Examples and exercises on controlling a monopolist

Example: a price ceiling

Suppose demand is given by p = 100

y and TC is TC(y) = 50y + 10.

Then

TR(y) = (100 y)y.

In the absence of regulation, the monopolist's output is the solution of

50 = 100 2y

or y = 25. The price is p = 75.

The efficient output is given by the solution of MC = AR, or

50 = 100 y,

or y = 50, with a price of p = 50.

If a price ceiling of 60 is imposed, the monopolist's MR is

60 if y < 40

100 2y if y 40,

as shown in the following figure by the red line. (The MR before the price ceiling is the pink line.)

When is this MR equal to MC (which is 50)? When y = 40 there is a jump: for smaller y, MR is higher than 50 and for larger y it is less than 50. Thus a candidate for the optimal output of the monopolist is y = 40. At this output the monopolist's profit is (40)(60) [(50)(40) + 10] = 390 > 0, so in fact this output is optimal.

In summary, in the presence of the price ceiling, the monopolist produces 40 units and sells them at the price of 60.

Example: Average cost pricing

The demand curve for a product is given by Q_d(p) = 200

2p. The cost function of a firm is TC(y) = 20y + 750.

How much does the firm produce if it is an unregulated monopolist? How much does it produce if it is subject to average cost pricing? How does the outcome compare with the efficient output?

Unregulated monopolist: Standard calculations show that an unregulated monopolist produces y = 80 and the price is p = 60.

Monopolist with average cost pricing: If it is forced to produce at the point that it makes zero profit, the firm produces where AC is equal to AR. We have

AC(y) = 20 + 750/y,

and

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

For AC(y) = AR(y) we need

20 + 750/y = (200y)/2,

40 + 1500/y = 200y,

1500 = 160y y²,

y² 160y + 1500 = 0,

y = (1/2)[160 ± {25600 6000}],

y = 150 or 10.

Thus there are two outputs at which AC = AR: 10 and 150.

The efficient output is that for which AR is equal to MC:

100 y/2 = 20,

y = 160.

Thus in this case the regulated firm produces less than the efficient output, whichever of the outputs at which AC = AR it produces.