Question

The total cost (TC) and inverse demand equations for a monopolist are: TC=100+5Q^2 P=200?5Q

a. What is the profit-maximizing quantity?

b. What is the profit-maximizing price?

c. What is the monopolist's maximum profit?

The demand equation for a product sold by a monopolist is Q=25?0.5P TC=225+5Q+0.25Q^2

a. Calculate the profit-maximizing price and quantity.

b. What is the monopolist's profit?

Answer 1

The profit-maximizing condition for monopoly is Marginal Revenue (MR) = Marginal Cost (MC).

First, TR = P * Q

So, given the inverse demand function, total revenue (TR) is

TR = (200 - 5Q) * Q

TR = 200Q - 5Q²

MR = d(TR) / dQ = 200 - 10Q

Now, given the total cost function, MC is

TC = 100 + 5Q²

MC = d(TC) / dQ = 10Q

a.

Now, if we equate MR and MC, we will get profit-maximizing quantity.

200 - 10Q = 10Q

20Q = 200

Q = 100

So, the profit-maximizing quantity is 10 units.

b.

The profit-maximizing price is P = 200 - (5 * 10) = 150.

So, the profit-maximizing price is $150.

c.

Profit = TR - TC

TR = (200 * 10) - (5 * 10 * 10) = 1500.

TC = 100 + (5 * 10 * 10) = 600.

So, the monopolist's maximum profit is (1500 - 600) = 900.

So, the monopolist's maximum profit is $900.

2.

Q = 25 - 0.5P

0.5P = 25 - Q

P = 50 - 2Q

TR = 50Q - 2Q²

MR = 50 - 4Q

TC = 225 + 5Q + 0.25Q²

MC = 5 + 0.5Q

a.

Profit-maximizing quantity is

50 - 4Q = 5 + 0.5Q

4.5Q = 45

Q = 10

So, the profit-maximizing quantity is 10 units.

P = 50 - 2 * (10) = 30.

So, the profit-maximizing price is $30.

b.

TR = (50 * 10) - (2 * 10 * 10) = 300

TC = 225 + (5 * 10) + (0.25 * 10 * 10) = 300.

So, the profit-maximizing profit is (300 - 300) = 0.

So, the monopolist's profit is 0.