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Week 10: Monopoly
PROFIT MAXIMIZATION BY A MONOPOLIST
A firm in a perfectly competitive market has an inconsequential impact on the market price and thus takes it as given. By contrast, a
monopolist sets the market price for its product. So what would stop the monopolist from setting an infinitely high price? The answer
is that the monopolist must take account of the market demand curve: The higher the price it sets, the fewer units of its product it will
sell; the lower the price it sets, the more units it will sell. Thus, the monopolist’s market demand curve is downward sloping, as shown
in Figure 1. The profit-maximizing monopolist’s problem is finding the optimal trade-off between volume (the number of units it sells)
and margin (the differential between price and marginal cost on the units it sells). The logic we develop to analyze this volume–margin
trade-off will apply in the non-monopoly market settings (oligopoly and monopolistic competition) that we study in later chapters.
Figure 1: The Monopolist’s Demand Curve Is the Market Demand Curve The market demand curve is D. To sell more, the monopolist must charge less.
But at what quantity will the monopolist maximize profit?
THE PROFIT-MAXIMIZATION CONDITION
Suppose a monopolist faces the market demand curve D in Figure 1. The equation of this demand curve is P(Q) = 12 - Q. (Q is
expressed in millions of ounces per year, and P is expressed in dollars per ounce.) To sell 2 million ounces, the monopolist would
charge a price of $10 per ounce. But to sell a higher quantity such as 5 million ounces, the monopolist would have to lower its price to
$7 per ounce. As we move along the monopolist’s demand curve, different quantities and their associated prices generate different
amounts of total revenue for the monopolist. Total revenue is price times quantity, so in this case the monopolist’s total revenue is
TR(Q) = P(Q) x Q = 12Q - Q2.
Let’s further suppose that the monopolist’s total cost of production is given by the equation TC(Q) = (1/2)Q 2. Table1 shows quantity,
price, total revenue, total cost, and profit for this monopolist. Figure 2(a) illustrates total revenue, total cost, and profit graphically,
revealing that TC increases as Q increases. By contrast, TR and profit first rise as Q increases but then fall. The monopolist’s profit is
maximized at the peak of the profit hill, which occurs at Q = 4 million ounces.
For quantities less than Q = 4 million, increasing the output increases total revenues more than it increases total cost, which moves the
firm up its profit hill. As Figure 2(b) shows, over this range of output, the monopolist’s marginal revenue exceeds its marginal cost: MR
= MC. For quantities greater than Q 4 million, producing less output increases profit. Over this range, decreasing quantity decreases
total cost faster than it decreases total revenue, which also moves the firm up its profit hill. Over this range of output, the monopolist’s
marginal revenue is less than its marginal cost: MR = MC.
Table 1: Total Revenue, Cost, and Profit for a Monopolist
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Figure 2: Profit Maximization by a Monopolist Panel (a): Total cost TC increases as Q increases. Total revenue TR first increases and then decreases, and
so does profit. The monopolist’s profit is maximized at Q = 4 million ounces. Panel (b): The monopolist’s profit-maximization condition is MR = MC,
where the marginal revenue and marginal cost curves intersect.
Let’s summarize what this discussion implies:
• If the firm produces a quantity at which MR > MC, the firm cannot be maximizing its profit because it could increase its output and
its profit would go up.
• If the firm produces a quantity at which MR < MC, the firm cannot be maximizing its profit because it could decrease its output and
its profit would go up.
• Thus, the only situation at which the monopolist cannot improve its profit by increasing or decreasing output is where marginal
revenue equals marginal cost. That is, if Q* denotes the profit-maximizing output, then
(
)
(
) ( )
Equation (1) is the profit-maximization condition for a monopolist. Figure 2(b) shows this condition graphically: the quantity at which
marginal revenue equals marginal cost occurs where MR and MC cross.
The profit-maximization condition in equation (1) is a general one, applying to both monopolists and perfectly competitive firms. Just
as in a perfectly competitive market, a price-taking firm maximizes profit by producing a quantity at which marginal cost equals
marginal revenue (MC = MR), the profit-maximizing monopolist must do the same.
