Chapter 23
Monopoly
We will now turn toward an analysis of the polar opposite of the extreme assumption of perfect
competition that we have employed thus far.1 Under perfect competition, we have assumed that
industries are composed of so many small firms that each firm has no impact on the economic
environment in which decisions are made. As a result, we could assume that individual firms in an
industry simply take the market price as given as they determine how much to produce in order to
maximize profits. In the case of a monopoly, on the other hand, the firm must make a decision not
only on how much to produce but also on what price to charge. There is, in the case of monopoly,
no “market” to set the price. In this sense, the monopolist has some control over her economic
environment (i.e. prices) that the competitive producer lacks.
While we will often talk about a “monopoly” as if it was a fixed concept, it is important
to keep in mind that monopoly power comes in more and less concentrated doses. Under perfect
competition, the demand that a firm faces for its product is perfectly elastic because of the existence
of many firms that produce the same product at the market price. Whenever a firm faces a demand
curve for its product that is not perfectly elastic, it has some market power. For instance, I might
produce a particular soft drink in a largely competitive market for soft drinks, but my soft drink
is nevertheless a bit distinctive. In a sense, my soft drink is therefore a separate product with a
separate market, but in another sense it is part of a larger market in which other firms produce
close but imperfect substitutes. The demand curve for my soft drink may then not be perfectly
elastic – giving me some market power, but that power is limited by the fact that there are close
substitutes in the larger soft drink market. If, on top of the existence of close substitutes, there
is free entry into the soft-drink market, my market power is limited even more. We will treat this
type of market in Chapter 26 as one characterized by “monopolistic competition.”
In other settings, of course, there is less of an availability of substitutes for a particular firm’s
product. If there are market entry barriers that keep potential competitors from producing substitutes, my monopoly power would then be considerably more pronounced, and the demand for
my product considerably less elastic. For now, we will simply treat monopolies as firms that face
downward sloping demand curves in an environment where barriers to entry keep other firms from
entering to produce substitute goods, and we will simply keep in mind that the elasticity of demand
1 This chapter presumes a basic understanding of demand and makes frequent references to the partial equilibrium
models of Chapters 14 and 15. It furthermore presumes a basic understanding of cost curves as derived in Chapter
11 and summarized in Section 13A.1 of Chapter 13.
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Chapter 23. Monopoly
for the monopoly’s product is closely connected to just how powerful a monopoly we are dealing
with. When we get to Chapter 26, it will become clear that the stark model of monopoly in this
chapter is an extreme model that rarely holds fully in the real world, but it gives us a good starting
point to talk about market power – just as perfect competition gives us a useful starting point to
talk about competition.
23A
Pricing Decisions by Monopolist
We begin our analysis of monopoly power by analyzing how the profit maximizing condition of
marginal revenue being equal to marginal cost translates into optimal firm decision making when
a firm faces a downward sloping demand curve. At first, we’ll assume that the firm is restricted
in its pricing policy in the sense that it can only set a single price per unit of output – a single
price that is charged to every consumer. We then proceed to think about how a monopoly might
want to differentiate the price it charges to different consumers – and under what conditions that
is possible. Finally, we will talk explicitly about what kinds of barriers to entry might in fact result
in real-world monopolies, and how the nature of the barrier to entry might determine the extent to
which we think monopoly power is a problem that requires government intervention.
Before moving on, however, recall the two ways in which we thought about profit maximization
for price taking firms in Chapter 11. We first set up the profit maximization problem under the
assumption that the competitive firm takes price as fixed and solved for the profit maximizing
production plan by finding the tangency between isoprofit curves with production frontiers. This
method no longer holds for monopolists – because the method presumed a fixed price that the price
setting firm simply took as given. We then developed a two-step profit maximization method –
with the first step focusing solely on the cost side (where firms attempt to minimize cost) and the
second step adding revenue considerations (given the price that competitive firms take as given).
Since output price plays no role in the cost-minimizing problem where the firm simply asks “what
is the least cost way of producing different levels of output”, this step is the same for monopolists.
The difference enters in the second step where we compare revenue to cost – with revenue for the
monopolist depending on the price that the monopolist chooses (rather than the price that is set
by the market). We can therefore use everything we learned about cost curves – marginal costs,
average costs, recurring fixed costs, etc. – and will thus focus on step 2 of the two-step profit
maximization method in analyzing monopoly decisions.
23A.1
Demand, Marginal Revenue and Profit
For competitive producers, price is the same as marginal revenue. Put differently, the competitive
producer knows that she can sell any amount of the good she could feasibly produce at the market
price, and so the marginal revenue she receives for each good she produces is simply the price set by
the interactions of producers and consumers in market equilibrium. She could, of course, choose to
sell her goods at a lower price, but that would not be profit maximizing. If, on the other hand, she
tries to sell her goods at a price above the market price, consumers will simply shop at a competitor.
While the market demand curve in competitive markets is therefore downward sloping, the demand
curve for each competitive producer is perfectly elastic at the market price.
For a monopolist, however, the market demand is the same as the firm’s demand since the
monopolist is the only producer in the market. As a result, the monopolist gets to choose a point
on the market demand curve — which involves a simultaneous choice of how much to produce and
23A. Pricing Decisions by Monopolist
857
how much to charge. When a monopolist decides to increase output, she therefore confronts the
following trade-off: On the one hand, she gets to sell more goods to consumers, but on the other
hand she sells all her goods at a lower price than before. Thus, as a monopolist increases output,
her marginal revenue is not equal to the price she charged initially because she will have to lower
price in order to sell the additional output.
23A.1.1
Marginal Revenue along a Market Demand Curve
Suppose we consider a demand curve first illustrated in Graph 18.3 in Chapter 18 and replicated
here as Graph 23.1a. The first unit produced by a monopolist facing such a market demand for
her goods can be sold for approximately $400. Thus, the marginal revenue for the first unit of
output is approximately $400. Next, suppose the monopolist was currently producing 199 units
of the output for $300.50 each. Were this monopolist to produce two additional units of output,
she would have to lower her price to $299.50 in order to sell all 201 goods. She would therefore
experience an increase in her total revenues of $599 for the 200th and 201st good, but she would
simultaneously lose $1 on each of the first 199 goods she is producing. Her marginal revenue from
producing two additional units is therefore $400, or approximately $200 for each of the two units.
Graph 23.1: Linear Demand and Marginal Revenue
Next, suppose that the monopolist was producing 399 units, selling each at $200.50, and suppose
she considered producing two additional units. She would then have to lower the price to $199.50
in order to sell the additional two units, earning an additional revenue of $399 on those units but
losing $399 on the units she previously produced because she had to lower the price by $1 for each
of the 399 units. Thus, her marginal revenue from producing two additional units is 0.
