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 $$Q_{1}=10-P$$ where \(Q_{1}\) is court hours per week and \(P\) is the fee per hour for each individual player. There are also "occasional" players with demand $$Q_{2}=4-0.25 P$$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. 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)? 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? 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 3000 serious players and 1000 occasional players. Would it still be profitable to cater to the occasional player? What would be the profitmaximizing annual dues and court fees? What would profits be per week?

Short answer

a. To maximize profits while catering only to serious players, the optimal fee per hour is around $5, leading to a weekly profit of about $12,500. \nb. By adjusting to include both serious and occasional players, the optimal fee per hour is roughly $2.67, which leads to a higher weekly profit of around $13,335.67. \nc. With an increased number of serious players (3,000) and fixed occasional players (1,000), the optimal fee per hour is still around $2.67, and the weekly profits rise significantly to around $23,335.67. It's thus still profitable to cater to the occasional player alongside the serious players with these demographics.

Step-by-step solution

Maximizing Profits Catering to Only 'Serious' Players

Starting with the demand formula for the serious players \(Q_{1}=10-P\), isolate P to get \(P=10-Q_1\). Multiply this with \(Q_1\) to form the Revenue function \(R_s = P * Q_{1} = (10-Q_{1})*Q_{1}\). The profit function will be \(R_s - FC\), where FC is the weekly fixed cost. Use the first derivative to find the value of \(Q_1\) for which the profit is maximized. The profit per week will be the maximum profit minus the fixed costs.

Maximizing Profits Catering to Both 'Serious' and 'Occasional' Players

First, we find the Revenue function for the occasional players using the same procedure as above and sum it with the revenue from the serious players. The profit function in this case will be the total revenue - fixed costs. Again, we use the first derivative to find the value of P which maximizes the profit. Calculate the overall profit per week by subtracting the fixed cost from the total revenue.

Adjusting to New Membership Demographics

Considering the change in the number of players, we adjust the previous steps to account for the additional serious players. We again find the value of P that maximizes profit and calculate the profit per week as total revenue minus fixed costs. We can then compare that with the profit when catering only to serious players to determine whether it's still profitable to cater to the occasional player.

Key concepts

Demand Curve

Understanding the demand curve is essential in determining how the quantity demanded by consumers changes with price. In this problem, we're looking at two distinct demand curves: one for 'serious' players and another for 'occasional' players. The demand curve for Serious players is represented as \( Q_1 = 10 - P \), where \( Q_1 \) signifies the number of court hours and \( P \) is the fee per hour. As the price per hour increases, the number of hours demanded decreases.
Similarly, for occasional players, the demand curve is \( Q_2 = 4 - 0.25P \). This reveals that occasional players are more sensitive to price changes as they’ll consume fewer hours even with minimal changes in court fees.
An important takeaway here is how demand curves can guide pricing strategies. By understanding these curves, businesses can decide pricing that encourages the desired type of consumer behavior, like attracting more serious players through dues and fees.

Marginal Cost

Marginal cost is the cost of producing one additional unit of a good. In this scenario, since the tennis club has plenty of courts, the marginal cost of court time is zero.
This means any additional hour of play doesn't cost anything extra to the club, aside from the existing fixed costs. Therefore, the club can fully focus on determining membership dues and court fees based on maximizing revenues without worrying about additional costs per extra hour of court time.
The absence of marginal costs simplifies profit maximization since revenue from each additional player hour directly contributes to covering fixed costs and generating profit.

Fixed Costs

Fixed costs are expenses that do not change with the level of goods or services produced, such as rent or salaries. In this exercise, the weekly fixed costs are fixed at $10,000.
Regardless of the number of players or hours played, these costs must be covered through revenue generated from membership dues and court fees.
For a tennis club, effective management of fixed costs is crucial. Keeping them in check while maximizing revenue is key to achieving sustainable profits, given that they don't vary with fluctuations in membership or court usage.

Revenue Function

The revenue function helps show how much income a business can earn from its pricing decisions. For serious players, the revenue function starting with the demand curve \( Q_1 = 10 - P \) is \( R_s = P \times Q_1 = (10 - Q_1) \times Q_1 \). This function represents the earning potential when setting different fees for court hours.
Similarly, by observing the revenue function for occasional players, you can construct a comparable model reflecting their specific demand curve, \( Q_2 = 4 - 0.25P \).
These revenue functions enable an assessment of various pricing strategies. By calculating derivatives, you can determine the optimal pricing to maximize profit, which involves identifying a point where the profit is the highest after considering fixed costs.