Chapter 10, Problem 10P

(a)

To determine

The price charged for the frequent golfers, and infrequent golfers, and the rounds used by each, and the profit of the club.

Explanation of Solution

The demand from the frequent golfer is given by ${\text{Q}}_{\text{F}}=24-0.3\text{P}$. Thus, the inverse demand function of the frequent golfers can be calculated as follows:

$\begin{array}{c}{\text{Q}}_{\text{F}}=24-0.3\text{P}\\ \text{0}\text{.3P}=24-{\text{Q}}_{\text{F}}\\ \text{P}=\frac{24-{\text{Q}}_{\text{F}}}{0.3}\\ =80-\frac{1}{0.3}{\text{Q}}_{\text{F}}\end{array}$

Marginal revenue equation can be derived as follows.

$\begin{array}{c}{\text{MR}}_{\text{f}}=\frac{\partial \left(\text{P}\times {\text{Q}}_{\text{F}}\right)}{\partial {\text{Q}}_{\text{F}}}\\ =\frac{\partial \left(80-\frac{1}{0.3}{\text{Q}}_{\text{F}}\right){\text{Q}}_{\text{F}}}{\partial {\text{Q}}_{\text{F}}}\\ =\frac{\partial \left(80{\text{Q}}_{\text{F}}-\frac{1}{0.3}{\text{Q}}_{\text{F}}^2\right)}{\partial {\text{Q}}_{\text{F}}}\\ =80-\frac{2}{0.3}{\text{Q}}_{\text{F}}\end{array}$

Since the marginal revenue will have double slope value, the marginal revenue curve of the frequent golfer will be ${\text{MR}}_{\text{f}}==80-\frac{2}{0.3}{\text{Q}}_{\text{F}}$.

The marginal cost is $20. The profit maximizing price of the frequent golfers can be calculated by equating the marginal revenue with the marginal cost as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 80-\frac{2}{0.3}{\text{Q}}_{\text{F}}=20\\ 80-20=\frac{2}{0.3}{\text{Q}}_{\text{F}}\\ 60=\frac{2}{0.3}{\text{Q}}_{\text{F}}\\ 2{\text{Q}}_F=60\times 0.3\\ {\text{Q}}_F=\frac{18}{2}\\ =9\end{array}$

Thus, the equilibrium quantity is 9. It can be plugged into the price equation to calculate the price as follows:

$\begin{array}{c}\text{P}=80-\frac{1}{0.3}{\text{Q}}_{\text{F}}\\ =80-\frac{1}{0.3}\times 9\\ =80-\frac{9}{0.3}\\ =80-30\\ =50\end{array}$

Thus, the profit maximizing price from the frequent golfers is equal to $50. The total profit that the club could make from the frequent golfers can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{frequent}}=\text{TR}-\text{TC}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(\text{Average cost}\times \text{Quantity}\right)\\ =\left(50\times 9\right)-\left(20\times 9\right)\\ =450-180\\ =270\end{array}$

Thus, the club could make a profit of $270 from the frequent golfers. The price charged from the frequent golfers is equal to $50 and the optimal quantity of rounds of gold demanded by frequent golfers is 9. Similarly, the price, quantity and the profit from the infrequent golfers can be calculated as follows:

The demand from the infrequent golfer is given by ${\text{Q}}_{\text{I}}=10-0.1\text{P}$. The inverse demand function of the infrequent golfers can be calculated as follows:

$\begin{array}{c}{\text{Q}}_{\text{I}}=10-0.1\text{P}\\ \text{0}\text{.1P}=10-{\text{Q}}_{\text{I}}\\ \text{P}=\frac{10-{\text{Q}}_{\text{I}}}{0.1}\\ =100-10{\text{Q}}_{\text{I}}\end{array}$

Marginal revenue equation can be derived as follows.

$\begin{array}{c}{\text{MR}}_{\text{I}}=\frac{\partial \left(\text{P}\times {\text{Q}}_{\text{I}}\right)}{\partial {\text{Q}}_{\text{I}}}\\ =\frac{\partial \left(100-10{\text{Q}}_{\text{I}}\right){\text{Q}}_{\text{I}}}{\partial {\text{Q}}_{\text{I}}}\\ =\frac{\partial \left(100{\text{Q}}_{\text{I}}-10{\text{Q}}_{\text{I}}^2\right)}{\partial {\text{Q}}_{\text{I}}}\\ =100-20{\text{Q}}_{\text{I}}\end{array}$

The marginal revenue curve of the frequent golfer is ${\text{MR}}_{\text{I}}=100-20{\text{Q}}_{\text{I}}$.

The marginal cost is $20. The profit maximizing price of the infrequent golfers can be calculated by equating the marginal revenue with the marginal cost as follows:

$\begin{array}{c}\text{MR}=\text{MC}\\ 100-20{\text{Q}}_{\text{I}}=20\\ 100-20=20{\text{Q}}_{\text{I}}\\ 80=20{\text{Q}}_{\text{I}}\\ {\text{Q}}_{\text{I}}=\frac{80}{20}\\ =4\end{array}$

Thus, the equilibrium quantity is 4. It can be plugged into the price equation to calculate the price as follows:

$\begin{array}{c}\text{P}=100-10{\text{Q}}_{\text{I}}\\ =100-\left(10\times 4\right)\\ =100-40\\ =60\end{array}$

Thus, the profit maximizing price from the infrequent golfers is equal to $60. The total profit that the club could make from the infrequent golfers can be calculated as follows:

$\begin{array}{c}{\text{Profit}}_{\text{infrequent}}=\text{TR}-\text{TC}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(\text{Average cost}\times \text{Quantity}\right)\\ =\left(60\times 4\right)-\left(20\times 4\right)\\ =240-80\\ =160\end{array}$

Thus, the club could make a profit of $160 from the infrequent golfers. The price charged from the infrequent golfers is $60 per round and the quantity demanded by them is only 4 rounds. Thus, the total profit that the firm could make can be calculated as follows:

$\begin{array}{c}\text{Total profit}={\text{Profit}}_{\text{Frequent}}+{\text{Profit}}_{\text{Infrequent}}\\ =270+160\\ =430\end{array}$

Thus, the club could make a total profit of $430 from the two group of consumers.