The marginal revenue curve for this monopolist is then depicted in panel (b) of Graph 23.1.
It begins at the same point as the demand curve because the marginal revenue of the first good
is approximately $400. When the monopolist is at approximately point B on the market demand
curve, we demonstrated above that her marginal revenue from producing an additional unit is
approximately $200, and when the monopolist is at approximately point A on her demand curve,
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her marginal revenue from producing an additional unit is approximately 0. Connecting these gives
us the blue line that shares the intercept of the demand curve but has twice the slope.
Exercise 23A.1 What is the marginal revenue of producing an additional good if the producer is at point
C on the demand curve in Graph 23.1?
23A.1.2
Price Elasticity of Demand and Revenue Maximization
You can already see in Graph 23.1 that marginal revenue is positive when price elasticity is below
-1, becomes zero as the price elasticity of demand approaches -1 (and becomes negative when price
elasticity lies between -1 and 0). This implies that total revenue for the monopolist increases as
she moves down the demand curve until she reaches the midpoint where price elasticity is equal to
-1, and total revenue falls if she moves beyond that midpoint into the range of the demand curve
where price elasticity is between -1 and 0. As a result, the maximum revenue the monopolist can
raise occurs at the midpoint of a linear demand curve where price elasticity is equal to -1.
Exercise 23A.2 Where does M R lie when price elasticity falls between -1 and 0?
This is closely related to our discussion of consumer spending and price elasticity in Chapter 18.
In Graph 18.4, we illustrated that consumer spending rises with an increase in price along the
inelastic portion of demand while it falls with an increase in price along the elastic portion of
demand. For the monopolist, consumer spending is the same as revenue. Thus, if a monopolist
finds herself on the inelastic portion of demand, she knows she can increase revenue by raising the
price. If, on the other hand, she finds herself on the elastic portion of demand, she can increase
revenue by lowering price. Consumer spending — and thus revenue — is therefore maximized when
price elasticity of demand is exactly -1.
Exercise 23A.3 Where does a monopolist maximize revenue if she faces a unitary elastic demand curve
such as the one in Graph 18.5?
23A.1.3
Profit Maximization for a Monopolist
Like all producers, however, monopolists do not maximize revenue — they try to maximize profit
which is economic revenue minus economic costs. Thus, in order for us to see what combination of
price and quantity a monopolist will choose (assuming she produces at all), we need to know not
only marginal revenue but also marginal cost.
First, suppose that the marginal cost of producing is zero. In that case, the monopolist’s M C
curve is a flat line that lies on the horizontal axis on Graph 23.1b, intersecting the M R curve at 400
units of output. If the monopolist has no variable costs, maximizing revenue and maximizing profit
is exactly the same thing — and so the monopolist would simply choose point A on the demand
curve where price elasticity is exactly equal to -1. By selling 400 units at $200 each, revenue and
profit (not counting recurring fixed costs) is then equal to $80,000. So long as recurring fixed costs
are not larger than $80,000, the monopolist would then choose to produce 400 units of output in
both the short and the long run.
Exercise 23A.4 True or False: If recurring fixed costs are $40,000, then the monopolist will earn $80,000
in short run economic profit and $40,000 in long run economic profit.
23A. Pricing Decisions by Monopolist
859
Next, suppose that the monopolist has the more common U-shaped M C curve depicted in
Graph 23.2a. If this monopolist produces a positive quantity, she will choose the quantity xM where
M C intersects M R and charge the price pM that allows her to sell everything she is producing. So
long as the short run average (variable) cost at xM is less than pM , this implies the monopolist will
in fact produce in the short run, and so long as average long run cost (including recurring fixed
costs) at the quantity xM lies below pM , she will produce in the long run.
Graph 23.2: Profit Maximization for a Monopolist
Exercise 23A.5 Suppose M C is equal to $200 for all quantities for a monopolist who faces a market
demand curve of the type in Graph 23.1. At what point on the demand curve will she choose to produce?
The first thing we can then observe is that, whenever M C is positive, a monopolist will choose to
produce on the elastic part of demand. This is because, for any positive M C, the intersection of M C
and M R must lie to the left of the intercept of M R with the horizontal axis — which in turn occurs
where price elasticity is exactly equal to -1. This should make intuitive sense: We know that, if a
monopolist ever finds herself on the inelastic portion of demand, she can raise revenue by increasing
price and producing less. If producing costs something, this implies that, whenever a monopolist
is on the inelastic portion of demand, she can raise revenue and reduce costs by producing less and
charging a higher price. As a result, it makes no sense for a monopolist to produce on the inelastic
portion of demand.
Exercise 23A.6 Suppose a deep freeze causes the Florida orange crop to be reduced by 50% causing the
price for oranges to increase. As a result, we observe that the total revenues of Florida orange growers
increases. Could the Florida orange industry be a monopoly? (Hint: The answer is no.)
Second, the concept of a “supply curve” that we developed for competitive firms does not make
any sense when we talk about monopolists. A supply curve illustrates the relationship between the
price set by the market and the quantity of output produced by a profit maximizing firm. But a
monopolist does not have a “market” that sets price — the monopolist herself sets the price. Thus,
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for any given demand curve and any technology that results in cost curves, the monopolist simply
picks a supply point.
23A.1.4
Monopoly and Dead Weight Loss
Finally, we can see in Graph 23.2 that the profit maximizing monopolist will produce an inefficiently
low quantity. In panel (b) of the graph, consumer surplus (assuming no income effects) can be
identified as area (a + b) and monopolist surplus (in the short run, or in the absence of recurring
fixed costs) as area (c + d). But there are additional units of output that could be produced at a
marginal cost below the value consumers place on that output. Such additional output could be
produced all the way up to the intersection of M C and demand at output x∗ , and additional surplus
of (e) could be produced if a benevolent social planner rather than a monopolist were in charge
of production. Thus, area (e) is a deadweight loss which arises because the monopolist strategically
restricts output in order to raise price to its profit maximizing level.
Notice that the deadweight loss does not arise because the monopolist makes a profit. Even if a
social planner forced the monopolist to produce the quantity x∗ and sell it at the appropriate price
along the demand curve, the monopolist might make a profit — it just would not be as large as it is
when the monopolist raises price to pM and restricts output. Rather, the deadweight loss emerges
from the fact that the monopolist is using her power to strategically restrict output in order to
raise price. The monopolist’s market power then causes self-interest to come into conflict with the
“social good” – at least when the social good is measured in efficiency terms – unless something
else interferes and causes the monopolist to produce more.
Exercise 23A.7 Suppose that demand is as depicted in Graph 23.1 and M C=0. What is the monopolist’s
profit maximizing output level and what is the efficient output level? What if M C=300?
