Chapter 11, Problem 16P

(a)

To determine

The reaction function of the company.

Explanation of Solution

The total revenue of the firm is the price multiplied with the quantity of the firm. The demand for the cable internet service is given as ${\text{q}}_{\text{c}}=100-3{\text{P}}_{\text{c}}+2{\text{P}}_{\text{r}}$ where ${\text{q}}_{\text{c }}{\text{and p}}_{\text{c}}$ represent the quantity demanded as well as the price of the cable internet services. ${\text{P}}_{\text{r}}$ represents the price of the internet services from the telephone. The demand for the telephone via internet is similar and it is ${\text{q}}_{\text{r}}=100-3{\text{P}}_{\text{r}}+2{\text{P}}_{\text{c}}$. Thus, the revenue of the internet service for the cable providers can be calculated as follows:

The marginal revenue curve of firm cable internet service can be derived as follows:

$\begin{array}{c}{\text{MR}}_{\text{C}}=\frac{\partial \left({\text{q}}_{\text{C}}\times {\text{p}}_{\text{C}}\right)}{\partial {\text{p}}_{\text{C}}}\\ =\frac{\partial \left(100-3{\text{p}}_{\text{C}}-2{\text{p}}_{\text{T}}\right){\text{p}}_{\text{C}}}{\partial {\text{p}}_{\text{C}}}\\ =\frac{\partial \left(100{\text{p}}_{\text{C}}-3{\text{p}}_{\text{C}}^2-2{\text{p}}_{\text{T}}{\text{p}}_{\text{C}}\right)}{\partial {\text{p}}_{\text{C}}}\\ =100-6{\text{p}}_{\text{C}}-2{\text{p}}_{\text{T}}\end{array}$

The marginal revenue is ${\text{MR}}_{\text{C}}=100-6{\text{p}}_{\text{C}}-2{\text{p}}_{\text{T}}$.

Through equating the marginal revenue with the marginal cost, the reaction curve of the cable internet service providing firm can be calculated. Here, the marginal cost of providing internet service is assumed to be zero and the reaction curve can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-6{\text{P}}_{\text{C}}+2{\text{P}}_{\text{r}}=0\\ 6{\text{P}}_{\text{C}}=100+2{\text{P}}_{\text{r}}\\ {\text{P}}_{\text{C}}=\frac{100+2{\text{P}}_{\text{r}}}{6}\\ {\text{P}}_{\text{C}}=16.67+0.33{\text{P}}_{\text{r}}\end{array}$

Thus, the reaction function of the cable internet service providing firm is ${\text{P}}_{\text{C}}=16.67+0.33{\text{P}}_{\text{r}}$.

Economics Concept Introduction

Bertrand competition: The Bertrand competition is the model of imperfect competition where the relation and interactions between the firms set their price and their consumers demands their quantities at the price set.

(b)

To determine

The reaction curve of the telephone internet service provider.

Explanation of Solution

The demand for the telephone via internet is similar and it is ${\text{q}}_{\text{r}}=100-3{\text{P}}_{\text{r}}+2{\text{P}}_{\text{c}}$. Thus, the revenue of the internet service for the cable providers can be calculated as follows:

The marginal revenue curve of firm cable internet service can be derived as follows:

$\begin{array}{c}{\text{MR}}_{\text{T}}=\frac{\partial \left({\text{q}}_{\text{T}}\times {\text{p}}_{\text{T}}\right)}{\partial {\text{p}}_{\text{T}}}\\ =\frac{\partial \left(100-3{\text{p}}_{\text{T}}-2{\text{p}}_{\text{C}}\right){\text{p}}_{\text{T}}}{\partial {\text{p}}_{\text{T}}}\\ =\frac{\partial \left(100{\text{p}}_{\text{C}}-3{\text{p}}_{\text{T}}^2-2{\text{p}}_{\text{C}}{\text{p}}_{\text{T}}\right)}{\partial {\text{p}}_{\text{T}}}\\ =100-6{\text{p}}_{\text{T}}-2{\text{p}}_{\text{C}}\end{array}$

The marginal revenue is ${\text{MR}}_{\text{C}}=100-6{\text{p}}_{\text{T}}-2{\text{p}}_{\text{C}}$.

