Chapter 10, Problem 6P

(a)

To determine

The inverse demand and the marginal revenue functions of an adult and a student.

Explanation of Solution

The demand from the adults is given as ${\text{Q}}_{\text{ad}}=5,000-10\text{P}$. The inverse demand function of the adults can be calculated as follows:

$\begin{array}{c}{\text{Q}}_{\text{ad}}=5,000-10\text{P}\\ \text{10P}=5,000-{\text{Q}}_{\text{ad}}\\ \text{P}=500-0.1{\text{Q}}_{\text{ad}}\end{array}$

Marginal revenue equation for adult can be derived as follows.

$\begin{array}{c}\text{MR}=\frac{\partial \left(\text{P}\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(500-0.1\text{Q}\right)\text{Q}}{\partial \text{Q}}\\ =\frac{\partial \left(500\text{Q}-0.1{\text{Q}}^2\right)}{\partial \text{Q}}\\ =500-0.2\text{Q}\end{array}$

Thus, the marginal revenue curve is $\text{MR}=500-0.2\text{Q}$.

The inverse demand curve of the students can be calculated as follows:

$\begin{array}{c}{\text{Q}}_{\text{stud}}=10,000-100\text{P}\\ \text{100P}=10,000-{\text{Q}}_{\text{stud}}\\ \text{P}=100-0.01{\text{Q}}_{\text{stud}}\end{array}$

Marginal revenue equation for adult can be derived as follows.

$\begin{array}{c}\text{MR}=\frac{\partial \left(\text{P}\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(100-0.01\text{Q}\right)\text{Q}}{\partial \text{Q}}\\ =\frac{\partial \left(100\text{Q}-0.01{\text{Q}}^2\right)}{\partial \text{Q}}\\ =100-0.02\text{Q}\end{array}$

Thus, the marginal revenue curve will be $\text{MR}=100-0.02\text{Q}$.

Economics Concept Introduction

Price discrimination: The price discrimination is the practice of charging different price from different consumers for the exact same commodity on the basis of the quantity bought or by the place or the group of individuals.

(b)

To determine

The profit maximizing output for adults and students.

Explanation of Solution

The profit maximizing output for adults and students can be calculated by equating the marginal revenue from each group with the marginal cost of production. The output of adults can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{500}-0.2\text{Q}=10\\ 490=0.2\text{Q}\\ \text{Q}=\frac{490}{0.2}\\ =2,450\end{array}$

Similarly, the profit maximizing output of students can be calculated as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ \text{100}-0.02\text{Q}=10\\ 90=0.02\text{Q}\\ \text{Q}=\frac{90}{0.02}\\ =4,500\end{array}$

Thus, the profit maximizing output is 2,450 for the adults and 4,500 for the students.

(c)

To determine

Profit maximizing price for adults and students.

Explanation of Solution

The profit maximizing price can be obtained by plugging the profit maximizing output into the inverse demand function as follows:

$\begin{array}{c}{\text{P}}_{\text{Adults}}=500-0.1\text{Q}\\ =500-\left(0.1\times 2,450\right)\\ =500-245\\ =255\end{array}$

Similarly, the profit maximizing price for the students can be calculated as follows:

$\begin{array}{c}{\text{P}}_{\text{students}}=100-0.01\text{Q}\\ =100-\left(0.01\times 4,500\right)\\ =100-45\\ =55\end{array}$

Thus, the profit maximizing price for adults is $255 and for students is $55.

(d)

To determine

Profit from adults and students.

Explanation of Solution

The profit from the adults can be calculated by subtracting the total cost from the total revenue as follows:

$\begin{array}{c}{\text{Profit}}_{\text{Adults}}=\text{TR}-\text{TC}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(\text{Average cost}\times \text{Quantity}\right)\\ =\left(255\times 2450\right)-\left(10\times 2450\right)\\ =624,750-24,500\\ =600,250\end{array}$

Similarly, the profit from the students can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{students}}=\text{TR}-\text{TC}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(\text{Average cost}\times \text{Quantity}\right)\\ =\left(55\times 4500\right)-\left(10\times 4500\right)\\ =247,500-45000\\ =202,500\end{array}$

Thus, the profit from adults is $600,250 and from students is $202,500. Thus, the total profit can be calculated as follows:

$\begin{array}{c}\text{Total profit}={\text{Profit}}_{\text{Students}}+{\text{Profit}}_{\text{Adults}}\\ =202,500+600,250\\ =802,750\end{array}$

Thus, the total profit is $802,750.

(e)

To determine

Change in calculation when there are only 5,000 seats.

Explanation of Solution

When there are only 5000 seats available in the market, the adults must be offered 2450 seats at the cost of $255 for each which would provide a profit of $600,250 to the firm. There will be 2550 seats remaining and the profit maximizing price from students for this seats can be calculated as follows:

$\begin{array}{c}{\text{P}}_{\text{students}}=100-0.01\text{Q}\\ =100-\left(0.01\times 2,550\right)\\ =100-25.5\\ =74.5\end{array}$

Thus, the remaining 2550 seats can be sold to the students at a price of maximum $74.50. Thus, the profit from the students can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{students}}=\text{TR}-\text{TC}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(\text{Average cost}\times \text{Quantity}\right)\\ =\left(74.50\times 2550\right)-\left(10\times 2550\right)\\ =189,975-25500\\ =164,475\end{array}$

Thus, the profit from adults is $600,250 and from students is $164,475. Thus, the total profit can be calculated as follows:

$\begin{array}{c}\text{Total profit}={\text{Profit}}_{\text{Students}}+{\text{Profit}}_{\text{Adults}}\\ =164,475+600,250\\ =764,725\end{array}$

Thus, the total profit is $764,725.