Exercise 23A.8 True or False: Depending on the shape of the M C curve, the efficient output level might
lie on the elastic or the inelastic portion of the demand curve.
Exercise 23A.9 True or False: In the presence of negative production externalities, a monopolist may
produce the efficient quantity of output.
Exercise 23A.10 True or False: If demand were not equal to marginal willingness to pay (due to the
presence of income effects on the consumer side), the deadweight loss area may be larger or smaller but
would would nevertheless arise.
23A.1.5
Monopoly Rent Seeking Behavior and Dead Weight Loss
We have demonstrated that monopolists are able to achieve economic profits if they have indeed
secured monopoly power in some way. We have furthermore demonstrated that this economic profit
comes at a social cost as the monopolist produces below the socially optimal level in order to raise
price above marginal cost – and we have denoted that social cost as deadweight loss. The actual
deadweight loss may, however, be larger than what we have derived thus far because firms may
engage in socially wasteful activity in order to secure and maintain the monopoly power that gives
them the opportunity to generate economic profits.
There are a variety of ways in which barriers to entry that lead to monopoly power can arise, and
we will say more about this later on in this chapter. One possibility, for instance, is that monopoly
power is granted through government intervention – with governments granting to a single firm
23A. Pricing Decisions by Monopolist
861
the exclusive right to produce a certain product. In such circumstances, firms may compete for
such government favor – in the process expending resources on lobbying politicians. The maximum
amount that a firm would be willing to invest in order to secure a government granted monopoly is
then equal to the present discounted value of the future profits the firm can expect to make from
exercising its monopoly power. It is therefore conceivable that firms will expend resources equal
to their monopoly profits in order to get the monopoly power, and it is similarly conceivable that
much of these resources are spent in socially wasteful ways. This is referred to as political “rent
seeking” – i.e. the seeking of “rents” or “profits” in the political arena. To the extent to which the
resources spent on political rent seeking are socially wasteful, this would add to deadweight loss
beyond what we have derived in our graphs thus far.
23A.2
Market Segmentation and Price Discrimination
So far, we have assumed that the monopolist is constrained in the sense that she can only charge a
single price to all of her customers. This is the case when a monopolist cannot effectively differentiate
between consumers and their marginal willingness to pay for her product, or when charging different
prices to different consumers is illegal. In this section, we will suppose that charging different prices
to different consumers — a practice known as price discrimination, is permitted and that the
monopolist can segment the set of consumers into those that are willing to pay relatively more and
those that are willing to pay relatively less. Even when a monopolist can segment the market into
different types of consumers, however, she must have some way of preventing resale to keep those
consumers that purchase the product at a low price from selling to those that are being offered the
same product at a higher price.
Below, we will illustrate three different ways in which monopolists may price discriminate under
different circumstances. We will begin with the case where monopolists can perfectly identify each
consumer’s demand and can offer each consumer a particular quantity at a particular overall price
for that quantity. This is known as perfect (or “first degree”) price discrimination. Then we will
consider a case where the monopolist, while still being able to identify each consumer’s demand
perfectly, can offer different per-unit prices to different customers who potentially want to buy multiple units of the good. We will call this imperfect (or “third degree”) price discrimination. Finally,
we will consider the case where a monopolist knows that there are different types of consumers with
different demands but she does not know what type each particular consumer is. We will see that
the monopolist can then construct price/quantity packages that cause customers to “reveal their
type”, a practice known as “second degree” price discrimination.
23A.2.1
Perfect (or “First Degree”) Price Discrimination
We can begin with another extreme assumption: Suppose that the monopolist knows all of her
customers extremely well and can thus perfectly ascertain each consumer’s willingness to pay for
her product. For example, suppose that I am an artist that has his own studio and gallery. I am
the only one who produces my unique type of art, and I know my customers personally and invite
them individually to sip snooty wine while pretentiously gazing at my art. To make the analysis as
simple as possible, let’s further suppose that each of my clients will buy a single piece of art from
me. (After all, my art is so special that owning a single piece produces complete intoxication as my
clients spend all their time simply gazing at their wall to view it.)
The demand curve for my art is then composed of many different individuals who each place
a certain value on one of my pieces of art. As I produce my art, I can therefore invite first the
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Chapter 23. Monopoly
individual who places the most value on my art — and who therefore sits at the very top of the
demand curve that I face. Suppose this individual of impeccable taste places a value of $10,000
on my art. In that case, I will charge that individual exactly $10,000. Next, I invite my second
biggest fan who might place a value of only $9,900 on my art. I can then sell a piece of art to
this individual for exactly $9,900. My marginal revenue for the first piece was $10,000, and my
marginal revenue for the second piece was $9,900. Since I can charge different prices to each of my
clients, I can therefore produce a second piece of art without foregoing any profit on the first piece.
As a result, the demand curve becomes my marginal revenue curve when I can price discriminate
perfectly between all my clients.
Graph 23.3 illustrates the behavior by a profit maximizing producer who can perfectly price
discriminate in this way. Since demand is equal to M R, this producer simply chooses to produce
xM where M C intersects demand. No single price is charged as each consumer is charged exactly
what she is willing to pay along the market demand curve. Consumers therefore attain no surplus —
and all the surplus, equal to the shaded area, accrues to the monopolist. In the process, the efficient
quantity is supplied — with any additional quantity costing more than the level at which it is valued
in society.
Graph 23.3: Perfect Price Discrimination
This form of perfect price discrimination is also referred to as first degree price discrimination.
While it leads to an efficient quantity of output, it clearly leaves consumers worse off than the
non-price discriminating outcome in the previous section. This is because consumers now attain
no consumer surplus while they do attain some consumer surplus (albeit at a lower output level)
when there is no price discrimination. Efficiency is, as we know, a statement about the maximum
overall surplus and says nothing about whether the distribution of the surplus is desirable.
Exercise 23A.11 * We simplified the analysis by assuming that each person will buy only 1 piece of art.
How would you extend the idea of perfect price discrimination (resulting in demand being equal to marginal
revenue) to the case where consumers bought multiple pieces? (The answer is provided in the next section.)
23A.2.2
Imperfect or “Third Degree” Price Discrimination
Perfect price discrimination assumes that a monopolist can not only identify perfectly each type of
consumer’s demand but can also charge an amount that is exactly equal to each consumer’s total
23A. Pricing Decisions by Monopolist
863
willingness to pay. In our somewhat artificial example of my art studio, we had assumed that each
consumer only demands one piece of art (implicitly assuming that the marginal value of the second
piece is zero for each consumer). As a result, perfect price discrimination meant that I simply
arrived at an individualized price equal to exactly each consumer’s willingness to pay for one piece
of art.
