Question

You are the manager of BlackSpot Computers, which competes directly with Condensed Computers to sell high-powered computers to businesses. From the two businesses’ perspectives, the two products are indistinguishable. The large investment required to build production facilities prohibits other firms from entering this market, and existing firms operate under the assumption that the rival will hold output constant. The inverse market demand for computers is P $5,900$Q, and both firms produce at a marginal cost of \($800$per computer. Currently, BlackSpot earns revenues of \)$4.25$million and profits (net of investment, R&D, and other fixed costs) of \($890,000$. The engineering department at BlackSpot has been steadily working on developing an assembly method that would dramatically reduce the marginal cost of producing these high-powered computers and has found a process that allows it tomanufacture each computer at a marginal cost of \)$500$. How will this technological advance impact your production and pricing plans? How will it impact BlackSpot’s bottom line?

Short answer

The profit of the firm will increase from $890000 to $1610000 due to the falling the marginal costs generated by the development of a few methods of assembling computers by the firm Black Spot.

Step-by-step solution

Given information

There are two companies that produce a similar product with the same marginal costs, and they don’t make any distinction between a leader and a follower firm. These features signify a Cournot oligopoly model. The relevant inverse demand function and marginal cost are given below.

The inverse demand function is $P=5,900‐Q$.

The initial marginal cost is $$800$.

There are two levels of output for each firm. So, $Q=Q_1+Q_2$with $Q_1$being the output of the firm Black Spot Computers and being the output of the firm Condensed Computers.

In the Cournot model, the following condition must be met for both firms.

$MR=MC$

To obtain the reaction function of Firm $1$, the demand function is multiplied by Q1, and then the marginal income is attained as follows.

Total revenue $=5,900‐\left(Q_1+Q_2\right)Q_1$

Total revenue role="math" localid="1653549056435" $=5,900Q_1‐Q_1^2‐Q_2{Q}_1$

$M{R}_1=\frac{d{R}_1}{d{Q}_1}=5,900‐2Q_1‐Q_2$

Equating the marginal revenue and costs and solving for Q1

Equating the marginal revenue and costs and solving for $Q_1$to obtain the reaction function of the firm Black Spot: [$7$pt], we get

$$\begin{gathered} 5,900‐2Q_1‐Q_2=800 \\ {Q}_1=\frac{5,900‐800‐Q_2}{2} \\ {Q}_1=2,550‐\frac{1}{2}{Q}^2 \end{gathered}$$

Obtaining the total income

To obtain the reaction function of Firm 2, the total income is attained by multiplying the inverse demand function by Q2. Then, the marginal income is derived as follows.

$$\begin{gathered} TotalRevenue=5,900‐\left(Q_1+Q_2\right)Q_2 \\ TotalRevenue=5,900Q_2‐Q_1{Q}_2‐Q_2^2 \\ M{R}_2=\frac{d{R}_2}{d{Q}_2}=5,900‐Q_1‐2Q_2 \end{gathered}$$

Equating the marginal revenue and costs and solving for Q2

Equating the marginal revenue and costs and solving for Q2 to obtain the reaction function of the firm Condensed Computers, we get

$$\begin{gathered} 5,900‐Q_1‐2Q_2=800 \\ {Q}_2=\frac{5,900‐800‐Q_1}{2} \\ {Q}_2=2,550‐\frac{1}{2}{Q}_1 \end{gathered}$$

Substituting the reaction function

The reaction function of Firm1 is put in the reaction function of Firm 2 to obtain the output level.

$$\begin{gathered} Q_2=2,550‐\frac{1}{2}2,550‐\frac{1}{2}{Q}_2 \\ {Q}_2=2,550‐1,275‐\frac{1}{4}{Q}_2\frac{1}{4}{Q}_2+Q_2 \\ =\frac{1}{4}{Q}_2+1,275‐\frac{1}{4}{Q}_2\frac{3}{4}{Q}_2 \end{gathered}$$

Multiplying both sides by 4, we get

$$\begin{gathered} 4\text{'}\frac{3}{4}{Q}_2=1,275\text{'}4Q_2 \\ =5,100/3 \\ =1700. \end{gathered}$$

Obtaining the output

The value of the output of Condensed Computers is put in the reaction function of Black Spot to obtain its output.

$$\begin{gathered} Q_1=2,550‐\frac{1}{2}\left(1,700\right) \\ {Q}_1=1,700 \end{gathered}$$

Now, the output levels of both firms are put in the inverse demand function to obtain the price that maximizes the benefits of the oligopoly.

$$\begin{gathered} P=5,900‐\left(1,700+1,700\right) \\ P=5,900‐3,400 \\ P=2,500 \end{gathered}$$

The benefits of Black Spot can be calculated by subtracting the costs from the income. However, there is a net profit of an investment, R&D, and other fixed costs of $\cancel{S}890,000$.

