Question

As the owner of the only tennis club in an isolated wealthy community, you must decide on membership dues and fees for court time. There are two types of tennis players. “Serious” players have demand

Q1 - 10 - P

where Q1 is court hours per week and P is the fee per hour for each individual player. There are also “occasional” players with demand

Q2 = 4 - 0.25P

Assume that there are 1000 players of each type. Because you have plenty of courts, the marginal cost of court time is zero. You have fixed costs of $10,000 per week. Serious and occasional players look alike, so you must charge them the same prices.

1. Suppose that to maintain a “professional” atmosphere, you want to limit membership to serious players. How should you set the annual membership dues and court fees (assume 52 weeks per year) to maximize profits, keeping in mind the constraint that only serious players choose to join? What would profits be (per week)?
2. A friend tells you that you could make greater profits by encouraging both types of players to join. Is your friend right? What annual dues and court fees would maximize weekly profits? What would these profits be?
3. Suppose that over the years, young, upwardly mobile professionals move to your community, all of whom are serious players. You believe there are now 3000 serious players and 1000 occasional players. Would it still be profitable to cater to the occasional player? What would be the profit-maximizing annual dues and court fees? What would profits be per week?

Short answer

1. The annual membership fee will be $2600, and the court fee will be zero. The total profit will be $40,000.
2. Yes, your friend is right. The annual dues will be $1098.5, and the court fee will be $3. The total profit will be $63,000.
3. The annual dues will be $1053, and the court fee will be $3.27. the total profit will be $147,455.

Step-by-step solution

Explanation for part (a)

The club owner will change the membership fee equal to the serious player's consumer surplus. The consumer surplus of serious players will be:

$$\begin{gathered} Q_1=10-P \\ MC=0 \\ CS=0.5\left(10-0\right)\left(10-0\right) \\ =0.5\times 10\times 10 \\ =\$50 \\ =\$2,600peryear \end{gathered}$$

Thus, the annual entry fee will be $2,600. The court fee is zero.

The total profit will be:

$$\begin{gathered} \mathrm{\pi }=50\left(1000\right)-10,000 \\ =50,000-10,000 \\ =\$40,000 \end{gathered}$$

The total profit will be $40,000 per week.

Explanation for part (b)

If there are both types of players, the club owner will be profitable by setting the court fee above the marginal cost and annual fee equal to the consumer surplus of the type of player with less demand.

The annual fee (T) will be:

$$\begin{gathered} {\text{T = 0.5Q}}_{\text{2}}\left(\text{16 - P}\right) \\ \text{= 0.5}\left(\text{4 - 0.25P}\right)\left(\text{16 - P}\right) \\ {\text{= 32 - 4P + 0.125P}}^{\text{2}} \end{gathered}$$

The total annual fee will be:

$$\begin{gathered} {\text{T}}_{\text{A}}\text{= 2000}\left({\text{32 - 4P + 0.125P}}^{\text{2}}\right) \\ {\text{= 64,000 - 8,000P + 250P}}^{\text{2}} \end{gathered}$$

Total revenue from court fee will be:

$$\begin{gathered} \text{TR = P}\left[\text{1,000}\left(\text{10 - P}\right)\text{+ 1000}\left(\text{4 - 0.25P}\right)\right] \\ {\text{= 14,000P - 1250P}}^{\text{2}} \end{gathered}$$

The total revenue from the annual fee and court fee will be:

${\text{TR = 64,000 + 6000P - 1000P}}^{\text{2}}$

The maximum TR will be:

$$\begin{gathered} \frac{\text{dTR}}{\text{dP}}\text{= 6000 - 200P = 0} \\ \text{P = \$ 3} \end{gathered}$$

The demand for both the players will be:

$$\begin{gathered} {\text{Q}}_{\text{1}}\text{= 10 - 3} \\ \text{= 7} \\ {\text{Q}}_{\text{2}}\text{= 4 - 0.25}\left(\text{3}\right) \\ \text{= 3.25} \end{gathered}$$

The profit will be:

$$\begin{gathered} TR=64,000+6000\left(3\right)-1000\left(3\right) \\ =\$73,000perweek \\ \mathrm{\pi }=73,000-10,000 \\ =\$63,000 \end{gathered}$$

The profit is greater than when the only serious player was a member. Thus, your friend is right.

Explanation for part (c)

There are 4,000 players; then the entry fee will be:

$$\begin{gathered} \text{T = 4000}\left({\text{32 - 4P + 0.125P}}^{\text{2}}\right) \\ {\text{= 128,000 - 16,000P + 500P}}^{\text{2}} \end{gathered}$$

The court fee will be:

$$\begin{gathered} \text{P}\left({\text{3000Q}}_{\text{1}}{\text{+ 1000Q}}_{\text{2}}\right)\text{= P}\left[\text{3000}\left(\text{10 - P}\right)\text{+ 1000}\left(\text{4 - 0.25P}\right)\right] \\ {\text{= 24,000 - 2350P}}^{\text{2}} \end{gathered}$$

The total revenue will be:

$$\begin{gathered} TR=128,000+18000P-2750P^2 \\ \frac{dTR}{dP}=18,000-5500P=0 \\ P=\$3.27perhour \end{gathered}$$

The total profit will be,

$$\begin{gathered} TR=128,000+18,000\left(3.27\right)-2750\left(3.27\right) \\ =\$157,455 \\ \mathrm{\pi }=157,455-10,000 \\ =\$147,455 \end{gathered}$$

The annual dues will be:

$\text{52}\left[\text{32 - 4}\left(\text{3.27}\right)\text{+ 0.125}{\left(\text{3.27}\right)}^{\text{2}}\right]\text{= \$ 1053}$

The annual profit will be,

$\text{52}\left(\text{147.455}\right)\text{= \$ 7.67}\text{million}\text{per}\text{year}$