A CLOSER LOOK AT MARGINAL REVENUE: MARGINAL UNITS AND INFRAMARGINAL UNITS
For a price-taking firm, marginal revenue equals the market price. For a monopolist, however, marginal revenue is not equal to market
price. To see why, let’s take another look at the demand curve D for our monopolist, in Figure 3. Suppose the monopolist initially
produces 2 million ounces, charging a price of $10 per ounce. The total revenue it gets at this price is 2 million x $10, which corresponds
to area I + area II. Now suppose the monopolist contemplates producing a larger output, 5 million ounces. To sell this quantity, it must
lower its price to $7 per ounce, as dictated by the market demand curve. The monopolist’s total revenue is now equal to area II + area
III. Thus, the change in the monopolist’s revenue when it increases output from 2 million ounces to 4 million ounces is area III minus
area I. Let’s interpret what each of these areas means:
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Area III represents the additional revenue the monopolist gets from the additional 3 million ounces of output it sells when it lowers its
price to $7: $7 x (5 - 2) million = $21 million. 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) x
2 million = $6 million. These 2 million ounces are called the inframarginal units.
Figure 3: The Change in Total Revenue When the Monopolist Increases Output To increase output from 2 million to 5 million ounces per year, the
monopolist must decrease price from $10 to $7 per ounce. The gain in revenue due to the increased output of 3 million units (the marginal units) is equal
to area III, while the revenue sacrificed on the 2 million units (the inframarginal units) it could have sold at the higher price is equal to area I. Thus, the
change in total revenue equals area III - area I.
When the monopolist lowers its price and raises its output, the change in total revenue, ΔTR, is the sum of the revenue gained on the
marginal units minus the revenue sacrificed on the inframarginal units: ΔTR = area III - area I = $21 million - $6 million = $15 million.
Or, put another way, the monopolist’s total revenues go up at a rate of $15 million/3 million ounces = $5 per ounce.
To derive a general expression for marginal revenue, note that in Figure 3
Area III = price x change in quantity = PΔQ
Area I = -quantity x change in price = -QΔP
Thus, the change in the monopolist’s total revenue is: ΔTR = area III - area I = PΔQ + QΔP.
If we divide this change in total revenue by the change in quantity, we get the rate of change in total revenue with respect to quantity,
or marginal revenue:
( )
Equation (2) indicates that marginal revenue consists of two parts. The first part, P, corresponds to the increase in revenue due to
higher volume—the marginal units. The second part, Q(ΔP/ΔQ) (which is negative, since ΔP is negative), corresponds to the decrease
in revenue due to the reduced price of the inframarginal units. Since Q(ΔP/ΔQ) < 0, then MR < P. That is, the marginal revenue is less
than the price the monopolist can charge to sell that quantity, for any quantity greater than 0.
When Q = 0, equation (2) implies that marginal revenue and price are equal. This makes sense in light of Figure 3. Suppose the
monopolist charges a price of $12 per ounce and thus sells zero output. To increase its output, the monopolist has to lower its price, but
starting at Q = 0, it has no inframarginal units. That is, per equation (2), marginal revenue equals price plus Q(ΔP/ΔQ), but when Q = 0,
Q(ΔP/ΔQ) = 0, and marginal revenue equals price.
Note that marginal revenue can either be positive or negative. It is negative if the increased revenue the firm gets from selling
additional volume is more than offset by the decrease in revenue caused by the reduction in price on units that it could have sold at a
higher price. In fact, the greater the quantity, the more likely it is that marginal revenue will be negative because the reduced price
(needed to sell more output) affects more inframarginal units.
AVERAGE REVENUE AND MARGINAL REVENUE
In previous chapters, we usually contrasted the average of something with the marginal of the same thing (e.g., average product versus
marginal product, average cost versus marginal cost). For a monopolist, it is important to contrast average revenue with marginal
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revenue because this will help explain why the monopolist’s marginal revenue curve MR is not the same as its demand curve D, as
shown in Figure 4(b) [and first illustrated in Figure 2(b)].
