Question

1.     A monopolist has demand and cost curves given by:

 

QD = 10,000 - 20P

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

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

ii.            Find the monopolist's profit.

Answer 1

$Q = 10,000 - 20P$

P = $\frac {(10,000 - Q)} {20} = 500 - 0.05Q$

Total Revenue, $TR = P × Q = 500Q - 0.05Q^2$

$TC = 1,000 + 10Q + 0.05Q^2$

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

MR = $\frac {dTR} {dQ} = 500 - 0.1Q$

MC = $\frac {dTC} {dQ} = 10 + 0.1Q$

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 = $TR - TC$

$= 500Q - 0.05Q^2 - (1,000 + 10Q + 0.05Q^2)$

= $490Q - 0.05Q^2 - 1,000$

= $(490 × 2,450) - (0.05 × 2,450 × 2,450) - 1,000$

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

= $899,375$