Question

Suppose there are three types of consumers who attend concerts at your university's performing arts center: students, staff, and faculty. Each of these groups has a different willingness to pay for tickets; within each group, willingness to pay is identical. There is a fixed cost of \(\$ 1,000\) to put on a concert, but there are essentially no variable costs. [LO 14.6\(]\) For each concert: \- There are 140 students willing to pay \(\$ 20\). \- There are 200 staff members willing to pay \(\$ 35\). \- There are 100 faculty members willing to pay \(\$ 50 .\) a. If the performing arts center can charge only one price, what price should it charge? b. What are profits at this price? c. If the performing arts center can price discriminate and charge two prices, one for students and another for faculty/staff, what are its profits? d. If the performing arts center can perfectly price discriminate and charge students, staff, and faculty three separate prices, what are its profits?

Short answer

a. Charge $35 and profit is $9,500. b. Profit is $9,500. c. Profits are $12,300. d. Profits are $13,800.

Step-by-step solution

Calculate Revenue at Different Prices

Compute the revenue if only one price is charged. Consider prices based on the willingness to pay of students ( $20), staff ($35), or faculty ($50). For each price, calculate the total number of tickets sold: - At $20: 140 students + 200 staff + 100 faculty = 440 tickets - At $35: 200 staff + 100 faculty = 300 tickets - At $50: 100 faculty = 100 tickets Calculate the total revenue for each scenario: - Revenue at $20: 440 * $20 = $8,800 - Revenue at $35: 300 * $35 = $10,500 - Revenue at $50: 100 * $50 = $5,000

Determine Optimal Single Price and Calculate Profit

Choose the price that maximizes revenue from Step 1. The highest revenue is obtained at a price of \(35 with revenue of \)10,500. Calculate profit using the formula:\[ \text{Profit} = \text{Revenue} - \text{Fixed Cost} \]- Profit at \(35 = \)10,500 - \(1000 (fixed cost) = \)9,500

Calculate Optimized Revenue with Two Prices

Consider price discrimination by separately charging faculty/staff and students: - If students are charged $20: 140 students * $20 = $2,800 - If faculty/staff are charged $35: 300 (200 staff + 100 faculty) * $35 = $10,500 Combine the revenue from both student and faculty/staff tickets: - Total Revenue = $2,800 + $10,500 = $13,300 - Profit = $13,300 - $1,000 = $12,300

Calculate Maximum Profit Through Perfect Price Discrimination

Consider charging each group according to their willingness to pay:- Students pay \(20: 140 * \)20 = \(2,800- Staff pay \)35: 200 * \(35 = \)7,000- Faculty pay \(50: 100 * \)50 = \(5,000Total revenue with perfect price discrimination would be:\[ \text{Total Revenue} = \)2,800 + \(7,000 + \)5,000 = \(14,800 \]- Profit = \)14,800 − \(1,000 = \)13,800.

Key concepts

Consumer Willingness to Pay

Understanding 'willingness to pay' is crucial in pricing strategies, especially when dealing with diverse customer groups. Each consumer group values products differently based on their needs, preferences, and financial circumstances.
In our concert scenario, we have distinct groups: students, staff, and faculty, each with a different maximum price they are ready to pay for a ticket.

For instance:

• Students: 140 individuals willing to pay $20 each.
• Staff: 200 individuals willing to pay $35 each.
• Faculty: 100 individuals willing to pay $50 each.

Recognizing these differences allows the performing arts center to tailor its pricing strategy to maximize attendance and revenue. While students have the lowest willingness to pay, faculty are comfortable paying the highest price. If only one ticket price is to be set, understanding these willingness levels helps to decide on a price that will maximize the total revenue from all ticket sales.

Revenue Calculation

Calculating revenue accurately is vital for understanding the financial performance of an event or service. Revenue is calculated by multiplying the number of tickets sold by the price per ticket.
Let's see how revenue varies depending on the concert ticket prices:

If a uniform price is applied:

• At \(20 per ticket, all 440 people from each group are likely to buy, resulting in revenue of \( 440 \times 20 = \\)8,800 \).
• At \(35, only staff and faculty (300 tickets) are likely to buy, generating \( 300 \times 35 = \\)10,500 \).
• If tickets are priced at \(50, only 100 faculty members buy, totaling \( 100 \times 50 = \\)5,000 \).

Among these options, charging \(35 maximizes revenue at \)10,500 since it covers two groups (staff and faculty) with significant numbers. This calculation is crucial because it precedes profit determination, considering costs to reveal the most financially viable price.

Profit Maximization

Profit maximization is the ultimate goal for businesses after fulfilling consumer needs. It's the definitive measure of a company’s success. Profit is calculated by subtracting total costs from total revenue.

For a single pricing strategy:

• If tickets are sold at \(35, the profit is \( \\)10,500 \text{ (revenue)} - \\(1,000 \text{ (fixed cost)} = \\)9,500 \).

Profit can increase with price discrimination, a method where different prices are set for different consumer groups based on their willingness to pay. When employing two prices:

• Students are charged \(20, generating \)2,800 in revenue.
• Staff and faculty are charged \(35, generating \)10,500.
• This brings the total revenue to \(13,300, leading to a profit of \( \\)13,300 - \\(1,000 = \\)12,300 \).

Finally, perfect price discrimination would involve charging each group their respective maximum willing price. This results in total revenues of \(14,800, which maximizes profit to \( \\)14,800 - \\(1,000 = \\)13,800 \).
By adapting prices to consumer willingness and maximizing revenues efficiently, businesses can significantly enhance profits.