More generally, consumers have downward sloping demand curves and thus place value on more
than one unit of output. Consider, for instance, two types of consumers whose demands are given as
D1 and D2 in panels (a) and (b) of Graph 23.4. Suppose further that the producer faces a constant
marginal cost of $10 per unit of output. Under perfect price discrimination, the producer would sell
200 units of the output to type 1 consumers and 100 units of the output to type 2 consumers, and
she would charge type 1 consumers the entire shaded blue area in panel (a) and type 2 consumers
the entire shaded magenta area in panel (b). Thus, when consumers place value on more than one
good, perfect price discrimination implies that the monopolist will not charge a per unit price but
rather a single price for all the units sold to a consumer together.
Graph 23.4: Imperfect (“Third Degree”) Price Discrimination
Exercise 23A.12 We might also think of the perfectly price discriminating firm as charging what is called
a “two-part tariff ” which consists of a fixed payment that is independent of the quantity a consumer buys
and a per-unit price for each unit purchased. Can you identify in the graph above which portion would be
the fixed payment and what would be the per unit price for each of the two consumers?
In many situations, this seems rather unrealistic. Instead, it might be that a monopolist who
can identify different types of consumers is restricted to charging a per-unit price for the goods —
a price that can differ for different types of consumers but remains constant for any amount a
particular consumer chooses to purchase. If this is the case, the monopolist can no longer perfectly
price discriminate but will rather price discriminate “imperfectly”. Such price discrimination is also
known as third degree price discrimination.
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Chapter 23. Monopoly
For our example in Graph 23.4, this would imply that the monopolist determines the marginal
revenue curve for each of the two types of consumers and then sets output where the constant M C
intersects M R. This leads the monopolist to charge the price p1 to type 1 consumers, with those
consumers choosing to consume x1 (in panel (a)). Similarly, a potentially different price p2 would be
charged to type 2 consumers who would then consume x2 (in panel (b)). Thus, when monopolists
can charge a per unit price that differs across identifiable consumer types, they will restrict output
below what it would be under efficient first degree price discrimination. As a result, a deadweight
loss will arise under imperfect (or third degree) price discrimination.
Exercise 23A.13 In our example of me running my art studio and selling to consumers who place value
only on the first piece of art they purchase, is there a difference between first and third degree price discrimination? Explain. (Hint: The answer is no.)
While we therefore know that deadweight loss will emerge under third degree price discrimination, it is not clear whether eliminating the ability by the monopolist to price discriminate in this
way will lead to greater or less deadweight loss. If such price discrimination were deemed illegal,
the monopolist would revert to charging a single price to all consumers – which would entail a
lower price for the high demanders and a higher price for the low demanders. Conceivably, this
uniform price could be such that low demanders will no longer consume any of the good, thus
leading to the effective closing of the market in the low demand consumer sector. The welfare losses
sustained by low demanders combined with the reduction in profit for monopolists would then have
to be weighed against the welfare gains by high demanders. Depending on the types of demand
the different consumers have, the elimination of third degree price discrimination could then lead
to either a welfare improvement (if the high demanders gain more than the low demanders and
the monopolist lose) or an additional welfare loss (if the low demanders and the monopolist lose
more than the high demanders gain). Without knowing the specifics in any particular case of third
degree price discrimination, it is therefore not possible to make a uniform efficiency-based policy
recommendation on how to treat monopolists who engage in third degree price discrimination.
Exercise 23A.14 Why do we not run into similar problems of ambiguity in thinking about the welfare
effects of first degree price discrimination?
23A.2.3
Non-Linear Pricing and “Second Degree” Price Discrimination
Sometimes there are external signals that a firm can use to infer the type of consumer she is facing.
Movie theaters know that students will generally have different demands than adults in the labor
force, and they may therefore offer student prices that are different from regular prices (and not
available to non-students). This is an example of third degree price discrimination. But in many
real world circumstances, firms do not have such external signals and therefore are unsure of what
types of consumers they face at any given moment. Put differently, it is often difficult to tell by
just looking at someone whether that person is a “high demander” or a “low demander” – even if
a firm knows how many high demanders there are relative to low demanders.
Even in such cases, however, the monopolist can try to find ways of increasing profit through
strategic pricing. But since the monopolist cannot tell what type of consumer she is facing, she
has to structure her pricing in such a way as to give the incentive to consumers to self-identify
who they are. This involves the setting of a single non-linear price schedule — or offering different
quantities of the good at different prices. Such a pricing strategy does not explicitly discriminate
between different consumers because all consumers are offered the same price schedule for different
23A. Pricing Decisions by Monopolist
865
quantities of the good. Rather, consumers end up paying different average prices based on their
choices once they see the non-linear price schedule posted by the monopolist.
Suppose, for instance, that the monopolist knows that she has two types of customers – just as
in Graph 23.4 in the previous section. But now she cannot tell in any particular instance which
type of consumer has entered her store; all she knows is that there is an equal number of both types
of consumers in the economy. In Graph 23.5a, we then illustrate the blue type 1 demand curve D1
and the magenta type 2 demand curve D2 within the same picture and again assume a constant
marginal cost of $10 per unit of output. If the monopolist could price discriminate perfectly, she
would want to offer 200 units of output to type 1 consumers and charge the entire area under D1 —
($2000 + a + b + c). Similarly, she would want to offer 100 units of the output to type 2 consumers
and charge the entire area under D2 — ($1000+a). This would result in no consumer surplus and
a surplus for the monopolist of (2a + b + c) assuming there is one consumer of each type.
Graph 23.5: “Second Degree” Price Discrimination
Exercise 23A.15 Explain how this represents separate “two-part tariffs” for the two consumer types (as
defined in exercise 23A.12).
When the monopolist cannot tell which consumers are type 1 and which are type 2, she cannot
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implement this perfect price discrimination (nor can she implement the third degree price discrimination from Graph 23.4). This is because type 1 consumers now have an incentive to simply pretend
to be type 2 consumers, purchase 100 units at the price ($1000+a) and get consumer surplus of
(b). Were the monopolist to offer the 100 and 200 unit packages at the prices suggested above, she
could look ahead and know that no one will pick the 200 unit package, leaving her with surplus of
only (2a).2
Exercise 23A.16 Why would the monopolist not be able to offer two per-unit prices as in Graph 23.4?
