2. The Snow City ski resort caters to both out-of-town skiers and local skiers.
ID: 1228077 • Letter: 2
Question
2. The Snow City ski resort caters to both out-of-town skiers and local skiers. The demand for ski tickets for each market segment is independent of the other market segment. The marginal cost of servicing a skier of either type is $40. Suppose the demand curves for the two market segments are:
Out-of-towners: Q0=1,000-5P
Locals: Q0=1,000-10P
(a) If the resort charges one price to all skiers, what is the profit maximizing price. Calculate how many tickets will be sold to each group. What is the total profit? (Show all of your work)
(b) If the company sells tickets at different prices to the two market prices what is the optimal price and quantity for each segment? What are the total profits for the resort? (Show all of your work)
Explanation / Answer
(a) Profit is maximized by equating total marginal revenue (MR) with marginal cost (MC).
Total Market demand is the horizontal summation of individual market demands:
Q = 1,000 - 5P + 1,000 - 10P
Q = 2,000 - 15P
15P = 2,000 - Q
P = (2,000 - Q) / 15
Total revenue, TR = P x Q = (2,000Q - Q2) / 15
MR = dTR / dQ = (2,000 - 2Q) / 15
Equating with MC,
(2,000 - 2Q) / 15 = 40
2,000 - 2Q = 600
2Q = 2,000 - 600 = 1,400
Q = 700
P = (2,000 - 700) / 15 = 1,300 / 15 = 86.67
Quantity (Out-of-towners) = 1,000 - (5 x 86.67) = 1,000 - 433.33 = 567 (integer value)
Quantity (locals) = 1,000 - (10 x 86.67) = 1,000 - 867 = 133 (integer value)
Total Profit = Revenue - Total cost = (P x Q) - (MC x Q) = Q x (P - MC)
= 700 x $(86.67 - 40) = 700 x $46.67 = $32,669
(b) Profit is maximized when MR1 = MR2 = MC
In out-of-towners market:
Q1 = 1,000 - 5P1
P1 = (1,000 - Q1) / 5 = 200 - 0.2Q1
TR1 = P1 x Q1 = 200Q1 - 0.2Q12
MR1 = dTR1 / dQ1 = 200 - 0.4Q1
Equating with MC,
200 - 0.4Q1 = 40
160 = 0.4Q1
Q1 = 400
P1 = 200 - (0.2 x 400) = 200 - 80 = 120
In Locals market:
Q2 = 1,000 - 10P2
P2 = (1,000 - Q2) / 10 = 100 - 0.1Q2
TR2 = P2 x Q2 = 100Q2 - 0.1Q22
MR2 = 100 - 0.2Q2
Equating with MC,
100 - 0.2Q2 = 40
0.2Q2 = 60
Q2 = 300
P2 = 100 - (0.1 x 300) = 100 - 30 = 70
Total profit ($) = (P1 x Q1) + (P2 x Q2) - TC = (P1 x Q1) + (P2 x Q2) - MC x (Q1 + Q2)
= (120 x 400) + (70 x 300) - 40 x (400 + 300)
= 48,000 + 21,000 - 40 x 700
= 69,000 - 28,000
= 41,000
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