Therefore, the profits of Black Spot are as follows.

$$\begin{gathered} =TotalRevenue‐TotalCost \\ P=Price\times Quantity‐TotalCost \\ P=\left(2,500\text{'}1,700\right)‐\left(800\text{'}1,700\right) \\ P=2,890,000 \end{gathered}$$

By subtracting the net profit, the value of the net investment, R&D, and other fixed costs of Black Spot is attained as follows:

$2,890,000‐890,000=2,000,000$

Obtaining the reaction function of Firm 1

Analyzing how it would be in the variation of the prices, output, and profit of Black Spot with a reduction of its marginal costs from $\cancel{S}800$to $\cancel{S}500$due to the development of a new assembly method of computers, the same step in which the marginal income is equal to the marginal costs is followed. Now, the new reaction functions for both firms are attained.

Reaction function of Firm 1 (Black Spot):

$$\begin{gathered} 5,900‐2Q_1‐Q_2 \\ =500Q_1 \\ =\frac{5,900‐500‐Q_2}{2}{Q}_1 \\ =2,700‐\frac{1}{2}{Q}_2 \end{gathered}$$

The marginal costs of Condensed Computers will remain constant at $\cancel{S}800$. So, the reaction function will not change. Therefore,

$$\begin{gathered} Q_2=2,550‐\frac{1}{2}2,700‐\frac{1}{2}{Q}_2 \\ {Q}_2=2,550‐1,350‐\frac{1}{4}{Q}_2\frac{1}{4}{Q}_2+Q_2 \\ =\frac{1}{4}{Q}_2+1,200‐\frac{1}{4}{Q}_2\frac{3}{4}{Q}_2 \\ =2,700‐\frac{1}{2}{Q}_2 \end{gathered}$$

Multiplying both sides by 4, we get

$$\begin{gathered} 4\times \frac{3}{4}{Q}_2 \\ =1,200\times 4Q_2 \\ =4,800/3 \\ =1,600 \end{gathered}$$

With the value of $Q_2$, the output level of Black Spot is obtained by substituting this value in the new reaction function with the new marginal costs.

$$\begin{gathered} Q_1=2,700‐\frac{1}{2}1,600 \\ {Q}_1=1,900 \end{gathered}$$

Therefore, with the reduction of marginal costs, BlackSpot will increase its production from 1700 to 1900, while the rival firm, Condensed Computers, will reduce its production from $1,700to1,600$.

Substituting in the inverse demand function

The new output levels for the two firms are put in the inverse demand function to obtain the new equilibrium price level for the oligopoly.

$$\begin{gathered} P=5,900‐\left(1,600+1,900\right) \\ P=5,900‐3,500 \\ P=2,400 \end{gathered}$$

The equilibrium price has been reduced from $2500 to $2400.

Now, to obtain the benefit of the company, the new total income of Black Spot is obtained.

$$\begin{gathered} TotalRevenue=Price\times Quantity \\ =2400\times 1900 \\ =4560000 \end{gathered}$$

Determining the benefits of the Black Spot signature

The benefits of the BlackSpot signature are as follows.

$$\begin{gathered} Profit=TotalRevenue-TotalCost \\ =\left(2,400\times 1,900\right)-\left(500\times 1,900\right) \\ =3,610,000 \end{gathered}$$

Given that the benefits were $2,000,000\cancel{S}$the difference between the gross profits will give the new net profit of the firm.

$3,610,000-2,000,000=1,610,000$

Therefore, the firm’s profit will increase from $890000 to $1610000. This happens as the marginal costs decrease after Black Spot develops a few methods of assembling computers.