180-7{\mathrm{Q}}_{\mathrm{E}}=0 \\ 7{\mathrm{Q}}_{\mathrm{E}}=180 \\ {\mathrm{Q}}_{\mathrm{E}}=25.7 \end{gathered}$$

Dimlit’s output will be:

$Q_D=\frac{90-25.7}{3}=\frac{64.3}{3}=21.4$

The equilibrium quantity and price will be:

$$\begin{gathered} Q_T=25.7+21.4 \\ =47.1 \\ P=100-47.1 \\ =\$52.9 \end{gathered}$$

The profit for each firm is calculated below:

$$\begin{gathered} {\mathrm{\pi }}_{\mathrm{E}}=\left(52.9\times 25.7\right)-10\left(25.7\right)-0.5{\left(25.7\right)}^2 \\ =1359.53-257-330.245 \\ =\$772.3\mathrm{million} \\ {\mathrm{\pi }}_{\mathrm{D}}=\left(52.9\times 21.4\right)-10\left(21.4\right)-0.5{\left(21.4\right)}^2 \\ =1132.06-214-228.98 \\ =\$689.08 \end{gathered}$$

Profit for Everglow will be $772.3 million, and for Dimlit will be $689.08 million.

Explanation for part (d)

As the firm are identical, then they will split the output equally. Now, the cost function will be:$C_i=10\left(\frac{Q}{2}\right)+\frac{1}{2}{\left(\frac{Q}{2}\right)}^2$. The total cost of the industry will be: $$\begin{gathered} TC=2Ci \\ =10Q+{\left(\frac{Q}{2}\right)}^2 \end{gathered}$$

The industry price and quantity are calculated below:

$$\begin{gathered} \text{P = 100 - Q} \\ {\text{TR = 100Q -Q}}^{\text{2}} \\ \text{MR = 100 - 2Q} \\ \text{C = 10Q +}{\left(\frac{\text{Q}}{\text{2}}\right)}^{\text{2}} \\ \text{MC = 10 + 0.5Q} \\ \text{MR = MC} \\ \text{100 - 2Q = 10 + 0.5Q} \\ \text{2.5Q = 90} \\ \text{Q = 36} \\ {\text{Q}}_{\text{E}}\text{= 18} \\ {\text{Q}}_{\text{D}}\text{= 18} \\ \text{P = 100 - 36} \\ \text{= \$ 64} \end{gathered}$$

The profit of each firm will be:

$$\begin{gathered} {\mathrm{\pi }}_{\mathrm{E}}=\left(64\times 18\right)-10\left(18\right)-0.5{\left(18\right)}^2 \\ =1152-180-162 \\ =\$810\mathrm{million} \\ {\mathrm{\pi }}_{\mathrm{D}}=\$810\mathrm{million} \end{gathered}$$

The profit for both the firm will be $810 million.