Chapter 11, Problem 11P

(a)

To determine

The competitive output and the price in the market.

Explanation of Solution

When the market is competitive, the price will be determined such that the marginal cost equals price. Here, the marginal cost is given to be $20, which means that the competitive price in the market will be $20. The output in the market can be calculated by substituting the price in the demand equation as follows:

$\begin{array}{c}\text{P}=80-\text{Q}\\ 20=80-\text{Q}\\ \text{Q}=60\end{array}$

Thus, the equilibrium price would be $20 per unit, and the quantity would be 60.

Economics Concept Introduction

Imperfect competition: 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 monopoly marginal revenue curve and the profit maximizing quantity.

Explanation of Solution

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

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

Thus, the marginal revenue curve will be $\text{MR}=80-2\text{Q}$. The monopoly output can be calculated by equating the marginal revenue to the marginal cost as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 80-2\text{Q}=20\\ 2\text{Q}=80-20\\ \text{Q}=\frac{60}{2}\\ =30\end{array}$

Thus, the marginal revenue curve can be illustrated as follows:

(i) The competitive output was 60 units, whereas the monopoly output is calculated to be just 30 units. Thus, the proportion of the two market condition outputs can be written as follows:

$\begin{array}{c}\text{Proportion of outputs}=\frac{\text{Output under monopoly}}{\text{Output under competitive market}}\\ =\frac{30}{60}\\ =\frac{1}{2}\end{array}$

Thus, the proportion is $\frac{1}{2}$ .

(ii) The monopoly price can be calculated by substituting the value of Q in the demand equation as follows:

$\begin{array}{c}\text{P}=80-\text{Q}\\ =80-30\\ =50\end{array}$

Thus, the market situation can be illustrated as follows:

In Figure 2, the horizontal axis measures quantity, and the vertical axis measures price.

(c)

To determine

Output and price under the Cournot competition.

Explanation of Solution

Under the Cournot competition, there will be two players in the market producing the identical products. Thus, the inverse demand function of the market would be $\text{P}=80-{\text{q}}_1-{\text{q}}_2$.

Marginal revenue of firm 1 can be derived as follows:

$\begin{array}{c}{\text{MR}}_1=\frac{\partial \left(\text{P}\times {\text{q}}_1\right)}{\partial {\text{q}}_1}\\ =\frac{\partial \left(80-{\text{q}}_{\text{1}}-{\text{q}}_2\right){\text{q}}_1}{\partial {\text{q}}_1}\\ =\frac{\partial \left(80{\text{q}}_1-{\text{q}}_1^2-{\text{q}}_2{\text{q}}_1\right)}{\partial {\text{q}}_1}\\ =80-2{\text{q}}_{\text{1}}-{\text{q}}_2\end{array}$

Thus, the marginal revenues of the firm 1 is ${\text{MR}}_1=80-2{\text{q}}_1-{\text{q}}_2$.

Marginal revenue of firm 2 can be derived as follows:

$\begin{array}{c}{\text{MR}}_2=\frac{\partial \left(\text{P}\times {\text{q}}_2\right)}{\partial {\text{q}}_2}\\ =\frac{\partial \left(80-{\text{q}}_{\text{1}}-{\text{q}}_2\right){\text{q}}_2}{\partial {\text{q}}_2}\\ =\frac{\partial \left(80{\text{q}}_2-{\text{q}}_1{\text{q}}_2-{\text{q}}_2^2\right)}{\partial {\text{q}}_2}\\ =80-{\text{q}}_{\text{1}}-2{\text{q}}_2\end{array}$

Thus, the marginal revenue of the firm 2 is ${\text{MR}}_2=80-{\text{q}}_1-2{\text{q}}_2$. Thus, the reaction function of firm 1 can be derived by equating the marginal revenue equal to the marginal cost as follows:

$\begin{array}{c}{\text{MR}}_1=\text{MC}\\ \text{80}-2{\text{q}}_1-{\text{q}}_2=20\\ 2{\text{q}}_1=80-20-{\text{q}}_2\\ {\text{q}}_1=\frac{60-{\text{q}}_2}{2}\\ =30-0.5{\text{q}}_2\end{array}$

Similarly, the reaction function of firm 2 can be calculated as follows:

$\begin{array}{c}{\text{MR}}_2=\text{MC}\\ \text{80}-{\text{q}}_1-2{\text{q}}_2=20\\ 2{\text{q}}_2=80-20-{\text{q}}_1\\ {\text{q}}_2=\frac{60-{\text{q}}_1}{2}\\ =30-0.5{\text{q}}_1\end{array}$

The output of the firm 1 can be calculated by substituting the reaction function of 2 in the reaction function of 1 as follows:

$\begin{array}{c}{\text{q}}_1=30-0.5{\text{q}}_2\\ =30-0.5\times \left(30-0.5{\text{q}}_1\right)\\ =30-15+0.25{\text{q}}_1\\ 0.75{\text{q}}_1=15\\ {\text{q}}_1=\frac{15}{0.75}\\ =20\end{array}$

Since both firms face the identical problem, both would produce 20 units each, and the total industrial output would become 40.

