Monopolies maximize profit by setting marginal cost equal to marginal
revenue.
LEARNING OBJECTIVE [ edit ]
Explain the monopolist's profit maximization function
KEY POINTS [ edit ]
For a monopoly, the first-order condition for the profit-maximizing quantity q is 0=∂q=p(q)+qp′
(q)−c′(q).
The monopoly's profits are π=p(q)q−c(q). The monopoly earns the revenue pq and pays
the cost c.
A monopoly chooses a quantity q where marginal revenue equals marginal cost, and charges the
maximum price p(q) that the market will bear at that quantity.
The monopoly restricts output and charges a higher price than would prevail under competition.
TERMS [ edit ]
first-order condition
A mathematical relationship that is necessary for a quantity to be maximized or minimized.
deadweight loss
A loss of economic efficiency that can occur when an equilibrium is not Pareto optimal.
Give us feedback on this content: FULL TEXT [edit ]
Even a monopoly is constrained by demand. A monopoly would like to sell lots of units at a
very high price, but a higher price (except under the most extreme conditions) necessarily
leads to a loss in sales. So how does a monopoly choose its price and quantity? A monopoly
can choose price, or a monopoly can choose quantity and let the demand dictate the price. It
is slightly more convenient to formulate
the theory in terms of quantity rather than
price, because costs are a function of
quantity. Thus, we let p(q) be the price
associated with quantity q, c(q) be the cost
to the firm of producing quantity q, and π
be profits.
The monopoly's profits are given by the
following equation:
π=p(q)q−c(q),
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The monopoly earns the revenue pq and pays the cost c, so profit is the difference between
these two numbers. This leads to the first-order condition for the profit-maximizing quantity
q:
0=∂q=p(q)+qp′(q)−c′(q) The above first-order condition must always be true if the firm is maximizing its profit - that
is, if p(q)+qp′(q)−c′(q) is not equal to zero, then the firm can change its price or quantity and
make more profit. The term p(q)+qp′(q) is known as marginal revenue. It is the derivative of
revenue p(q) with respect to quantity. The term c′(q) is marginal cost, which is the derivative
of c(q) with respect to quantity.Thus, a monopoly chooses a quantity q where marginal
revenue equals marginal cost, and charges the maximum price p(q) that the market will bear
at that quantity.
Consider the example of a monopoly firm that can produce widgets at a cost given by the
following function:
c(q)=2+3q+q2 If the firm produces two widgets, for example, the total cost is 2+3(2)+22=12. The price of
widgets is determined by demand: p(q)=24-2p When the firm produces two widgets it can charge a price of 24-2(2)=20 for each widget. The
firm's profit, as shown above, is equal to the difference between the quantity produces
multiplied by the price, and the total cost of production: p(q)q−c(q). How can we maximize
this function?
Using the first order condition, we know that when profit is maximized, 0=p(q)+qp′(q)−c′
(q). In this case: 0=(24-2p)+q(-2)-(3+2q)=21-6q Rearranging the equation shows that q=3.5. This is the profit maximizing quantity of
production.
Consider the diagram illustrating monopoly competition . The key points of this diagram are
fivefold. First, marginal revenue lies below the demand curve. This occurs because marginal
revenue is the demand, p(q), plus a negative number. Second, the monopoly quantity equates
marginal revenue and marginal cost, but the monopoly price is higher than the marginal cost.
Third, there is a deadweight loss, for the same reason that taxes create a deadweight loss: The
higher price of the monopoly prevents some units from being traded that are valued more
highly than they cost. Fourth, the monopoly profits from the increase in price, and the
monopoly profit is illustrated. Fifth, since—under competitive conditions—supply equals
marginal cost, the intersection of marginal cost and demand corresponds to the competitive
outcome. We see that the monopoly restricts output and charges a higher price than would
prevail under competition. Price
MC
P
ATC
Economic
profit
D=AR=P
MR
Q
Quantity
Monopoly Diagram
This graph illustrates the price and quantity of the market equilibrium under a monopoly.