Chapter 9. Proﬁt
Maximization by a
In this chapter, we will
• Analyzing production and pricing deci-
sions in a market with a monopolist
• Compare market equilibrium in monopoly
market and perfectly competitive mar-
• See why monopoly markets arise and ex-
plore the notion of barriers to entry.
1. Proﬁt Maximization by a
Monopoly is the industry structure when there
is only one ﬁrm in the industry – the oppo-
site to perfect competition. We call this ﬁrm
A ﬁrm in perfectly competitive markets has
a negligible impact on the market price and
thus takes the price as given. By contrast,
a monopolist sets the market price for its
In doing so, the monopolist must take ac-
count of the market demand. The higher it
sets its price, the fewer units it will sell. The
monopolist’s demand curve is the market de-
mand curve. See Figure 9.1.
The proﬁt-maximizing monopolist’s problem
is to ﬁnd the optimal trade-oﬀ between vol-
ume (the number of units it sells) and margin
(the diﬀerence between price and marginal
Figure 9.1 The Monopolist’s Demand
Curve Is the Market Demand Curve
A Monopolist Does Not Have a Sup-
A perfectly competitive ﬁrm takes the mar-
ket price as given and chooses a proﬁt-maximizing
quantity. The resulting s(p) is the ﬁrm’s sup-
However, for a monopolist, there is no such
thing as given market price, since how much
output the monopoly produces will aﬀect the
As a result, the ﬁrm can choose either price
or output. But it does not have the freedom
to choose both at the same time.
If monopolist chooses price, then consumers
decide how much they are willing to buy at
this price, and the monopolist should supply
If monopolist chooses quantity, then con-
sumers decide at what price they are willing
to pay for that quantity, and the monopolist
will sell at this price.
The Proﬁt-Maximization Condition
Suppose a monopolist faces the demand
p(Q) = 12 − Q,
where Q is measured in millions of ounces,
and p in dollar per ounce. To sell more, the
monopolist has to cut its price.
Monopolist’s total revenue is price times quantity,
T R(Q) = p(Q)×Q = (12−Q)×Q = 12Q−Q2.
Now let’s further assume that the monopo-
list’s cost function is
T C(Q) = Q2.
Figure 9.2(a) illustrates total revenue (T R),
total cost (T C) and proﬁt graphically.
We can ﬁnd that T C always increases with
Q, T R and proﬁt ﬁrst rises as Q increases
but then fall.
Proﬁt is maximized at the peak of the proﬁt
hill, which occurs at Q = 4 million.
For quantities less than Q = 4 million, in-
creasing Q increases T R more than it in-
creases T C, which moves the ﬁrm up its
As Figure 9.2(b) shows, when Q < 4 mil-
lion, the monopolist’s marginal revenue ex-
ceeds its marginal cost (M R > M C).
For Q > 4 million, producing less output in-
creases its proﬁt, as now M R < M C.
Figure 9.2 Proﬁt Maximization by a
• If the ﬁrm produces a quantity at which
M R > M C, the ﬁrm can’t be maximiz-
ing its proﬁt, because it can increase its
proﬁt by producing more.
• If the ﬁrm produces a quantity at which
M R < M C, the ﬁrm can’t be maximiz-
ing its proﬁt, because it can increase its
proﬁt by producing less.
• Thus, the proﬁt-maximizing quantity Q∗
M R(Q∗) = M C(Q∗).
The above equation is the proﬁt-maximization
condition for a monopolist.
Note that this condition applies to both monopoly
and perfect competition.
A Closer Look at Marginal Revenue:
Marginal units and Inframarginal Units
In the case of perfect competition, M R = p.
Then M R = M C implies p = M C, which is
the optimal condition we used in the previous
However, for a monopolist, M R = p. To see
why, let’s see Figure 9.3.
Suppose that initially the monopolist pro-
duces Q = 2, charging p = 10. The T R
corresponds to area I+area II.
Now suppose that the monopolists contem-
plates increasing its output to Q = 5. To sell
this much, it must lower its price to p = 7.
Now the T R corresponds to area II + area III.
