In a monopolistic market, there is only one firm that produces a product. There is absolute product differentiation because there is no substitute. One characteristic of a monopolist is that it is a profit maximizer. Since there is no competition in a monopolistic market, a monopolist can control the price and the quantity demanded. The level of output that maximizes a monopoly's output is calculated by equating its marginal cost to its marginal revenue.

Marginal Cost and Marginal Revenue

The marginal cost of production is the change in the total cost that arises when there is a change in the quantity produced. In calculus terms, if the total cost function is given, the marginal cost of a firm is calculated by taking the first derivative with respect to the quantity.

The marginal revenue is the change in the total revenue that arises when there is a change in the quantity produced. The total revenue is found by multiplying the price of one unit sold by the total quantity sold. For example, if the price of a good is $10 and a monopolist produces 100 units of a product per day, its total revenue is $1,000. The marginal revenue of producing 101 units per day is $10. However, the total revenue per day increases from $1,000 to $1,010. The marginal revenue of a firm is also calculated by taking the first derivative of the total revenue equation.

How to Calculate Maximized Profit in a Monopolistic Market

In a monopolistic market, a firm maximizes its total profit by equating marginal cost to marginal revenue and solving for the price of one product and the quantity it must produce.

For example, suppose a monopolist's total cost function is

P=10Q+Q2where:P=priceQ=quantity\begin{aligned} &P = 10Q + Q^2 \\ &\textbf{where:}\\ &P=\text{price}\\ &Q=\text{quantity}\\ \end{aligned}P=10Q+Q2where:P=priceQ=quantity

Its demand function is

P=20QP = 20 - QP=20Q

and the total revenue (TR) is found by multiplying P by Q:

TR=P×QTR = P \times QTR=P×Q

Therefore, the total revenue function is:

TR=25QQ2TR = 25Q - Q^2TR=25QQ2

The marginal cost (MC) function is:

MC=10+2QMC = 10 + 2QMC=10+2Q

The marginal revenue (MR) is:

MR=302QMR = 30 - 2QMR=302Q

The monopolist's profit is found by subtracting total cost from its total revenue. In terms of calculus, the profit is maximized by taking the derivative of this function,

π=TR+TCwhere:π=profitTR=total revenueTC=total cost\begin{aligned} &\pi = TR + TC \\ &\textbf{where:}\\ &\pi=\text{profit}\\ &TR=\text{total revenue}\\ &TC=\text{total cost}\\ \end{aligned}π=TR+TCwhere:π=profitTR=total revenueTC=total cost

and setting it equal to zero.

Therefore, the quantity supplied that maximizes the monopolist's profit is found by equating MC to MR:

10+2Q=302Q10 + 2Q = 30 - 2Q10+2Q=302Q

The quantity it must produce to satisfy the equality above is 5. This quantity must be plugged back into the demand function to find the price for one product. To maximize its profit, the firm must sell one unit of the product for $20. The total profit of this firm is 25, or

TRTC=10075TR - TC = 100 - 75TRTC=10075

(For related reading, see: A History of U.S. Monopolies.)