Figure 4: Total, Average, and Marginal Revenue The demand curve D and the average revenue curve AR coincide. The marginal revenue curve MR lies
below the demand curve. The slope of the demand curve is ΔP/ΔQ = -1; for example, if price decreases by $3 per ounce (from $10 to $7), quantity
increases by 3 million ounces per year (from 2 million to 5 million). When price P = $7 per ounce and quantity Q = 5 million ounces per year:
Panel (a)—Total revenue TR =P x Q = 7 x 5 = $35 million per year.
Panel (b)—Average revenue AR = TR/Q = 35/5 = $7 per ounce.
Marginal revenue MR = P + Q(ΔP/ΔQ) = 7 + 5(Δ1) = $2 per ounce.
The total revenue curve in panel (a) reaches its maximum when Q = 6, the same quantity at which MR = 0 in panel (b).
The monopolist’s average revenue is the ratio of total revenue to quantity: AR = TR/Q. Since total revenue is price times quantity, AR =
(P x Q)/Q = P. Thus, average revenue is equal to price. And, since the price P(Q) the monopolist can charge to sell any quantity of
output Q is determined by the market demand curve, the monopolist’s average revenue curve coincides with the market demand
curve: AR(Q) = P(Q).
Combining these insights with the discussion in the preceding section, we can see that, if output is positive (Q > 0):
• Marginal revenue is less than price (MR < P).
• Because average revenue is equal to price, marginal revenue is less than average revenue (MR < AR).
• Since the average revenue curve coincides with the demand curve, the marginal revenue curve must lie below the demand curve.
Figure 4 shows the relationships among price, quantity, total revenue, average revenue, and marginal revenue. The relationship
between average revenue and marginal revenue is consistent with other average–marginal relationships we have seen elsewhere in the
book. When the average of something is falling, the marginal of that thing must be below the average. Because market demand slopes
downward (i.e., is falling) and the average revenue curve corresponds to the demand curve, the marginal revenue curve must be below
the average revenue curve.
Problem -- Marginal and Average Revenue for a Linear Demand Curve
Suppose that the equation of the market demand curve is P = a - bQ.
What are the expressions for the average and marginal revenues curves?
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Solution
Average revenue coincides with the demand curve. Thus, AR = a - bQ. Per equation (2), marginal revenue is
Now note that ΔP/ΔQ=-b (since P = a - bQ is in the general form of a linear equation). Substituting into the equation above:
MR(Q) = a - bQ + Q( - b) = a – 2Qb
Thus, the marginal revenue curve for a linear demand curve is also linear. In fact, it has the same P-intercept as the demand curve (i.e.,
at a), with twice the slope. This implies that the marginal revenue curve intersects the Q-axis halfway between the origin and the
horizontal intercept of the demand curve, which occurs at Q = a/(2b). For quantities greater than this halfway point, marginal revenue
not only lies below the demand curve, it is also negative. Notice that the shape of the marginal curve in Figure 4(b) is consistent with
these properties.
THE PROFIT-MAXIMIZATION CONDITION SHOWN GRAPHICALLY
Figure 5 illustrates the profit-maximization condition MR = MC for our monopolist. The marginal revenue curve MR is decreasing and
lies below the demand curve D (which is also the average revenue curve) for all positive output levels. The marginal cost curve MC is a
straight line from the origin, as is the average cost curve AC. For all positive output levels, the marginal cost curve lies above the
average cost curve.
Figure 5: The Monopolist’s Profit-Maximization Condition The profit-maximizing output is 4 million ounces per year, where MC = MR. To sell that
output, the monopolist will set a price of $8 per ounce (as indicated by the demand curve D). Total revenue is areas B + E + F. Total cost is area F. Profit
(total revenue minus total cost) is areas B + E. Consumer surplus is area A.
