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Chapter 10: Problem 6

Promoters of a major college basketball tournament estimate that the demand for tickets on the part of adults is given by \(Q_{a d}=5,000-10 P,\) and that the demand for tickets on the part of students is given by \(Q_{s t}=10,000-100 P .\) The promoters wish to segment the market and charge adults and students different prices. They estimate that the marginal and average total cost of seating an additional spectator is constant at \(\$ 10\) a. For each segment (adults and students), find the inverse demand and marginal revenue functions. b. Equate marginal revenue and marginal cost. Determine the profit-maximizing quantity for each segment. c. Plug the quantities you found in (b) into the respective inverse demand curves to find the profit-maximizing price for each segment. Who pays more, adults or students? d. Determine the profit generated by each segment, and add them together to find the promoter's total profit. e. How would your answers change if the arena where the event was to take place had only 5,000 seats?

Short Answer

Expert verified
Adults pay more than students ($255 vs. $55). Total profit is $801,250 without seating constraints.

Step by step solution

01

Finding the Inverse Demand Functions

For adults, the demand function is \( Q_{ad} = 5000 - 10P \). To find the inverse demand, solve for \( P \):\[ P = 500 - 0.1Q_{ad} \]For students, the demand function is \( Q_{st} = 10000 - 100P \). To find the inverse demand:\[ P = 100 - 0.01Q_{st} \]
02

Deriving Marginal Revenue Functions

To find the marginal revenue (MR), differentiate the total revenue (TR) function \( TR = P imes Q \) with respect to \( Q \). For adults:\[ MR_{ad} = 500 - 0.2Q_{ad} \]For students:\[ MR_{st} = 100 - 0.02Q_{st} \]
03

Equating Marginal Revenue and Marginal Cost

Given that the marginal cost (MC) is constant at \(10\), we equate MR to MC for both segments. Solving for \(Q\):For adults: \( 500 - 0.2Q_{ad} = 10 \) leading to \( Q_{ad} = 2450 \).For students: \( 100 - 0.02Q_{st} = 10 \) leading to \( Q_{st} = 4500 \).
04

Determine Profit-Maximizing Prices

Plug the quantities back into the inverse demand functions to find prices:For adults: \( P = 500 - 0.1 \times 2450 = 255 \).For students: \( P = 100 - 0.01 \times 4500 = 55 \).Hence, adults pay more than students.
05

Calculating Profit for Each Segment

Profit is calculated using \( \text{Profit} = (P - MC) \times Q \).For adults: \( \text{Profit}_{ad} = (255 - 10) \times 2450 = \\(598750 \).For students: \( \text{Profit}_{st} = (55 - 10) \times 4500 = \\)202500 \).Total profit: \( \$801250 \).
06

Adjusting for a Capacity Constraint of 5000 Seats

The total seats sold in the current solution are \(2450 + 4500 = 6950\), which exceeds the arena capacity. Under a constraint of 5000 seats, promoters need to adjust quantities or switch to a single price strategy, optimizing either over a combined demand function or proportionately reducing quantities to stay within capacity limits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Demand Function
The demand function is crucial to understanding how different pricing strategies can maximize profits for businesses. In essence, a demand function expresses the relationship between the price of a good and the quantity demanded by consumers. For a college basketball tournament, like the one discussed, two separate demand functions help determine ticket prices: one for adults and another for students.
  • For adults: \( Q_{ad} = 5000 - 10P \), describes how many tickets adults will buy at various prices.
  • For students: \( Q_{st} = 10000 - 100P \), outlines the purchasing behavior of students based on ticket pricing.
These demand functions allow promoters to forecast ticket sales and adjust prices to achieve optimal revenue depending on different demographics.
Inverse Demand Curve
The inverse demand curve is another important concept, as it shows how the price of tickets needs to be adjusted based on the targeted sale quantities. By rearranging the terms in the demand function to solve for price (\(P\)), we can create an inverse demand function.
  • For adults, the inverse demand function: \( P = 500 - 0.1Q_{ad} \), indicates price based on the number of tickets sold.
  • For students: \( P = 100 - 0.01Q_{st} \), similarly adjusts the price for student sales.
Understanding these relationships helps determine the maximum price consumers are willing to pay for certain quantities, which is essential in setting adjusted prices that capture maximum consumer willingness to pay in each segment.
Marginal Revenue
Marginal revenue plays a pivotal role in determining the revenue gained from selling one additional unit. Calculating marginal revenue involves differentiating the total revenue function (TR) concerning quantity (\(Q\)).
The marginal revenue function can be expressed as:
  • For adults: \( MR_{ad} = 500 - 0.2Q_{ad} \)
  • For students: \( MR_{st} = 100 - 0.02Q_{st} \)
This indicates how revenue changes with each additional ticket sold and is used to find the equilibrium quantity where profit is maximized. By equating marginal revenue to marginal cost, businesses can find the optimal quantity to sell in maximizing profit.
Profit Maximization
Profit maximization is the process of setting prices and quantities that lead to the highest possible profits. For this tournament, the profit-maximizing quantity needs to be calculated for both adults and students by setting the marginal revenue equal to the given marginal cost (MC), which is constant at \(10\) for both segments. This is calculated as follows:
  • For adults: Set \( 500 - 0.2Q_{ad} = 10 \) resulting in a quantity of \( Q_{ad} = 2450 \).
  • For students: Set \( 100 - 0.02Q_{st} = 10 \), giving a quantity of \( Q_{st} = 4500 \).
Subsequently, you can use these quantities to determine price by plugging them back into the inverse demand functions. Ultimately, this strategy ensures that each market segment pays an optimized price, maximizing the promoters' total profit efficiently.

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Most popular questions from this chapter

Rockway \& Daughters Piano Co. wishes to sell a piano to everyone. But some consumers are budgetconscious, and others are not, and unfortunately, Rockway cannot tell which is which. So, Rockway produces a premium line of pianos that it markets under the Rockway name, and a similar line of pianos that it markets under the Dundee name. While the cost of producing these pianos is quite similar, all consumers agree that Rockway pianos are of higher quality than Dundee pianos, and would be willing to pay more for a Rockway. Budget-conscious consumers feel that Dundee pianos are worth $$ 6,000,\( and Rockways are worth \) 8,000 .\( Performance artists believe that Dundee pianos are worth \)\$ 7,000\( and Rockways are worth $$ 12,000\). a. Suppose Rockway & Daughters prices its Dundee pianos at $$ 5,000\( and its Rockway pianos at $$ 10,500 .\) Are these prices incentive compatible - that is, will more price-conscious consumers purchase the Dundee line, while more performance-oriented players choose the Rockway? Explain. b. How much must Rockway \& Daughters reduce the price of its Rockway line in order to achieve incentive compatibility? c. Suppose instead that Rockway \& Daughters tries to achieve incentive compatibility by raising the price of its Dundee line. Can it do so? And if so, how? d. If you ran Rockway \& Daughters Piano Co., which method would you use to achieve incentive compatibility: the one you outlined in (b), or the one you outlined in (c)? Why?

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