You are the manager of a monopoly and your demand and cost f

You are the manager of a monopoly, and your demand and cost functions are given by P = 300 – 3Q and C(Q) = 1,500 + 2Q², respectively.

a. What price–quantity combination maximizes your firm’s profits?

Price: $
Quantity:  units

b. Calculate the maximum profits.

d. What price–quantity combination maximizes revenue?

Price: $
Quantity:  units

e. Calculate the maximum revenues.

Solution

A. Profits = TR – TC , where TR = P*Q

= 300Q – 3Q² – 1500 – 2Q²

To max Profits take first order derivative of the profit function

D / dQ = 300 – 6Q – 4Q =0

= 300 – 10Q = 0

Q = 300/10

Q = 30

At Q = 30

P = 300 – 3*30 = 210

B. TR = 210 * 30 = 6300

TC = 1500 + 2*900 = 3300

Profits = 6300 – 3300 = 3000

D. Revenue = P * Q

= 300Q - 3Q2

For max revenue, dRevenue/dQ = 0, hence

300 - 6Q = 0

Q = 50

Price = 300 - 3Q = 300 - 150 = 150

E. TR = 50 * 150 = 7500