Figure 9.3 The Change in Total Revenue
When Monopolist Increases Output
Area III represents the additional revenue the
monopolist gets from the additional 3 mil-
lion ounces it sells when it lowers its price
to p = $7 : $7 × (5 − 2) million=$21 mil-
lion. The extra 3 million ounces are called
the marginal units.
Area I represents the revenue the monopolist
sacriﬁces on the 2 million ounces it could
have sold at the higher price of $10 : ($10 −
$7) × 2 million =$6 million. These 2 million
ounces are called the inframarginal units.
When the monopolist lowers its price and
raises its output, the change in total revenue,
∆T R, is the sum of the revenue gained on the
marginal units minus the revenue lost on the
∆T R = area III − area I = $21−$6 = $15 million
To derive a general expression for marginal
revenue, note that in the Figure:
Area III = price × change in quanity = p∆Q.
Area I = −change in price×quantity = −∆pQ.
∆T R = Area III − Area I = p∆Q + ∆pQ.
∆T R ∆p
⇒ MR = =p+Q .
Note that the marginal revenue can be ei-
ther positive or negative. It is negative if
the increased revenue ﬁrm gets from selling
more is more than oﬀset by the decrease in
revenue due to price reduction.
Average Revenue and Marginal Rev-
The monopolist’s average revenue is the ra-
tio of total revenue to quantity:
AR = ⇒ AR = = p.
Now we can conclude that if Q > 0,
(1) M R < p, i.e., marginal revenue curve is
below the demand curve;
(2) M R < AR.
Figure 9.4 shows the relationships among
price, quantity, total revenue, average rev-
enue, and marginal revenue.
Figure 9.4 Total, Average and Marginal
An Example with Linear Demand
Suppose that the monopolist faces a linear
p(Q) = 12 − Q.
The cost function is
c(Q) = Q2 ⇒ M C = Q.
Derive the monopolist’s proﬁt-maximizing out-
put, the corresponding price and proﬁt.
Solution: We will go over it in class.
M R for a general linear demand
Suppose that the market demand is given by
p = a − bQ, with a > 0, b > 0. Then
T R = p × Q = (a − bQ) × Q = aQ − bQ2,
⇒ MR = = a − 2bQ.
Next we compare the demand curve
p = a − bQ,
and the M R curve
M R = a − 2bQ.
We can ﬁnd that
(1) They have the same vertical intercept a.
This is because when Q = 0, p = M R = a.
(2) The horizontal intercept of demand curve
is twice that of the M R curve, or M R curve
is twice as steep as the demand curve.
0 = p = a − bQ ⇒ Q = a−p .
0 = M R = a − 2bQ ⇒ Q = a−p .
2. The Welfare Economics of
The Monopoly Equilibrium Diﬀers from
the Perfectly Competitive Equilibrium
Figure 9.5 shows the equilibrium in a per-
fectly competitive market. The competitive
equilibrium price is p = $5, where the indus-
try supply curve S intercepts the demand D.
The equilibrium quantity is Q = 1000.
Now suppose that the industry was monop-
olized. The new equilibrium quantity is Q =
600, where the M R curve intercepts the M C
curve. The corresponding equilibrium price
is p = $9.
We can see from Figure 9.5 how the monopoly
equilibrium (point J) and the competitive
equilibrium (point K) diﬀer: the monopoly
price is higher, and the monopolist supplies
Figure 9.5 Monopoly versus Perfect
Monopoly Deadweight Loss
How does the diﬀerence between the monopoly
and competitive equilibria aﬀect economic
beneﬁts in this market?
From Figure 9.5, the consumer surplus (CS)
in monopoly is area A. The monopolist’s
producer surplus is the accumulation of the
diﬀerence between the monopolist’s price and
its marginal cost. This corresponds to area
B + E + H. Thus the net economic ben-
eﬁt (or social surplus SS) in the monopoly
equilibrium is A + B + E + H.
In the perfectly competitive equilibrium, con-
sumer suplus is area A + B + F . Producer
surplus is area E + G + H. Social surplus is
A + B + E + F + G + H.
The table in Figure 9.5 compares the con-
sumer surplus, producer surplus (proﬁt) and
net economic beneﬁt (social surplus) under
monopoly and perfect competition. It shows
that consumers surplus and social surplus is
higher under perfect competition than monopoly.