The marginal revenue curve of the firm will be $\text{MR}=100-6{\text{P}}_{\text{r}}+2{\text{P}}_{\text{C}}$. Through equating the marginal revenue with the marginal cost, the reaction curve of the DSL internet service providing firm can be calculated. Here, the marginal cost of providing internet service is assumed to be zero and the reaction curve can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-6{\text{P}}_{\text{r}}+2{\text{P}}_{\text{C}}=0\\ 6{\text{P}}_{\text{r}}=100+2{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{r}}=\frac{100+2{\text{P}}_{\text{C}}}{6}\\ {\text{P}}_{\text{r}}=16.67+0.33{\text{P}}_{\text{C}}\end{array}$

Thus, the reaction function of the telephone DSL internet service providing firm is ${\text{P}}_{\text{r}}=16.67+0.33{\text{P}}_{\text{C}}$.

(c)

To determine

Price, quantity, and profits of each firm when the ATC is zero.

Explanation of Solution

The equilibrium quantity and price can be calculated with the help of the two reaction functions. The reaction function of the firm r can be substituted in the reaction function of firm c to calculate the price as follows:

$\begin{array}{c}{\text{P}}_{\text{C}}=16.67+0.33{\text{P}}_{\text{r}}\\ =16.67+0.33\times \left(16.67+0.33{\text{P}}_{\text{C}}\right)\\ =16.67+5.50+0.11{\text{P}}_{\text{C}}\\ 0.89{\text{P}}_{\text{C}}=22.17\\ {\text{P}}_{\text{C}}=\frac{22.17}{0.89}\\ =24.9\end{array}$

Thus, the price charged should be 25 for the cable internet service firm,  since the DSL internet firm is also facing the similar problem. The DSL firm should also charge the same price. It can be verified as follows:

$\begin{array}{c}{\text{P}}_{\text{r}}=16.67+0.33{\text{P}}_{\text{C}}\\ =16.67+0.33\times \left(16.67+0.33{\text{P}}_{\text{r}}\right)\\ =16.67+5.50+0.11\text{Pr}\\ 0.89{\text{P}}_{\text{r}}=22.17\\ {\text{P}}_{\text{r}}=\frac{22.17}{0.89}\\ =24.9\end{array}$

By substituting the value of P in the demand function, the quantities can be calculated for each firm as follows:

$\begin{array}{c}{\text{q}}_{\text{c}}=100-3{\text{P}}_{\text{c}}+2{\text{P}}_{\text{r}}\\ =100-\left(3\times 25\right)+\left(2\times 25\right)\\ =100-75+50\\ =150-75\\ =75\end{array}$

Thus, the quantity that the cable internet service provider provides is 75, since the DSL firm faces the similar problem. It also serves 75 customers at a price of $25. The ATC of the firm is zero. Therefore, the total revenue of each firm is their total profit and the total profit of each firm can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{c}}={\text{Q}}_{\text{C}}\times {\text{P}}_{\text{C}}\\ =75\times 25\\ =1,875\end{array}$

Thus, each firm would make a profit of $1,875.

(d)

To determine

Price, quantity, and profits of each firm when the cable internet becomes faster.

Explanation of Solution

The demand for the cable internet service is given as ${\text{q}}_{\text{c}}=100-2{\text{P}}_{\text{c}}+3{\text{P}}_{\text{r}}$ where the ${\text{q}}_{\text{c }}{\text{and p}}_{\text{c}}$ represent the quantity demanded as well as the price of the cable internet services. ${\text{P}}_{\text{r}}$ represents the price of the internet services from the telephone. The demand for the telephone via internet is ${\text{q}}_{\text{r}}=100-4{\text{P}}_{\text{r}}+{\text{P}}_{\text{c}}$. Thus, the revenue of the internet service for the cable providers can be calculated as follows:

$\begin{array}{c}\text{TR}={\text{Q}}_{\text{C}}\times {\text{P}}_{\text{C}}\\ ={\text{P}}_{\text{C}}\times \left(100-2{\text{P}}_{\text{C}}+3{\text{P}}_{\text{r}}\right)\\ =100{\text{P}}_{\text{C}}-2{\text{P}}_{\text{C}}{}^2+3{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}\end{array}$

The total revenue is $100{\text{P}}_{\text{C}}-2{\text{P}}_{\text{C}}{}^2+3{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}$.

Marginal revenue can be calculated as follows:

$\begin{array}{c}\text{MR}=\frac{\partial \text{Total revenue}}{\partial {\text{P}}_{\text{C}}}\\ =\frac{\partial 100{\text{P}}_{\text{C}}-2{\text{P}}_{\text{C}}{}^2+3{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}}{\partial {\text{P}}_{\text{C}}}\\ =100-4{\text{P}}_{\text{C}}+3{\text{P}}_{\text{r}}\end{array}$

The marginal revenue is $\text{MR}=100-4{\text{P}}_{\text{C}}+3{\text{P}}_{\text{r}}$.

