Question

1)For a monopolistic firm demand and total cost (TC) functions are as follows:
q = 230 – (1/2) P and TC = 20 + (1/2) q^2
What is the equilibrium level of P (P*) that maximizes the profit of firm?

2)) If the total cost function (TC) of a competitive firm is TC = 20 + 0.2 Q^2 and the price is (P) = 400 what is the equilibrium level of quantity (Q*) that maximizes profit?

Answer 1

1) Answer

$P^* = \$276$

Solution

For the monopolistic firm to maximize profit, it will produce at the level of output where $MR = MC.$

From the demand function $q = 230 - \dfrac {1} {2} p$ , the inverse demand function is found by making $p$ the subject of formula.

$=> 2q = 460 - p$

$=> p = 460 - 2q$

But, $P = AR$

$=> AR = 460 - 2q$

$TR = AR × q$

$= (460 - 2q) × q$

$= 460q - 2q^2$

$MR = \dfrac {d} {dq} (TR)$

$= \dfrac {d}{dq} (460q - 2q^2)$

$= 460 - 4q$

Also,

$MC = \dfrac {d}{dq} (TC)$

$= \dfrac {d}{dq} (20 + \dfrac {1}{2} q^2)$

$= q$

Now,

$MR = MC$

$=> 460 - 4q = q$

$460 = 4q + q$

$460 = 5q$

$\therefore q = 92 \space units$

Substituting 92 units for q in the inverse demand function gives:

$p = 460 - 2(92)$

$= 460 - 184$

$= \$276$

Therefore, the equilibrium level of price that maximizes the firm's profit, $P^* = \$276$

2) Answer

$Q^* = 1,000 \space units$

Solution

The profit maximizing condition is $MR = MC.$

For a competitive firm, $P = AR = MR$

$\therefore \space since \space P = \$400,\space it \space follows \space that \space MR = \$400$

Now,

$MC = \dfrac {d}{dQ} (TC)$

$= \dfrac {d} {dQ} (20 + 0.2Q^2)$

$= 0.4Q$

$Since \space MR = MC,$

$=> \$400 = 0.4Q$

$=> Q = 1,000 \space units$

$\therefore$ the equilibrium level of output that maximizes the firm's output, $Q^* = 1,000 \space units$