Figure 5 illustrates three important points about the equilibrium in a monopoly market:
First, the monopolist’s profit-maximizing price ($8) exceeds the marginal cost of the last unit supplied ($4). This differs from the
outcome in a perfectly competitive market, in which price equals the marginal cost of the last unit supplied.
Second, the monopolist’s economic profits can be positive. This is in contrast to a perfectly competitive firm in a long-run equilibrium,
because the monopolist does not face the threat of free entry that drives economic profits to zero in competitive markets.
Third, even though the monopolist raises price above marginal cost and earns positive economic profits, consumers still enjoy some
benefits at the monopoly equilibrium. The consumer surplus at the equilibrium in Figure 5 is the area between price and the demand
curve, or area A, which equals $8 million. The total economic benefit at the monopoly equilibrium is the sum of consumer surplus and
the monopolist’s producer surplus, which is equal to areas A + B + E, or $32 million per year.
Problem -- Applying the Monopolist’s Profit-Maximization Condition
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The equation of the monopolist’s demand curve in Figure 5 is P = 12 - Q, and the equation of marginal cost is MC = Q, where Q is
expressed in millions of ounces.
What are the profit-maximizing quantity and price for the monopolist?
Solution
To solve this problem, (1) find the marginal revenue curve, (2) equate marginal revenue to marginal cost to find the profit-maximizing
quantity, and (3) substitute this quantity back into the demand curve to find the profit-maximizing price.
The monopolist’s demand curve has the same form as the demand curve (P = a - bQ). Therefore, as in that exercise, our monopolist’s
marginal revenue curve has the same vertical intercept as the demand curve (i.e., 12) and twice the slope: MR = 12 - 2Q. The profitmaximization condition is MR = MC, or 12 = 2Q - Q. Thus, the profit-maximizing quantity is Q = 4 (i.e., 4 million ounces). Substituting
this result back into the equation for the demand curve, we find that the profit-maximizing price P = 12 - 4 = 8 (i.e., $8 per ounce).
A MONOPOLIST DOES NOT HAVE A SUPPLY CURVE
A perfectly competitive firm takes the market price as given and chooses a profit maximizing quantity. The fact that the perfect
competitor views price as exogenous allows us to construct the firm’s supply schedule, by taking each possible market price and
associating it with the corresponding profit-maximizing quantity.
For the monopolist, however, price is endogenous, not exogenous. That is, the monopolist determines both quantity and price.
Depending on the shape of the demand curve, the monopolist might supply the same quantity at two different prices or different
quantities at the same price. The unique association between price and quantity that exists for a perfectly competitive firm does not
exist for a monopolist. Thus, a monopolist does not have a supply curve.
WHY DO MONOPOLY MARKETS EXIST?
NATURAL MONOPOLY
A market is a natural monopoly if, for any relevant level of industry output, the total cost a single firm producing that output would
incur is less than the combined total cost that two or more firms would incur if they divided that output among them. A good example
of a natural monopoly is satellite television broadcasting. If, for example, two firms split a market consisting of 50 million subscribers,
each must incur the cost of buying, launching, and maintaining a satellite to provide service to its 25 million subscribers. But if a single
firm serves the entire market, the satellite that served 25 million subscribers can just as well serve 50 million subscribers. That is, the
cost of the satellite is fixed: It does not go up as the number of subscribers goes up. A single firm needs just one satellite to serve the
market, while two independent firms would need two satellites to serve the same number of subscribers overall.
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
incumbent firm to earn positive economic profits, while at the same time making it unprofitable for newcomers to enter the industry.
Perfectly competitive markets have no barriers to entry: When incumbent firms earn positive profits, new firms enter the industry,
driving profits to zero. But barriers to entry are essential for a firm to remain a monopolist. Without the protection of barriers to entry, a
monopoly or cartel that earned positive economic profits would attract new market entry, and competition would then dissipate
industry profit.