By setting the MR equal to MC which is zero, the reaction curve of the cable internet firm can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 100-4{\text{P}}_{\text{C}}+3{\text{P}}_{\text{r}}=0\\ 4{\text{P}}_{\text{C}}=100+3{\text{P}}_{\text{r}}\\ {\text{P}}_{\text{C}}=\frac{100+3{\text{P}}_{\text{r}}}{4}\\ {\text{P}}_{\text{C}}=25+0.75{\text{P}}_{\text{r}}\end{array}$

The total revenue for the DSL firm can be calculated as follows:

$\begin{array}{c}\text{TR}={\text{Q}}_{\text{r}}\times {\text{P}}_{\text{r}}\\ ={\text{P}}_{\text{r}}\times \left(100-4{\text{P}}_{\text{r}}+{\text{P}}_{\text{C}}\right)\\ =100{\text{P}}_{\text{r}}-4{\text{P}}_{\text{r}}{}^2+{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}\end{array}$

The total revenue is $100{\text{P}}_{\text{r}}-4{\text{P}}_{\text{r}}{}^2+{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}$.

Marginal revenue can be calculated as follows:

$\begin{array}{c}\text{MR}=\frac{\partial \text{Total revenue}}{\partial {\text{P}}_{\text{C}}}\\ =\frac{\partial 100{\text{P}}_{\text{r}}-4{\text{P}}_{\text{r}}{}^2+{\text{P}}_{\text{C}}{\text{P}}_{\text{r}}}{\partial {\text{P}}_{\text{r}}}\\ =100-8{\text{P}}_{\text{r}}+{\text{P}}_{\text{C}}\end{array}$

By setting the MR equal to MC which is zero, the reaction curve of the DSL internet firm can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 100-8{\text{P}}_{\text{r}}+{\text{P}}_{\text{C}}=0\\ {\text{8P}}_{\text{r}}=100+{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{r}}=\frac{100+{\text{P}}_{\text{C}}}{8}\\ {\text{P}}_{\text{r}}=12.5+0.125{\text{P}}_{\text{C}}\end{array}$

The equilibrium quantity and price can be calculated with the help of the two reaction functions. The reaction function of the firm r can be substituted in the reaction function of firm c to calculate the price as follows:

$\begin{array}{c}{\text{P}}_{\text{C}}=25+0.75{\text{P}}_{\text{r}}\\ {\text{P}}_{\text{C}}=25+0.75\times \left(12.50+0.125{\text{P}}_{\text{C}}\right)\\ {\text{P}}_{\text{C}}=25+9.375+0.093{\text{P}}_{\text{C}}\\ {\text{P}}_{\text{C}}-0.093{\text{P}}_{\text{C}}=34.375\\ {\text{P}}_{\text{C}}=\frac{34.375}{0.907}\\ =37.9\end{array}$

Thus, the price charged should be 37.9 for the cable internet service firm and DSL internet firm price can be calculated as follows:

$\begin{array}{c}{\text{P}}_{\text{r}}=12.50+0.125{\text{P}}_{\text{C}}\\ =12.50+0.125\times \left(37.9\right)\\ =12.50+4.74\\ =17.24\end{array}$

By substituting the value of P in the demand function, the quantities can be calculated for each firm as follows:

$\begin{array}{c}{\text{q}}_{\text{c}}=100-2{\text{P}}_{\text{c}}+3{\text{P}}_{\text{r}}\\ =100-\left(2\times 37.78\right)+\left(3\times 17.17\right)\\ =100-75.56+51.51\\ =151.51-75.56\\ =75.95\end{array}$

Thus, the quantity that the cable internet service provider provides is 76 and the DSL firm can be calculated as follows:

$\begin{array}{c}{\text{q}}_{\text{r}}=100-4{\text{P}}_{\text{r}}+{\text{P}}_{\text{c}}\\ =100-\left(4\times 17.17\right)+37.78\\ =100-68.68+37.78\\ =137.78-68.68\\ =69.10\end{array}$

The output of DSL firm is 69 consumers. The ATC of the firm is zero and the total revenue of each firm is their total profit and the total profit of each firm can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{c}}={\text{Q}}_{\text{C}}\times {\text{P}}_{\text{C}}\\ =76\times 37.93\\ =2,882.68\end{array}$

Similarly, the profit of DSL firm can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{r}}={\text{Q}}_{\text{r}}\times {\text{P}}_{\text{r}}\\ =69\times 17.24\\ =1,189.56\end{array}$

Thus, cable services increases their price and profit through providing increased speed for their internet services. The DSL firm faces a decline in their price and profitability.