In order to induce type 1 consumers to behave differently than type 2 consumers, the monopolist
must therefore come up with a different set of price/quantity packages. For instance, the monopolist
might continue to offer 100 units at the price ($1000+a) while reducing the price of 200 units
to ($2000+a + c). This would equalize the surplus a type 1 consumer will get under the two
packages and would therefore make it optimal for type 1 consumers to pick 200 units. (In fact, the
monopolist has to charge a price just under ($2000+a + c) for 200 units in order to insure that type
1 consumers will in fact strictly prefer the 200 unit package over the 100 unit package.) As a result,
the monopolist would be able to expect a surplus of (2a + c) which is larger than the surplus of
(2a) she could expect under the previous price/quantity combinations.
Exercise 23A.17 In exercise 23A.12, we introduced the notion of a “two-part tariff ”. Can you express the
pricing suggested above in terms of two-part tariffs?
In panel (b) of Graph 23.5, however, we can see that the monopolist can do even better by
making the package targeted at type 2 consumers less attractive and thus charging more for the
package containing 200 units. Consider, for instance, the scenario under which the monopolist offers
a package with 90 units and another with 200 units. Type 2 consumers will be willing to buy the
90 units at a price of ($900+d). But now the monopolist can charge ($2000+d + f + g + h) for the
200 unit package — giving an overall surplus of (2d + f + g + h). The surplus of (2a + c) in panel
(a) is the same as a surplus of (2d + 2g + h) in panel (b) — which implies that the monopolist’s
surplus has changed by (f − g) as she switched from offering the 100 unit package to offering only
a 90 unit package. Area (f ) is larger than area (g) — so profit has increased.
But once the monopolist recognizes that she can earn higher profit by reducing the attractiveness
of the package targeted at type 2 consumers, she can do even better. In panel (b) of the graph,
the vertical magenta distance represents the approximate loss in profit from type 2 consumers if
the monopolist decreases the type 2 package by another unit (from 90 to 89) while the vertical
blue distance represents the approximate increase in profit from type 1 consumers that can now be
charged a higher price for the 200 unit package. The monopolist can increase profit by reducing the
type 2 package so long as the vertical magenta distance is shorter than the vertical blue distance.
Thus, a forward looking monopolist would reduce the type 2 package to a quantity x∗ (where the
two distances are equal to one another). This is represented in panel (c) of the graph.
Exercise 23A.18 What price will the profit maximizing monopolist charge for x∗ and for 200 units in
panel (c) of Graph 23.5?
Exercise 23A.19 * We have assumed in our example that there is an equal number of type 1 and type 2
consumers in the economy. How would our analysis change if the monopolist knew that there were twice as
many type 1 consumers as type 2 consumers?
2 In Chapter 24, we will introduce the idea of a “sequential game” in which some players move first. We could
then say that the monopolist plays such a sequential game with consumers – setting her pricing schedule in stage 1
knowing that consumers will optimize in stage 2.
23A. Pricing Decisions by Monopolist
867
Exercise 23A.20 In Chapter 22, we analyzed situations in which there is asymmetric information between
consumers and producers (as in the insurance market). Can you see how the problems faced by an insurance
company that does not know the risk-types of its consumers are similar to the problem faced by the monopolist
who is trying to second-degree price discriminate?
The above example is just one of many that might arise for a monopolist who seeks to offer
different per-unit prices to different customers whose type she cannot identify. We will see further
examples later on. In the real world, the “packages” offered to different types of consumers may
also vary in ways that are related to not just quantity but also quality. For instance, in the airline
industry, fares for the same flights are often priced quite differently for business travelers and leisure
travelers, with business travelers facing fewer restrictions on when and how they can change their
tickets. If these topics are of interest, you should consider taking a course in industrial organization.
23A.3
Barriers to Entry and Remedies for Inefficient Monopoly Behavior
So far, we have simply assumed that a particular firm has a monopoly in the market for good x.
But how does a firm get such monopoly status in the first place? And how does it hold onto it?
We began to discuss this a bit in our brief section on political rent seeking and its implications
for deadweight loss. In this section, we will try to dig a bit deeper and point out more explicitly
that there must exist some barrier to entry of new firms in order for a monopoly to be able to
earn long run positive profits. Such a barrier might emerge simply from the technological nature of
production, from different types of legal barriers to entry that we introduced when thinking about
political rent seeking or through other channels.
23A.3.1
Technological Barriers to Entry and Natural Monopolies
In our discussion of perfectly competitive firms, we never considered the case of a firm that has
increasing returns to scale for all output quantities. Rather, we focused on firms that may have
increasing returns to scale in their production process for low levels of output but eventually face
decreasing returns to scale as output increases. It is because of this assumption that M C and AC
curves eventually sloped up. But, while we argued in Chapter 11 that the logic of scarcity requires
that marginal product of each input eventually diminishes, there is no particular reason that the
production process itself cannot have increasing returns to scale.
Exercise 23A.21 Review the logic of how a production process can have diminishing marginal product of
all inputs while still exhibiting increasing returns to scale.
Now suppose the production process for good x always has increasing returns to scale. This
implies, as we illustrated in Graph 12.9, that the M C curve is always downward sloping and
always lies below AC, which further implies that any price taking firm will either produce nothing
at a particular price or will produce an infinite quantity of the good. But, in a world of scarcity,
consumers will not demand an infinite quantity of the good at a positive price — which implies that
the assumption of price taking behavior on the part of the firm is not reasonable under increasing
returns to scale. It is for this reason that no competitive industry can have firms whose production
process always has increasing returns to scale.
Similar logic applies when a production process has a large initial or a significant recurring fixed
cost together with a constant marginal cost, a case which is illustrated in Graph 23.6a. This can
arise in many different contexts. For instance, a large investment in research and development may
868
Chapter 23. Monopoly
be required prior to the production of a vaccine, but, once the research is complete, the vaccine can
be produced easily at constant M C. Or a utility company might have to invest a large amount in
laying electricity lines within a city in order to then be able to provide electricity to everyone at a
constant M C. Or a software company might work for years to produce a piece of software that can
then be offered at virtually no marginal cost by having customers download it from the internet.
Graph 23.6: A Natural Monopoly
A natural monopoly is then defined as a firm that faces an AC curve that declines at all output
quantities. This declining AC curve can be due to increasing returns to scale everywhere or due
to the presence of a recurring fixed cost with constant marginal cost. In either case, we cannot
identify a “supply curve” that is equal to the M C curve above AC because M C never lies above
AC. It is therefore “natural” for a single firm to emerge as a monopoly.
Exercise 23A.22 Can you see in Graph 23.6a that a price taking firm facing a downward sloping AC
curve would produce either no output or an infinite amount of the output depending on what the price is?
Exercise 23A.23 Suppose the technology is such that AC is U-shaped but the upward sloping part of the Ushape happens at an output level that is high relative to market demand. Can the same “natural monopoly”
situation arise?
