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=12−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 are1,000 players of each type. Because you have plenty of​ courts, the marginal cost of court time is ​$0.

You have fixed costs of ​$12,000 per week. Serious and occasional players look​ alike, so you must charge them the same prices.

a. 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)?  ​(round your answers to two decimal​ places)

The annual membership fee is:

The court fee is:

Weekly profit is:

b. 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? (round your answers to two decimal places)

The annual membership fee is:

The court fee is:

Weekly profit is:

c. Suppose that over the years, young, upwardly mobile professionals move to your community, all of whom are serious players. You believe there are now 3500 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? (round your answers to two decimal places)

The annual membership fee is:

The court fee is:

Weekly profit is:

Answer 1

A. In order to limit membership to serious players, the club owner should charge an entry fee, T, equal to the total consumer surplus of serious players and a usage fee P equal to marginal cost of zero.

With individual demands of Q1=120-P, individual consumer surplus is equal to:

=.5*12*12

=$72

For 52 weeks,

=72*52=3744 per year.

An entry fee of $3744 maximizes profits by capturing all consumer surpluses. The profitmaximizing court fee is set to zero, because marginal cost is equal to zero. The entry fee of $3744 is higher than what the occasional players are willing to pay (higher than their consumer surplus at a court fee of zero); therefore, this strategy will limit membership to the serious players only.

Weekly profits would be  

72*1000-12000

=60000

B. When there are two classes of customers, serious and occasional players, the club owner maximizes profits by charging court fees above marginal cost and by setting the entry fee (annual dues) equal to the remaining consumer surplus of the consumer with the lesser demand, in this case, the occasional player. The entry fee, T, equals the consumer surplus remaining after the court fee.

T = (Q2 – 0)(16 – P)(1/2),

where Q2=4-.25P.

Entry fee=0.5(4-0.25P)(16-P)=32-4P-0.125P2

Since there are total of 2000 players. Total entry fee

=2000*(32-4P-0.125P2)

64000-8000P+250P2

Revenues from court fees equal

P(1000Q1+1000Q2)

=P[1000(12−P)+1000(4−0.25P)]

=P(16000-1250P)

=16000P-1250P2

So, total revenue

=64000-8000P+250P2+16000P-1250P2

=64000+8000P-1000P2

This will be max when

dTR/dP=0

Differentiating, we get

8000-2000P=0

P=4 per hour.

Serious players will play 12-P=8 hours per week, and occasional players will demand 4-0.25(4)=3 hours of court time per week. Total revenue is then 64,000+8000*4-1000*42= $80,000 per week.

Net ptofit=80000-12000=68000. This is higer than 60000, as shown in part 1, and hence the friend is right.

C. An entry fee of $72 per week would attract only serious players. With 3500 serious players, total revenues would be 72*3500=$252,000 and profits would be $242,000 per week.

With both serious and occasional players, we may follow the same procedure as in part b. Entry fees would be equal to 4500 times the consumer surplus of the occasional player

4500*(32-4P-0.125P2)

=144000-18000P-562.5P2

Court fees

P(3500Q1+1000Q2)

=P[3500(12-P)+1000(4−0.25P)]

=P(42000-3500P+4000-250P)

=P(46000-3750P)

=46000P-3750P2

Total revenue=144000-18000P-562.5P2+46000P-3750P2

=144000+28000P-4312.5P2

Differentiating and equating to zero

28000-8625P=0

P=3.25

At this fee,

Total revenue=144000+28000*3.25-4312.5*3.252

=189449.

This is lower than revenue with serious players only. So, we will keep the fee at 72 and profit will be 242000.