Chapter 11, Problem 8P

(a)

To determine

The market demand curve, price, and output under the monopolist.

Explanation of Solution

The market demand curve for Otters is given as $\text{P}=100-0.5\text{Q}$. This means that the vertical intercept of the demand curve is at 100. Since the slope is 0.5, the horizontal intercept will be twice the value of the vertical intercept which means at 200.

The marginal revenue of the market can be derived as follows:

$\begin{array}{c}\text{MR}=\frac{\partial \left(\text{P}\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(100-0.5\text{Q}\right)\text{Q}}{\partial \text{Q}}\\ =\frac{\partial \left(100\text{Q}-0.5{\text{Q}}^2\right)}{\partial \text{Q}}\\ =100-\text{Q}\end{array}$

The market marginal revenue is $\text{MR}=100-\text{Q}$. Since the marginal cost of production is given as $20, the output that the monopolist would make can be calculated by equating MR with MC as follows:

 $\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-\text{Q}=20\\ \text{Q}=100-20\\ =80\end{array}$

By substituting the value of Q in the demand equation, the monopoly price can be obtained as follows:

$\begin{array}{c}\text{P}=100-0.5\text{Q}\\ =100-\left(0.5\times 80\right)\\ =100-40\\ =60\end{array}$

Thus, the market demand curve can be illustrated as follows:

The monopoly output and price are 80 units and $60 per unit.

Economics Concept Introduction

Imperfect competition: The imperfect competition is the market structure where there are many sellers selling the differentiated products and there will be information asymmetry in the market which provides some market control to the producers.

(b)

To determine

The output in the market by A.

Explanation of Solution

When firm J announces that half of the monopoly output would be brought by J, firm J would bring 40 units. By knowing this, the residual demand for A can be calculated as follows:

$\begin{array}{c}\text{P}=100-0.5\times \left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)\\ =100-0.5\times \left(40+{\text{q}}_{\text{A}}\right)\\ =100-20-0.5{\text{q}}_{\text{A}}\\ =80-0.5{\text{q}}_{\text{A}}\end{array}$

Inverse demand equation (Price) for firm A can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{A}}=\frac{\partial \left(\text{P}\times {\text{q}}_{\text{A}}\right)}{\partial {\text{q}}_{\text{A}}}\\ =\frac{\partial \left(80-0.5{\text{q}}_{\text{A}}\right){\text{q}}_{\text{A}}}{\partial {\text{q}}_{\text{A}}}\\ =\frac{\partial \left(80{\text{q}}_{\text{A}}-0.5{\text{q}}_{\text{A}}^2\right)}{\partial {\text{q}}_{\text{A}}}\\ =80-{\text{q}}_{\text{A}}\end{array}$

Thus, the marginal revenue curve of A will be ${\text{MR}}_{\text{A}}=80-{\text{q}}_{\text{A}}$. Thus, by equating the marginal revenue with the marginal cost, the output by A can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 80-{\text{q}}_{\text{A}}=20\\ {\text{q}}_{\text{A}}=80-20\\ =60\end{array}$

Thus, A's profit maximizing output would be 60 units.

(c)

To determine

Industrial output, price, and each seller's profit.

Explanation of Solution

The output by J is 40 units and the output by A is 60 units, which means that the total industrial output would be the summation of these two which is equal to 100 units. The price can be calculated by substituting the values of the two quantities in the demand function as follows:

$\begin{array}{c}\text{P}=100-0.5\text{Q}\\ =100-0.5\times \left(40+60\right)\\ =100-50\\ =50\end{array}$

Thus, the price of the market would be $50 per unit. Thus, the profit of each seller can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{J}}=\left(\text{P}-{\text{ATC}}_{\text{J}}\right)\times {\text{q}}_{\text{J}}\\ =\left(50-20\right)\times 40\\ =30\times 40\\ =1,200\end{array}$

Similarly, the profit of A can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{A}}=\left(\text{P}-{\text{ATC}}_{\text{A}}\right)\times {\text{q}}_{\text{A}}\\ =\left(50-20\right)\times 60\\ =30\times 60\\ =1,800\end{array}$

Thus, the profit of A is $1,800 and that of J is $1,200.

(d)

To determine

Industrial output, price, and each seller's profit.

Explanation of Solution

When firm J knows that the output brought by A is 60 units, the residual demand for the J can be calculated as follows:

$\begin{array}{c}\text{P}=100-0.5\times \left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)\\ =100-0.5\times \left(60+{\text{q}}_{\text{J}}\right)\\ =100-30-0.5{\text{q}}_{\text{J}}\\ =70-0.5{\text{q}}_{\text{J}}\end{array}$

Inverse demand equation (Price) for firm J can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{J}}=\frac{\partial \left(\text{P}\times {\text{q}}_{\text{J}}\right)}{\partial {\text{q}}_{\text{J}}}\\ =\frac{\partial \left(70-0.5{\text{q}}_{\text{J}}\right){\text{q}}_{\text{J}}}{\partial {\text{q}}_{\text{J}}}\\ =\frac{\partial \left(70{\text{q}}_{\text{j}}-0.5{\text{q}}_{\text{J}}^2\right)}{\partial {\text{q}}_{\text{J}}}\\ =70-{\text{q}}_{\text{J}}\end{array}$