Panels (b) and (c) of Graph 23.6 then add demand and M R curves to the cost curves from
panel (a). In panel (b), demand is relatively “high”, and the usual profit maximizing single price
pM read off the demand curve at quantity xM where M C and M R intersect results in a positive
profit for the monopoly firm. In panel (c), on the other hand, demand is relatively “low” — causing
the monopoly to make a loss if it simply produced where M R intersects M C.
In order for a firm facing the situation in panel (c) to make a positive profit, it would therefore
have to price output differently — thus employing one of the price discrimination strategies discussed
in the previous section. In the absence of being able to identify different consumer types, this implies
that, in order to produce, the firm would have to engage in a form of pricing that involves more
than just a single per-unit price. The most common such strategy for natural monopolists (in the
absence of price regulation) is to charge a fixed fee plus a per unit price, which we referred to as a
23A. Pricing Decisions by Monopolist
869
“two part tariff” in exercise 23A.12. In the case of utility companies, for instance, there might be
a fixed service fee per month plus a price per unit of electricity consumed.
Because the technological constraints are such that multiple firms in such industries would
entail higher per unit costs, governments have often favored regulation of natural monopolies over
alternative policies to address the deadweight loss from monopoly pricing. Such regulation typically
focuses on pricing policies that guarantee a “fair market return” for the natural monopolist while
moving production closer to the socially optimal level. Given that the fixed cost is a sunk cost once
the monopolist is operating, efficiency would require output where M C crosses the demand curve.
But because AC lies above M C, forcing the natural monopolist to price the output at M C would
imply negative profits for the monopolist (when profit is defined as long run profit that includes
recurring fixed costs).
Exercise 23A.24 In a graph similar to Graph 23.6b, illustrate the negative profit that arises when the
monopolist is forced to price at M C.
Exercise 23A.25 Suppose the fixed cost is a one-time fixed entry cost that is sufficiently large to result
in a picture like panel (c). True or False: If the government pays the fixed cost for the firm, it will not
have to regulate the firm in order to make sure the firm makes a profit – but he monopoly outcome will be
inefficient.
For instance, suppose the monopolist faces high recurring fixed costs. Then regulators who
attempt to achieve efficient output levels in natural monopolies might aim to set price at M C and
allow monopolists to charge an additional “fixed fee” that each customer has to pay independent
of the level of consumption. For instance, an electricity provider might charge a fixed “hook up”
fee for connecting a household to the service and then a per unit price for each unit of electricity
consumed, or a phone company might charge a fixed monthly fee plus a per minute charge for
phone calls made. The fixed fees can then be set in such a way as to make the natural monopoly
profitable even though the per-unit prices do not cover any of the fixed costs.
Exercise 23A.26 Is this an example of a two-part tariff ? Does it result in efficiency?
While it is easy to see how this type of regulation works in principle, in practice the regulator
unfortunately does not have all the required information to implement the optimal two-part pricing.
In particular, the regulator does not typically know the cost functions of the natural monopoly, and
the natural monopolist has every incentive to inflate her costs to the regulator in order to obtain
higher fixed fees and higher per-unit prices. There are examples in the real world of natural monopolists devising clever schemes involving fake billing from secondary firms in order to show higher
costs than they actually incur, and it is not always easy for regulators to identify such falsifications
of cost records. The monopolist furthermore has no particular incentive to find innovative ways of
lowering costs through technological innovations even if she is perfectly honest in how she reports
the costs she actually incurs.
For some of these reasons, more recent policy approaches have made an effort to introduce
competition into some industries that face these cost curves by having the government pay the
fixed costs that cause AC curves to be downward sloping. In the utility industry, for instance,
the government could lay (and maintain) the electricity lines to all the houses in a city and then
allow any utility company to use these lines in order to “ship” electricity to individual houses. It
is much like the government laying a system of roads that different trucking companies can use to
deliver goods. With the fixed costs paid by the government, individual electricity suppliers then
have only variable costs — and thus flat or upward sloping M C curves. It then becomes once
870
Chapter 23. Monopoly
again possible for many different electricity providers to compete for households, with households
choosing a provider based on quality of service and price.
Exercise 23A.27 Suppose that instead a private company is charged with laying all the infrastructure and
then charges competing electricity firms to use the electrical grid. How might this raise a different set of
efficiency issues related to monopoly pricing? Would these issues still arise if the government auctioned off
the right to build an electricity grid to a single private company?
23A.3.2
Legal Barriers to Entry
While monopoly power can certainly arise from technological barriers that prevent several firms
from operating simultaneously, it may alternatively arise from legal barriers. Such legal barriers
might arise because of general patent and copyright laws that grant the exclusive right to produce
particular products (for a certain number of years) to those firms that were awarded the patent
or copyright. The motivation behind such laws is not to encourage the formation of monopolies
but rather to provide incentives for innovations by ensuring that innovators can profit from their
activities for some period. We will discuss the role of patents and copyrights in more detail in
Chapter 26.
Patent and copyright laws are not, however, the only legal barriers to entry. As we have seen,
free entry (in the absence of technological barriers) tends to drive economic profits to zero. Thus, if
a firm can successfully lobby the government to protect it from competitors, it will invest resources
to accomplish this if the required resources are smaller than the present discounted value of the
monopoly profits the firm can expect to earn if legal barriers to entry were erected. As we have
already mentioned, to the extent to which such lobbying involves socially wasteful activities, the
deadweight loss from government created monopolies may therefore exceed the loss due to the
decline in production that results under monopoly profit maximization.
Monopoly power has been granted by governments to a variety of firms throughout history. In
the 15th and 16th centuries, for instance, the British Crown awarded exclusive rights to shipping
companies to establish trade routes in the West Indies and other parts of the world. More recently,
airlines routes were regulated in a similar manner, with airlines being assigned exclusive rights to
certain routes within the US (prior to airline deregulation). The same was true until the 1970s
in the trucking industry and the phone industry. Today, the US Post Office continues to hold the
exclusive right to deliver first class mail, although the government now permits carriers like UPS
and FedEx to deliver express packages and large ground packages. In each of these cases, you
should be able to see how the firm that attained the exclusive rights to serve a particular market
benefits from the governments’ entry barriers – and how it might have a vested interest in engaging
in socially wasteful lobbying activities in order to retain its monopoly power.
23A.3.3
Restraining Monopoly Power
While governments have, as we have mentioned, been prime culprits – for better or worse – of
granting monopoly power to certain firms, the increasing awareness of potential social losses from
the exercise of monopoly power has also led to government policies aimed at restraining monopolies.
