Question

A monopolist with the cost function C=1/2Q^2 faces a demand curve Q=12-P,

A. What will be his equilibrium price and quantity?

B. If for some reason the firm behaves as if it were in a perfectly competitive industry, what will equilibrium price and quantity?

C. How much money will the firm require to forgo monopoly profits and behave competitively instead?

Answer 1

A) Given that $C=\frac{1}{2}Q^2$ and $Q=12-P$

To find the price, P, we take the inverse of Q, so that

$P=12-Q$

The total revenue,$TR=P×Q$

$TR=(12-Q)Q=12Q-Q^2$

And

$MR=TR'=12-2Q$

$MC=TC'=Q$

At equilibrium, $MR=MC$

$\therefore 12-2Q=Q$

$Q^*=4$

$P^*=12-4=8$

B) In a perfectly competitive industry, $P=MC$ , which implies that the firm would charge a lower price but now a sells a higher quantity.

$\therefore8=Q^*$

And $P^*=12-8=4$

C)

The monopolist's profit would have been

$\pi=TR-TC$

At Q^* = 4

$\pi=12(4)-(4^2)-\frac{1}{2}(4^2)$

$\pi=24$

The monopolist's profit, if it behaves like a perfectly competitive industry, would be

$\pi=12(8)-(8^2)-\frac{1}{2}(8^2)$

$\pi=0$

Therefore, the monopolist would be giving up all of it's profit (24) to behave like a perfectly competitive industry.