Question

Promoters of a concert featuring a popular indie band estimate that the demand for tickets on the part of adults is given by Q_A = 5,000 – 10P, and the demand for tickets on the part of students is given by Q_S = 10,000 – 100P. The promoters wish to segment the market and charge adults and students different prices. They estimate that the cost of seating each spectator is constant at $10.

1. For each segment (adults & students), find the inverse demand and marginal revenue functions.
2. Equate marginal revenue and marginal cost. Determine the profit-maximizing quantity for each segment.
3. 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?
4. Determine the profit generated by each segment and add them together to find the promoter’s total profit.
5. How would your answers change if the arena where the event was to take place had only 5,000 seats? (Note that you may have to revisit some of your earlier equations to successfully answer part ‘e’)

Answer 1

(a)

For Adults,

QA = 5,000 - 10P

10P = 5,000 - QA

P = 500 - 0.1QA (inverse demand)

TR = P x QA = 500QA - 0.1QA²

MR = dTR/dQA = 500 - 0.2QA

For Students,

QS = 10,000 - 100P

100P = 10,000 - QS

P = 100 - 0.01QS (inverse demand)

TR = P x QS = 100QS - 0.01QS²

MR = dTR/dQS = 100 - 0.02QS

(b)

For Adults,

500 - 0.2QA = 10

0.2QA = 490

QA = 2450

For Students,

100 - 0.02QS = 10

0.02QS = 90

QS = 4500

(c)

For Adults,

P = 500 - 0.1 x 2450 = 500 - 245 = 255

For Students,

P = 100 - 0.01 x 4500 = 100 - 45 = 55

(d)

For Adults,

Profit = QA x (P - MC) = 2450 x (255 - 10) = 2450 x 245 = 600250

For Students,

Profit = QB x (P - MC) = 4500 x (55 - 10) = 4500 x 45 = 202500

Total profit = 600250 + 202500 = 802750

NOTE: As per Chegg Answering Policy, 1st 4 parts are answered.