Wet-n-Wild Indoor Water Park offers family fun year-round in the Northstar state

Question

Wet-n-Wild Indoor Water Park offers family fun year-round in the Northstar state to locals and out-of-state visitors to the nearby Mall of America. The demand for day-passes to the water park for each market segment is independent of the other market segment. The marginal cost of providing service to each visitor is $5 per day. Suppose the daily demand curves for the two market segments are:

Locals: QL = 3000-200P OR P = 15-.005QL
Out of town: QO = 3000-100P OR P=30-.01QO

a. If Wet-n-Wild Indoor Water Park charges one price to all visitors, what is the profit maximizing price? How many day-passes will be sold per day?

b. If Wet-n-Wild Indoor Water Park charges one price to locals, what is the profit maximizing price for locals? How many day-passes will be sold per day to locals?

c. If Wet-n-Wild Indoor Water Park charges one price to out-of-towners, what is the profit maximizing price for out-of-town guests? How many day-passes will be sold per day to out-of-town guests?

d. Compare the prices from uniform pricing to the prices from price discrimination.

Please show all work.

Explanation / Answer

(a) Let Q = QL + QO (number of all visitors)

Then,

Q = 3,000 - 200P + 3,000 - 100P

Q = 6,000 - 300P

P = (6,000 - Q) / 300

Profit is maximized by equating Marginal revenue (MR) with MC.

Total revenue (TR) = P x Q = (6,000Q - Q2) / 300

MR = dTR / dQ = (6,000 - 2Q) / 300 = (3,000 - Q) / 150

Equating with MC,

(3,000 - Q) / 150 = 5

3,000 - Q = 750

Q = 2,250

P = (6,000 - Q) / 300 = (6,000 - 2,250) / 300 = 3,750 / 300 = 12.5

(b) For local visitors,

TR = P x Quantity = 15QL - 0.005QL2

MR = dTR / dQL = 15 - 0.01QL

Equating with MC,

15 - 0.01QL = 5

0.01QL = 10

QL = 1,000

P = 15 - (0.005 x 1,000) = 15 - 5 = 10

(c) For out-of-town visitors:

TR = P x Quantity = 30QO - 0.01QO2

MR = dTR / dQO = 30 - 0.02QO

Equating with MC,

30 - 0.02QO = 5

0.02QO = 25

QO = 1,250

P = 30 - (0.01 x 1,250) = 30 - 12.5 = 17.5

(d)

Uniform price = 12.5

Discriminatory price for Locals = 10, which is lower than uniform price

Discriminatory price for out-of-town visitors = 17.5, which is higher than uniform price.

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