The question of when and under what circumstances government intervention is desirable is a
complicated one. The tendency of monopolies to limit output in order to raise price has the
clear deadweight loss implications that we have discussed. At the same time, patent-protection of
innovation may have led to the emergence of products that might otherwise never have seen the light
23A. Pricing Decisions by Monopolist
871
of day – implying the creation of social surplus despite the fact that, at any given moment, more
surplus could be gained by forcing monopolies to produce more. (We will have more to say about
this in Chapter 26.) And the existence of increasing returns to scale in certain industries implies
that natural monopolies may lower per unit costs even as they attempt to use their monopoly status
to raise price above marginal cost.
We will show in end-of-chapter exercise 23.9 that some of the potential remedies that one might
think of applying to monopolies are either ineffective or counterproductive. These include per
unit taxes and profit taxes. We have already discussed (in our treatment of regulation of natural
monopolies) that attempts to directly regulate the pricing of monopoly goods run into informational
constraints because regulators typically do not know the real costs of firms and because such
regulation would give little incentive for cost innovations by monopolies. This does not imply that
regulation in some circumstances is not the appropriate policy, but it does imply that regulation
is no panacea in all cases. In some instances, governments have forced the break-up of monopolies
(as in the case of large oil companies many years ago or large phone companies more recently),
and in other cases they have found ways of addressing the root causes of natural monopolies by
disconnecting the fixed cost infrastructure from the marginal cost provision of services. And in other
cases, governments have actively blocked mergers of large companies that might have resulted in
excessive monopoly power. Finally, there has been an increasing trend toward deregulation of
industries where regulation itself (such as in the airline industry) created monopolies to begin with.
If these topics seem interesting to you, you might consider taking a course on antitrust economics
or law and economics.
In many circumstances, however, the most effective tool for restraining monopoly power has
little to do with direct government actions and more to do with the fact that, when a monopoly
does exercise its power to create profit, there is a powerful incentive for entrepreneurs to find new
ways to challenge that monopoly power. A firm may, for instance, have captured a large portion
of the market, perhaps for no other reason than being first and making early, strategically smart
decisions (as in the case of Microsoft and its Windows operating system). There is no doubt
that such firms will use their monopoly power to their advantage, but they may also be more
cognizant of the threat of competitors (that may find ways of producing substitutes) than our
simple static models of monopoly behavior predict. The more a firm exercises its monopoly power,
the greater is the incentive for others to find ways of producing such substitutes, and a forward
looking monopolist should take that into account when setting current prices, as we will see in
upcoming chapters. Sometimes barriers to entry that may seem rock-solid at one time can fall
quickly with new technological innovations, as, for instance, with the sudden emergence of cell
phone technology, internet calling and cable provision of telephone service that are challenging
traditional phone companies. In such environments, governments can play in important role in
insuring that existing firms (such as traditional phone companies) do not successfully erect barriers
of entry through legislation or regulation (by prohibiting, for instance, cable companies or internet
providers from providing telephone service). Just as there exists a powerful incentive for innovators
to find ways of breaking barriers to entry by existing firms, there is a similarly powerful incentive
on the part of existing firms to find other ways of shoring up these barriers to entry in order to
preserve market power.
Exercise 23A.28 In the 1970s when OPEC countries raised world prices for oil substantially by exercising
their market power, the Saudi oil minister is said to have warned them: “Remember, the Stone Age did
not end because we ran out of stones.” Explain what he meant and how his words relate to constraints that
monopolies face.
872
23B
Chapter 23. Monopoly
The Mathematics of Monopoly
From a mathematical point of view, monopolies engage in the same optimization problem that
competitive firms undertake except that monopolies have additional choice variables. Both types
of firm face some cost function that emerges from the cost minimization problem and tells them
the total cost c(x) of producing any quantity x. We should note at the outset that for much of our
development below we will assume that dc(x)/dx = c – i.e. the firm faces a constant marginal cost.
This simplifies some of the analysis in convenient ways, and we will explore different marginal cost
schedules in some of the end-of-chapter exercises.
Exercise 23B.1 Explain why the cost minimization problem in the firm’s duality picture is identical for
firms regardless of whether they are monopolies or perfect competitors.
A monopoly that is restricted to charging a single per-unit price then solves the problem
max π = px − c(x) subject to p ≤ p(x),
x,p
(23.1)
where the price the monopolist charges when trying to sell the quantity x cannot be greater than
the price for that quantity given by the inverse demand function p(x). The perfect competitor’s
problem could be written in exactly the same way, except that for the perfect competitor the inverse
demand function is simply p(x) = p∗ , where p∗ is the market price. Thus, price ceases to be a choice
variable when price is set by the competitive market, but it is a choice variable for a monopolist
who faces a downward sloping demand curve.
Since the monopolist will set price as high as she can while still selling all the goods she produces,
the inequality in equation (23.1) will bind – i.e. p = p(x). The monopolist’s problem can therefore
be re-written as
max π = p(x)x − c(x).
x
(23.2)
Note that by choosing the optimal quantity xM , the monopolist implicitly chooses the optimal
price pM = p(xM ) once we have substituted the constraint into the objective function of the
optimization problem. And because of the resulting one-to-one mapping from quantity to price,
the monopolist’s problem could alternatively be written as
max π = px(p) − c(x(p))
p
(23.3)
where x(p) is the market demand function (as opposed to the inverse market demand function
p(x) in the previous problem.) Whether we view the monopolist as choosing quantity as in equation
(23.1) or price as in equation (23.3), the same monopoly quantity and price will emerge.
When a monopolist is not restricted to charging a single per-unit price, she has additional
decisions to make as we have seen in our discussion of price discrimination in Section A. The exact
nature of that choice problem then depends on what the firm knows and what pricing strategies
are available to the firm. If the firm can identify consumer types prior to consumption choices
by consumers, first and third degree price discrimination become possible (assuming re-sale can
be prevented), and if the firm only knows the distribution of consumer types in the population,
second degree price discrimination becomes possible. Different forms of such discrimination are
furthermore restricted by the types of pricing schedules that firms are permitted to post, as we will
see a little later in the chapter. Fundamentally, however, the firm is still just maximizing profit by
making production choices and potentially by engaging in strategic price differentiation.
23B. The Mathematics of Monopoly
23B.1
873
Demand, Marginal Revenue and Profit
Suppose that the market demand facing a monopolist is of the form
x(p) = A − αp,
(23.4)
which gives rise to an inverse market demand
1
A
− x.
(23.5)
α
α
For consistency, we will use this market demand specification repeatedly, both in this chapter as
well as in the following chapters that deal with other market structures within which firms might
operate.
p(x) =
23B.1.1
Marginal Revenue and Price Elasticity
For the monopolist, total revenue is then equal to price times output, where price is determined by
the inverse market demand curve; i.e.
