# 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?

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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.**

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