The diﬀerence is area F + G. This diﬀer-
ence is the deadweight loss (DW L) due to
The DWL arises because the monopolist does
not produce the socially optimal output level.
When Q ∈ (600, 1000), consumer’s willing-
ness to pay (social beneﬁt) exceeds ﬁrm’s
marginal cost (social cost), but the monopo-
list chooses not to produce because ﬁrm beneﬁt
is less than ﬁrm cost.
Consider a market where demand is charac-
p = 2 − Q.
We further assume that the monopolist’s cost
function is c(Q) = 1 Q2. Then M C = Q.
We want to compare Case 1: Monopoly with
Case 2: Competition. We will go over it in
Compare monopoly with competition
Variable Competition Monopoly
Price 1 4
Quantity 1 2
Proﬁt 1 1
Consumer surplus 1 2
Social surplus 1 8
Figure 9.6 Monopoly
Figure 9.7 Monopolist forced to behave
Question: Is the DWL in table 9.5 the true
monopoly DWL loss?
Because a monopolist often earns above nor-
mal economic proﬁts, you might expect that
ﬁrms would have an incentive to acquire monopoly
For example, during the 1990s, cable televi-
sion companies spent millions lobbying Congress
to preserve regulations that limit the ability
of satellite broadcasters to compete with tra-
ditional cable service.
Activities aimed at creating or preserving monopoly
power are called rent-seeking activities.
Expenditures on rent-seeking activities can
represent an important social cost of monopoly
that the table in Figure 9.5 does not reﬂect.
What is the maximum amount that a ﬁrm is
willing to spend to become such a monopo-
list as in Figure 9.5?
Suppose that a competitive ﬁrm earns zero
proﬁt, than a ﬁrm would be willing to spend
up to B + E + H to become the monopolist.
The upper bound of social waste of monopoly
is (B+E+H)+(F +G), if all the rent-seeking
activities does not generate any net beneﬁt
to the society.
3. Why Do Monopoly Markets
Now we have seen that monopoly equilibrium
can create a deadweight loss. But how do
monopolies arise in the ﬁrst place?
A market is a natural monopoly if, for rel-
evant output levels, a single ﬁrm can sup-
ply the market more eﬃciently than multiple
Figure 9.8 shows a natural monopoly market.
Suppose that the industry is to supply Q =
9000. If supplied by a single ﬁrm, AC = $1.
If supplied equally by two ﬁrms, AC = $1.2.
If one ﬁrm can serve a market at lower cost
than two or more ﬁrms, we would expect
that, without government interference, the
market would eventually become monopo-
Figure 9.8 Natural Monopoly Market
Barriers to Entry
A natural monopoly is an example of a more
general phenomenon known as barriers to entry.
Barriers to entry are factors that allow an in-
cumbent ﬁrm to earn positive economic prof-
its, while at the same time making it unprof-
itable for newcomers to enter the industry.
Barriers to entry are essential for a ﬁrm to
remain a monopolist. Barriers to entry can
be structural, legal, or strategic.
Structural barriers to entry exist when incum-
bent ﬁrms have cost or marketing advan-
tage that would make it unattractive for a
new ﬁrm to enter the industry and compete
We will look at two of such factors.
(1) Economies of scale.
When new ﬁrm enters, each ﬁrm (incumbent
and entrant) both have to produce fewer
outputs, and AC would be higher due to
economies of scale, and the market becomes
(2) Network externality.
The auction site Ebay is attractive to both
buyers and sellers because its volume. The
more buyers there are, the more attractive
Ebay is to sellers. More sellers will do busi-
ness on Ebay, making Ebay more attractive
to buyers, in turn luring more buyers there.
Legal barriers to entry exist when an incum-
bent ﬁrm is legally protected against compe-
tition. Patents are an important legal barrier
to entry. Government regulation can also
create legal barriers to entry.
Strategic barriers to entry result when an in-
cumbent ﬁrm takes explicit steps to deter
An example would be the development of
a reputation over time as a ﬁrm that will
aggressively defend its market against en-
croachment by new entrants (e.g., by start-
ing a price war if a new ﬁrm chooses to come
into the market)