# Maximizing Profit for a Monopolist

The monopolist has the ability to determine the price and/or the amount of the good or service that is available to consumers. Profit-maximizing firms produce where their marginal cost is equal to their marginal revenue.
All profit-maximizing firms produce where their marginal cost (MC) (the cost of producing one more unit) is equal to their marginal revenue (MR) (the revenue received from selling an additional unit). This $\mathrm{MR}=\mathrm{MC}$ rule is the same for monopolists as is it is for perfectly competitive firms. That is where the firm maximizes profit. How it sets its price, however, does differ in the two markets. It stems from the demand curve, the representation of the relationship between price and demand, that individual firms face in each market structure.
A monopolist has market power and can therefore affect the market price. Because the monopolist is the only supplier, its individual demand curve is also the market demand curve. As a result, it faces a downward-sloping demand curve that follows the law of demand: if the monopolist wants to sell more, it has to lower its price.

The law of demand states that as the price of a good decreases, the quantity demanded will increase, all other things being equal. Even though the monopolist has market power and can set its price, its pricing decision is still subject to the law of demand. As the only supplier in the market, the monopolist's demand curve is the downward-sloping market demand curve. So, in order to sell more, the monopolist must lower price assuming they are a single-price monopolist. To sell an extra unit the monopolist not only has to lower the price for that unit, it has to also lower the price on all units that it sells. This means that the contribution made to total revenue (MR) by the next unit sold is less than the contribution to total revenue (MR) made by the sale of the previous units. The result is that the marginal revenue curve for the monopolist is downward-sloping.

Thus, for the monopolist $\mathrm{MR}\;<\;\mathrm P$ .

The contribution to total revenue for selling an additional unit is less than the price charged for the unit (less by the amount it loses as a result of lowering the price on all units that it was selling already).

Like a perfectly competitive firm, a monopolist determines the profit maximizing level of output where $\mathrm{MC}=\mathrm{MR}.$ . However, unlike a perfectly competitive firm, the monopolist does not face a given market price. With market power, the monopolist gets to set the price. Having set its production to an amount that maximizes profits, it then charges what consumers are willing to pay based on the market demand curve. For example, a monopolist sets its production where $\mathrm{MC}=\mathrm{MR}$ , at 200,000 units. Given the monopolist is producing 200,000 units, it sets a price where the quantity demanded is also equal to 200,000 units (this is the price from the demand curve at the profit-maximizing level of output). The point where the quantity demanded meets the amount produced is at point B on the market demand curve, which corresponds to a price of \$7.

In order to calculate its profit, the monopolist must compare its price to its average total costs at the profit maximizing level of output or quantity.

The equation for determining the amount of profit for a monopolist is:
$Profit\;=\;(P\;-\;ATC)\;\times\;Q$
where P is price, ATC is average total cost, and Q is quantity. The monopolist will chooses a level of output where $\mathrm{MR}=\mathrm{MC}$ and will generate a profit as long as $\mathrm P>\mathrm{ATC}$ at that level of output.
The monopolist's profit can also be calculated from the graph using geometry. The area of a rectangle is its base multiplied by its height. The base of the rectangle is the line from the vertical axis to the profit maximizing quantity = 200,000. The height of the rectangle is the line $\mathrm P\;-\;\mathrm{ATC}\;=\;7\;-\;4\;=\;3$ . The base multiplied by the height is $200,000\;\times\;3\;=\;\600,000$ .

A monopolist will set production at its profit-maximizing quantity and then determine the market price. The monopolist will produce where marginal cost equals marginal revenue, then set the price where quantity demanded equals the monopolist's output.