Thus, the marginal revenue curve of J will be ${\text{MR}}_{\text{J}}=70-{\text{q}}_{\text{J}}$. Thus, by equating the marginal revenue with the marginal cost, the output by J can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 70-{\text{q}}_{\text{J}}=20\\ {\text{q}}_{\text{J}}=70-20\\ =50\end{array}$

Thus, J's profit maximizing output would be 50 units. The output by J is 50 units and the output by A is 60 units, which means that the total industrial output would be the summation of these two which is equal to 110 units. The price can be calculated by substituting the values of the two quantities in the demand function as follows:

$\begin{array}{c}\text{P}=100-0.5\text{Q}\\ =100-0.5\times \left(50+60\right)\\ =100-55\\ =45\end{array}$

Thus, the price of the market would be $45 per unit. Thus, the profit of each seller can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{J}}=\left(\text{P}-{\text{ATC}}_{\text{J}}\right)\times {\text{q}}_{\text{J}}\\ =\left(45-20\right)\times 50\\ =25\times 50\\ =1,250\end{array}$

Similarly, the profit of A can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{A}}=\left(\text{P}-{\text{ATC}}_{\text{A}}\right)\times {\text{q}}_{\text{A}}\\ =\left(45-20\right)\times 60\\ =25\times 60\\ =1,500\end{array}$

Thus, the profit of A is $1,500 and that of J is $1,250.

(e)

To determine

Cournot equilibrium in the economy.

Explanation of Solution

The market demand curve is $\text{P}=100-0.5\left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)$.

The marginal revenue of firm J can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{J}}=\frac{\partial \left(\text{P}\times {\text{q}}_{\text{J}}\right)}{\partial {\text{q}}_{\text{J}}}\\ =\frac{\partial \left(100-0.5\left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)\right){\text{q}}_{\text{J}}}{\partial {\text{q}}_{\text{J}}}\\ =\frac{\partial \left(100{\text{q}}_{\text{J}}-0.5{\text{q}}_{\text{J}}^2-0.5{\text{q}}_{\text{A}}{\text{q}}_{\text{J}}\right)}{\partial {\text{Q}}_{\text{J}}}\\ =100-{\text{q}}_{\text{J}}-0.5{\text{q}}_{\text{A}}\end{array}$

 Thus, the marginal revenue curve for J will be ${\text{MR}}_{\text{J}}=100-{\text{q}}_{\text{J}}-0.5{\text{q}}_{\text{A}}$.

The marginal revenue of firm A can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{A}}=\frac{\partial \left(\text{P}\times {\text{q}}_{\text{A}}\right)}{\partial {\text{q}}_{\text{A}}}\\ =\frac{\partial \left(100-0.5\left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)\right){\text{q}}_{\text{A}}}{\partial {\text{Q}}_{\text{A}}}\\ =\frac{\partial \left(100{\text{q}}_{\text{A}}-0.5{\text{q}}_{\text{A}}^2-0.5{\text{q}}_{\text{J}}{\text{q}}_{\text{A}}\right)}{\partial {\text{q}}_{\text{A}}}\\ =100-{\text{q}}_{\text{A}}-0.5{\text{q}}_{\text{J}}\end{array}$

 Thus, the marginal revenue curve for J will be ${\text{MR}}_{\text{A}}=100-0.5{\text{q}}_{\text{J}}-{\text{q}}_{\text{A}}$. Thus, the reaction function of firm J can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{J}}=\text{MC}\\ 100-{\text{q}}_{\text{J}}-0.5{\text{q}}_{\text{A}}=20\\ {\text{q}}_{\text{J}}=100-20-0.5{\text{q}}_{\text{A}}\\ =80-0.5{\text{q}}_{\text{A}}\end{array}$

Similarly, A can be calculated as follows:

$\begin{array}{c}{\text{MR}}_{\text{J}}=\text{MC}\\ 100-0.5{\text{q}}_{\text{J}}-{\text{q}}_{\text{A}}=20\\ {\text{q}}_{\text{A}}=100-20-0.5{\text{q}}_{\text{J}}\\ =80-0.5{\text{q}}_{\text{J}}\end{array}$

Thus, by substituting the reaction function of A in J  the equilibrium output can be provided as follows:

 $\begin{array}{c}{\text{q}}_{\text{J}}=80-0.5{\text{q}}_{\text{A}}\\ {\text{q}}_{\text{J}}=80-0.5\times \left(80-0.5{\text{q}}_{\text{J}}\right)\\ {\text{q}}_{\text{J}}=80-40+0.25{\text{q}}_{\text{J}}\\ {\text{q}}_{\text{J}}-0.25{\text{q}}_{\text{J}}=40\\ {\text{q}}_{\text{J}}=\frac{40}{0.75}\\ =53.33\end{array}$

Since A also faces the identical problem, the output of A will also be 53.33. Thus, the total output is 106.66 and this can be substituted in the market demand in order to calculate the market price as follows:

$\begin{array}{c}\text{P}=100-0.5\left({\text{q}}_{\text{A}}+{\text{q}}_{\text{J}}\right)\\ =100-0.5\times \left(53.33+53.33\right)\\ =100-\left(0.5\times 106.66\right)\\ =100-53.33\\ =46.67\end{array}$

Thus, the Cournot equilibrium price is $46.67 and the output is 53.33. Since part d calculated the outputs and prices that are different, they are not Cournot equilibrium prices and outputs.