A monopolist faces a demand curve of Q = 80-2P. Its cost function is C = 2Q. What is its optimum level of output and price to maximise profits?
If a firm has a monopoly on the market for a particular product, then it can set the market demand curve to be equal to its own demand curve. The firm is described as a price-maker, but supply must by definition meet demand (the firm being the only firm to produce the product and supply to consumers) so that the firm cannot set the price above the maximum that a consumer is willing to pay. If it did, the nature of the market would necessarily change from a monopolistic market to one with competition.
We are given that the demand curve of the firm, and hence of the market is
Q = 80 - 2P (1)
where P is the price of the product and Q is the demand for the product in unit time, and that the total cost function is
TC = 2Q (2)
To maximise profits, the firm should set marginal revenue equal to marginal costs, that is
MR = MC
where MR is the derivative of total revenue TR = P*Q with respect to Q, that is
MR = d(TR)/dQ = d(P*Q)/dQ
and MC is the derivative of the total cost TC with respect to Q, that is
MC = d(TC)/dQ
To calculate the MR, we first need the price P in terms of the quantity demanded Q. Rearranging the demand curve given by equation (1) we have that
P = (80 - Q)/2 = 40 - Q/2 (3)
To calculate the TR, we multiply this by Q, giving
TR = P*Q = 40Q - Q^2/2
To calculate the MR, we differentiate this with respect to Q, giving
MR = d(TR)/dQ = 40 - Q (4)
Now to calculate the MC so that we can equate MR and MC to find the Q that corresponds to maximum profits for the monopolistic firm.
We are given that the total cost
TC = 2Q
and we can calculate MC simply by differentiating by Q, as per the TR and MR, giving
MC = d(2Q)/dQ = 2 (5)
Equating results (4) and (5) we have that MR = MC when
40 - Q = 2 implying
Q = 40 - 2 = 38 items
Therefore the demand ( = supply here) when the firm is operating at maximum profits is 38 items. Substituting this back into the rearranged demand curve (3) (where we have P as a function of Q) we get
P = 40 - Q/2 = 40 - 38/2 = 40 - 19 = $21
The firm should produce Q = 38 items per unit time at a price of P = $21 per item.