Question

Wu owns a local business that provides email updates on surf conditions. She is the only supplier of these email updates in Misty Palms and Gaoling, which gives her a monopoly in both cities. The marginal cost of producing another update is zero (and we'll ignore fixed costs). The inverse demand for these updates in Misty Palms is P = 79 - Q and the inverse demand in Gaoling is P = 42 - 4Q . Suppose Wu charges different uniform prices in MP and Gaoling. If Wu sets each price such that she is maximizing her total profits, what are Wu's total profits?

Answer 1

She will maximize profits according to the rule: MR = MC

Misty Palms: P = 79 - Q
So, TR = P*Q = (79-Q)*Q = 79Q - Q2
MR = d(TR)/dQ = 79 - 2Q
So, MR = MC gives,
79 - 2Q = 0
So, 2Q = 79
So, Q = 79/2 = 39.5
P = 79 - Q = 79 - 39.5 = 39.5

Gaoling: P = 42 - 4Q
So, TR = P*Q = (42 - 4Q)*Q = 42Q - 4Q2
So, MR = d(TR)/dQ = 42 - 2(4Q) = 42 - 8Q
Now, MR = MC gives,
42 - 8Q = 0
So,8Q = 42
So, Q = 42/8 = 5.25
P = 42 - 4Q = 42 - 4(5.25) = 42 - 21 = 21

Total profit = Total revenue form both places - total cost = (39.5*39.5) + (21*5.25) - 0 = 1,560.25 + 110.25 = 1,670.5

Wu's total profits are 1,670.5