Barriers to entry can be structural, legal, or strategic. Structural barriers to entry exist when incumbent firms have cost or marketing
advantages that would make it unattractive for a new firm to enter the industry and compete against them. The interaction of
economies of scale and market demand that gives rise to a natural monopoly market is an example of a structural barrier to entry. The
Internet auction market provides an example of another type of structural entry barrier, this one based on positive network
externalities. As noted in Chapter 5, positive network externalities arise when a firm’s product is more attractive to a given consumer
the more the product is used by other consumers. The auction site of market leader eBay is attractive to auction buyers because there
are so many items offered for sale and there are often several sellers of the same item. Auction sellers like eBay because there are so
many buyers. The sheer volume of transactions on eBay, in and of itself, is an important part of eBay’s appeal. This network externality
creates a significant barrier to entry. A newcomer seeking to establish its own Internet auction site (to make money, as eBay does,
through commissions on transactions) would face an enormous challenge: Lacking the critical mass that eBay possesses, it would
simply not be as attractive a site. This barrier to entry explains why some very savvy Internet companies, including Amazon.com and
Yahoo, found it difficult to establish their own auction sites to compete against eBay.
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Week 10: Monopoly
Legal barriers to entry exist when an incumbent firm is legally protected against competition. Patents are an important legal barrier to
entry. Government regulations can also create legal barriers to entry. For example, between 1994 and 1999, the company Network
Solutions had a government-sanctioned monopoly in the business of registering domain names on the Internet.
Strategic barriers to entry result when an incumbent firm takes explicit steps to deter entry. An example of a strategic barrier to entry
would be the development of a reputation over time as a firm that will aggressively defend its market against encroachment by new
entrants (e.g., by starting a price war if a new firm chooses to come into the market). Polaroid’s aggressive response to Kodak’s entry
into the instant photography market in the 1970s is an illustration of this strategy.
Problems:
1. The market demand curve for a monopolist is given by P = 40 - 2Q.
a) What is the marginal revenue function for the firm?
b) What is the maximum possible revenue that the firm can earn?
a)
Since the demand curve is written in inverse form and is linear, the MR curve has the same vertical intercept and twice the
slop as the demand curve. Thus, MR = 40 – 4Q.
b)
Total revenue will be maximized when MR = 0, or when Q = 10. At that quantity, the price will be P = 40 – 2Q = 20. Total
revenue is PQ = 20(10) = 200.
2. A monopolist operates in an industry where the demand curve is given by Q = 1000 - 20P. The monopolist’s constant marginal cost is
$8. What is the monopolist’s profit-maximizing price?
Recall that the MR curve can easily be derived from the demand curve when the latter is written in the inverse form. The inverse
demand curve is P = 50 – (Q/20) so the marginal revenue curve is P = 50 – (Q/10) (using the fact that the slope of the MR curve is twice
that of the inverse demand curve, with the same intercept). Using the rule MR=MC, we get 50 – (Q/10) = 8, so Q = 420. Plugging this
back into the demand curve (or the inverse demand curve) we can calculate the profit maximizing price, P = 29.
3. Suppose a monopolist faces the market demand function P = a - bQ. Its marginal cost is given by MC = c + eQ. Assume that a > c and
2b + e > 0.
a) Derive an expression for the monopolist’s optimal quantity and price in terms of a, b, c, and e.
b) Show that an increase in c (which corresponds to an upward parallel shift in marginal cost) or a decrease in a (which corresponds to
a leftward parallel shift in demand) must decrease the equilibrium quantity of output.
c) Show that when e
a)
0, an increase in a must increase the equilibrium price.
The monopolist will operate where
MR  a  2bQ .
MR  MC .
With demand
P  a  bQ , marginal revenue is given by
Setting this equal to marginal cost implies
a  2bQ  c  eQ
ac
Q
2b  e
At this quantity price is
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Week 10: Monopoly
 ac 
P  a b

 2b  e 
ab  ae  bc
P
2b  e
b)
Since
Q
increasing
c)
c
or decreasing
a
will reduce the numerator of the expression, reducing
Since e  0 and
P
increasing
a
ac
2b  e
Q.
ab  ae  bc
2b  e
will increase the numerator for this expression. This will therefore increase the equilibrium price.
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