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 price for the individual round of golf.

Explanation of Solution

When the club is not possible to tell which consumer is a frequent golfer and which consumer is an infrequent golfer, the club should set a price which is equal to the price of $60. The marginal cost and the marginal revenue of the infrequent consumers equals at $60 which means it maximizes the profit of the club.

(c)

To determine

The price for the individual round of golf in discount plan.

Explanation of Solution

When the club is planning to maximize their profit through a discount plan, then the club should follow the method of quantity discounting where the minimum quantity limit should be set. The price should be $50 per round when the minimum of 9 rounds is demanded to be eligible for the discounts. This is the profit maximizing price from the frequent golfers. Those who are not demanding this minimum quantity should be charged $60 per round.

(d)

To determine

The consumer surplus of the frequent golfers under the two schemes.

Explanation of Solution

The inverse demand curve of the frequent golfers can be illustrated as follows:

When the individual rounds are priced $60 for each round, the frequent golfers demands for 6 rounds and when the discounted price is offered, they go for 9 rounds. Thus, the consumer surplus from the individual pricing will be equal to the area of A which can be calculated as follows:

$\begin{array}{c}{\text{Consumer Surplus}}_{\text{Frequent-individual price}}=\frac{1}{2}\times \left(80-60\right)\times \left(6-0\right)\\ =\frac{1}{2}\times 20\times 6\\ =60\end{array}$

Thus, the consumer surplus of the frequent golfers under the individual pricing is $60. When the price is discounted, the consumer surplus becomes the area of A+B+C and it can be calculated as follows:

$\begin{array}{c}{\text{Consumer Surplus}}_{\text{Frequent-Discounted price}}=\frac{1}{2}\times \left(80-50\right)\times \left(9-0\right)\\ =\frac{1}{2}\times 30\times 9\\ =135\end{array}$

Thus, the consumer surplus of the frequent golfers from the discounted pricing is $135. This means that the consumer surplus is maximum for the frequent golfers under the discounted pricing.

(e)

To determine

The consumer surplus of the infrequent golfers under the two schemes.

Explanation of Solution

The inverse demand curve of the frequent golfers can be illustrated as follows:

When the individual rounds are priced $60 for each round, the infrequent golfers demands for 4 rounds and when the discounted price is offered, they go for 5 rounds. Thus, the consumer surplus from the individual pricing will be equal to the area of A which can be calculated as follows:

$\begin{array}{c}{\text{Consumer Surplus}}_{\text{Infrequent-individual price}}=\frac{1}{2}\times \left(100-60\right)\times \left(4-0\right)\\ =\frac{1}{2}\times 40\times 4\\ =80\end{array}$

Thus, the consumer surplus of the infrequent golfers under the individual pricing is $80. When the price is discounted, the infrequent consumers have to go for 9 rounds. When the price is $50, the ordinary infrequent golfers would go for 5 rounds and thus, the consumer surplus for 5 rounds can be calculated as follows:

$\begin{array}{c}{\text{Consumer Surplus}}_{\text{Infrequent-Discounted price}}=\frac{1}{2}\times \left(100-50\right)\times \left(5-0\right)\\ =\frac{1}{2}\times 50\times 5\\ =125\end{array}$

Thus, the consumer surplus of the infrequent golfers from the discounted pricing is $125. But this option will not be possible because the discounted price is applicable only when the infrequent consumers demand for 9 rounds. This means that the infrequent golfers has to go for 4 additional rounds which creates a negative consumer surplus to the infrequent golfers equal to the area of D. The area of D can be calculated as follows:

$\begin{array}{c}\text{Negative consumer surplus}=\frac{1}{2}\times \left(10-50\right)\times 4\\ =\frac{1}{2}\times -40\times 4\\ =-80\end{array}$

Thus, the total surplus of the infrequent golfers from the discounted plan can be calculated by summating the positive and negative consumer surpluses as follows:

$\begin{array}{c}{\text{Total consumer surplus}}_{\text{Infrequent-discounted price}}=\text{CS}+\left(-\text{CS}\right)\\ =125+\left(-80\right)\\ =125-80\\ =45\end{array}$

Thus, the total consumer surplus from the discounted plan to the infrequent golfers is only $45 which is lower than that of the individual pricing. Thus, the infrequent golfers would go for the individual pricing strategy.

(f)

To determine

The successful plan for the frequent and infrequent golfers.

Explanation of Solution

When the club offers the individual pricing and the discounted plans, the frequent golfers would go for the discounted plan as it maximizes their consumer surplus and the infrequent golfers go for the individual plan as it is the plan that maximizes their consume surplus. Thus, self selecting option is more successful in the club.

(g)

To determine

The successful plan for the frequent and infrequent golfers when the type of player is indicated.

Explanation of Solution

When the club is able to identify the type of players, the club could charge $50 for frequent golfers and $60 for infrequent golfers rather than offering a self select option. But the profit made by the club would be exactly the same as self select option.