Examples and exercises on monopoly and entry

Example

The demand function for a good is given by Q_d(p) = 200

2p. The cost function of every potential firm is TC(y) = 20y + F .

If there is a single firm, acting as a profit-maximizing monopolist setting a single price, how much does it produce? What is the price? For what values of F , if any, can another firm that takes the monopolist's output as given make a profit?

We have

TR(y) = y(200y)/2.

MR(y) = 100 y.

Further, MC(y) = 20. So the monopolist produces y such that

100 y = 20,

y = 80.

The price is p such that

80 = 200 2p

or p = 60. The monopolist's profit is

(60)(80) (20)(80) F = 3200 F .

What demand does an entrant face? The monopolist sells 80 units, so the units remaining for an entrant are

Q_R(p) = 120 2p.

Thus for an entrant, total revenue is

TR(y) = y(120y)/2.

Thus

MR(y) = 60 y.

So the entrant produces y_E such that

60 y_E = 20,

y_E = 40.

The price it sells at is

(120y_E)/2 = 40,

so its profit is

(40)(40) (20)(40) F = 800 F .

Thus the firm is a natural monopoly if F > 800.