A monopolist has demand and cost curves given by QD 10000

A monopolist has demand and cost curves given by:

QD = 10,000 - 20P

TC = 1,000 + 10Q + .05Q2

A. Find the monopolist\'s profit-maximizing quantity and price.

B. Find the monopolist\'s profit.

Solution

Q = 10,000 - 20P

P = (10,000 - Q) / 20 = 500 - 0.05Q

Total Revenue, TR = P x Q = 500Q - 0.05Q²

TC = 1,000 + 10Q + 0.05Q²

(A) A monopolist maximizes profits by equating marginal revenue (MR) with marginal cost (MC).

MR = dTR / dQ = 500 - 0.1Q

MC = dTC / dQ = 10 + 0.1Q

Equating MR with MC:

500 - 0.1Q = 10 + 0.1Q

0.2Q = 490

Q = 2,450

P = 500 - 0.05Q = 500 - (0.05 x 2,450) = 500 - 122.5 = 377.5

(B) Profit = TR - TC

= 500Q - 0.05Q² - (1,000 + 10Q + 0.05Q²)

= 490Q - 0.05Q² - 1,000

= (490 x 2,450) - (0.05 x 2,450 x 2,450) - 1,000

= 1,200,500 - 300,125 - 1,000

= 899,375