(i) The competitive output was 60 units, whereas the cournot output is calculated to be just 40 units. Thus, the proportion of the two market condition outputs can be written as follows:

$\begin{array}{c}\text{Proportion of outputs}=\frac{\text{Output under Cournot}}{\text{Output under competitive market}}\\ =\frac{40}{60}\\ =\frac{2}{3}\end{array}$

Thus, the proportion is $\frac{2}{3}$ .

(ii) The Cournot price can be calculated by substituting the value of Q in the demand equation as follows:

$\begin{array}{c}\text{P}=80-\text{Q}\\ =80-40\\ =40\end{array}$

Thus, the market situation can be illustrated as follows:

In Figure 3, the horizontal axis measures quantity, and the vertical axis measures price.

(d)

To determine

Output and price under the 3 player Cournot competition.

Explanation of Solution

Under the three-player cournot competition, the inverse demand function of the market would be $\text{P}=80-{\text{q}}_1-{\text{q}}_2-{\text{q}}_3$.

Marginal revenue of firm 1 can be derived as follows:

$\begin{array}{c}{\text{MR}}_1=\frac{\partial \left(\text{P}\times {\text{q}}_1\right)}{\partial {\text{q}}_1}\\ =\frac{\partial \left(80-{\text{q}}_{\text{1}}-{\text{q}}_2-{\text{q}}_3\right){\text{q}}_1}{\partial {\text{q}}_1}\\ =\frac{\partial \left(80{\text{q}}_1-{\text{q}}_1^2-{\text{q}}_2{\text{q}}_1-{\text{q}}_3{\text{q}}_1\right)}{\partial {\text{q}}_1}\\ =80-2{\text{q}}_{\text{1}}-{\text{q}}_2-{\text{q}}_3\end{array}$

Thus, the marginal revenues of the firm 1 is ${\text{MR}}_1=80-2{\text{q}}_1-{\text{q}}_2-{\text{q}}_3$.

Marginal revenue of firm 2 can be derived as follows.

$\begin{array}{c}{\text{MR}}_2=\frac{\partial \left(\text{P}\times {\text{q}}_2\right)}{\partial {\text{q}}_2}\\ =\frac{\partial \left(80-{\text{q}}_{\text{1}}-{\text{q}}_2-{\text{q}}_3\right){\text{q}}_2}{\partial {\text{q}}_2}\\ =\frac{\partial \left(80{\text{q}}_2-{\text{q}}_1{\text{q}}_2-{\text{q}}_2^2-{\text{q}}_3{\text{q}}_2\right)}{\partial {\text{q}}_2}\\ =80-{\text{q}}_{\text{1}}-2{\text{q}}_2-{\text{q}}_3\end{array}$

Thus, the marginal revenue of the firm 2 is ${\text{MR}}_2=80-{\text{q}}_1-2{\text{q}}_2-{\text{q}}_3$.

Marginal revenue of firm 2 can be derived as follows:

$\begin{array}{c}{\text{MR}}_3=\frac{\partial \left(\text{P}\times {\text{q}}_3\right)}{\partial {\text{q}}_3}\\ =\frac{\partial \left(80-{\text{q}}_{\text{1}}-{\text{q}}_2-{\text{q}}_3\right){\text{q}}_3}{\partial {\text{q}}_3}\\ =\frac{\partial \left(80{\text{q}}_2-{\text{q}}_1{\text{q}}_3-{\text{q}}_2{\text{q}}_3-{\text{q}}_3^2\right)}{\partial {\text{q}}_3}\\ =80-{\text{q}}_{\text{1}}-{\text{q}}_2-2{\text{q}}_3\end{array}$

Thus, the marginal revenue of the firm 2 is ${\text{MR}}_3=80-{\text{q}}_1-{\text{q}}_2-2{\text{q}}_3$.

All the firms face the marginal cost. The quantity of individual firs is equal $\left({\text{q}}_1={\text{q}}_2={\text{q}}_3\right)$. Thus, the reaction function of firm 1 can be derived as by equating the marginal revenue equal to the marginal cost as follows.

$\begin{array}{c}{\text{MR}}_1=\text{MC}\\ \text{80}-2{\text{q}}_1-{\text{q}}_2-{\text{q}}_3=20\\ \text{80}-2{\text{q}}_1-{\text{q}}_1-{\text{q}}_1=20\\ 80-4{\text{q}}_1=20\\ {\text{q}}_1=\frac{60}{4}\\ =15\end{array}$

Similarly, the reaction function of other firms will be the same. Thus, each firm would produce 15 units each. Thus, the fraction of output would be equal to $\frac{3}{4}$.

(e)

To determine

Fraction of output when there are N firms in the market.

Explanation of Solution

When there are N identical firms in the market, then there will be N fractions produced in the market. Thus, the fraction of output brought to the market by each firm can be written as $\frac{\text{N}}{\text{N}+1}$.

(f)

To determine

Impact when there are N firms in the market.

Explanation of Solution

When there are N identical firms in the market, each firm would produce an identical output, which leads the market to converge into a competitive equilibrium. Thus, the price would also tend to move toward the competitive price in the market. Thus, the consumer surplus would increase, and the deadweight loss would decrease in the market.