Question

If the demand curve of a profit maximizing monopolist is given as Q = 40 − 0.2P and cost function as C = 30 + 30Q, find equilibrium output level, monopolist price, and profit

Answer 1

Given

$Q = 40 − 0.2P\\C = 30 + 30Q$

For profit monopolist ,the equilibrium output is determined at point where marginal revenue is equal to marginal cost.

$TR=P×Q\\Q=40-0.2P\\0.2p =40-Q\\P=200-5Q$

$TR=(200-5Q)Q\\TR=200Q-5Q^2\\MR=200-10Q......(1)$

$TC=30+30Q\\MC=30......(2)$

At equilibrium

$MR=MC\\200-10Q=30\\170=10Q\\Q=17$

Substitute Q to get value of P

$P=200-5Q\\=200-5(17)\\P=115$

Profit=Total revenue-Total cost

$=(P×Q)-(TC)\\=(200-5Q)Q-(30+30Q)\\=200Q-5Q^2-30+30Q\\=230Q-5Q^2-30$

Substitute for Q=17

Profit

$=230×17-5(17)^2-30\\=3910-1445-30\\=2435$