1
A
1
A
− x x = x − x2 .
(23.6)
T R = p(x)x =
α
α
α
α
In Section A, we argued verbally that the marginal revenue curve for a monopolist has the same
intercept as the inverse demand curve but twice the slope. This is easily verified mathematically,
with marginal revenue simply the derivative of T R with respect to output
dT R
A
2
= − x.
(23.7)
dx
α
α
More generally, we can write the inverse demand function as p(x) and total revenue as T R =
p(x)x. Using this expression, we can differentiate T R with respect to x to get
MR =
dp
x.
(23.8)
dx
Now suppose we multiply the second term in equation (23.8) by (p(x)/p(x)). Then we can write
the expression for M R as
dp x
.
(23.9)
M R = p(x) 1 +
dx p(x)
M R = p(x) +
Recall that the price elasticity of demand for an inverse demand function p(x) is given by
ǫD = (dx/dp)(p(x)/x), which is just the inverse of the second term in parenthesis in the above
equation. Thus, we can write the expression for M R as
1
M R = p(x) 1 +
.
(23.10)
ǫD
Suppose, for instance, that we are currently at the mid-point of a linear demand curve (such as
the one in Graph 23.1a) where the price elasticity of demand is equal to -1. Equation (23.10) then
tells us that marginal revenue at that point is equal to 0, precisely as we derived in panel (b) of
Graph 23.1.
874
Chapter 23. Monopoly
Exercise 23B.2 Use equation (23.10) to verify the vertical intercept of the marginal revenue curve in
Graph 23.1b.
23B.1.2
Revenue Maximization
In order to maximize total revenue T R, the monopolist would simply set M R equal to zero. Using
equation (23.10) for M R, it follows immediately that revenue is maximized when ǫD = −1. With
the linear demand specified in equation (23.4), this implies an output level of A/2.
Exercise 23B.3 Set up a revenue maximization problem for the firm. Then verify that this is indeed the
revenue maximizing output level and that, at that output, ǫD = −1.
23B.1.3
Profit Maximization
The monopolist’s profit maximization problem differs from revenue maximization in that costs are
taken into account. This problem, already introduced at the beginning of this section, can be
written as
max π = p(x)x − c(x),
x
(23.11)
where c(x) is the total cost function (that is derived from the production function as described
in our producer theory chapters earlier in the text).3 Taking first order conditions, we get
M R = p(x) +
dc(x)
dp
x=
= M C.
dx
dx
(23.12)
Exercise 23B.4 Can you use equation (23.10) to now prove that, so long as M C > 0, the monopolist will
produce where ǫD < −1?
For instance, suppose that market demand is linear as specified in equation (23.4) and c(x) = cx.
Then our M R = M C condition implies
2
A
− =c
α
α
(23.13)
which further implies a monopoly output xM and price pM of
xM =
A − αc
A + αc
and pM =
.
2
2α
(23.14)
Exercise 23B.5 Illustrate that profit maximization approaches revenue maximization as M C = c approaches zero.
Exercise 23B.6 Verify for the example of our linear demand curve and constant marginal cost c that it
does not matter whether the firm maximizes profit by choosing x or p (as in the problems defined in equations
(23.1) and (23.3) above).
3 Recall that the cost function is really a function of output x as well as input prices. We are suppressing the
input price notation since input markets are not a focus for us here.
23B. The Mathematics of Monopoly
23B.1.4
875
Constant Elasticity Demand and Monopoly Markups
Another way to write the optimal monopoly price emerges from substituting our elasticity-based
expression for M R from equation (23.10) into the M C = M R condition of equation (23.12); i.e.
1
= M C.
p 1+
ǫD
(23.15)
p − MC
−1
.
=
p
ǫD
(23.16)
Re-arranging terms, we then get
The difference between price and M C – i.e. (p − M C) – is called the monopoly markup because
it represents how much the monopolist “marks its price up” above marginal cost where we would
expect competitive firms to produce. The left hand side of equation (23.16) is called the monopoly
markup ratio – which is simply the markup relative to the price charged by the monopolist. (The
markup ratio is also called the Lerner Index.) Since the price elasticity term ǫD is negative, this
cancels the negative sign on the right hand side and makes the mark-up itself positive.
Suppose, then, that instead of facing a linear demand curve for which price elasticity differs
at each point, a monopolist faces a constant-elasticity demand curve of the form x = αp−ǫ for
which the price elasticity of demand is −ǫ everywhere. Equation (23.16) then tells us that the
monopolist’s markup ratio is inversely proportional to the price elasticity of demand. This implies
that the markup ratio (and the markup itself) approaches zero as the price elasticity of demand
approaches minus infinity. That certainly makes intuitive sense: as the price elasticity of demand
approaches minus infinity, the monopolist faces a demand curve that increasingly looks like the
demand curve faced by a perfect competitor. When working with the family of constant elasticity
demand curves, the price elasticity of demand is therefore a nice measure of the degree of monopoly
power that the firm actually has.
23B.2
Price Discrimination when Consumer Types are Observed
In Section A of the chapter, we differentiated between three different types of price discrimination
that monopolists might employ depending on what they know about their consumers and the
degree to which the monopolist can prevent consumers from undermining the price discrimination.
In cases where monopolists can identify demand by each consumer, the firm can perfectly (or firstdegree) price discriminate and capture the consumers’ entire surplus – as long as something prevents
consumers from selling the goods to each other. When monopolists are restricted to charging perunit prices but are not restricted to charging the same per unit price to all consumers (whose demand
they can again identify), we illustrated how they can employ third degree price discrimination –
again assuming that consumers cannot engage in re-sale. Finally, if monopolists know that different
consumers have different demands but cannot identify which consumer is which type, we saw that
the firm can second-degree price discriminate by designing (non-linear) price/quantity combinations
that cause consumers to self select into packages based on their type. We will begin in this section
with the mathematically easier cases of first and third degree price discrimination where we assume
that firms observe consumer types prior to setting pricing policies.
876
23B.2.1
Chapter 23. Monopoly
Perfect or First Degree Price Discrimination
As we illustrated in Section A, first degree price discrimination implies that the firm will charge
the consumer her marginal willingness to pay for each of the goods she purchases. Suppose that
a monopolist faces a constant marginal cost M C and let pc = M C represent the per unit price
we would expect under perfect competition. For a particular consumer n, let CS n represent the
consumer surplus n would receive under competitive pricing, with the consumer choosing to consume
where pc crosses her demand curve Dn . One way to think of perfect price discrimination is to think
of the monopolist as continuing to charge a per-unit price of pc but supplementing this with a fixed
fee that the consumer has to pay before she can purchase anything at all. Notice that this fixed fee
is a sunk cost for the consumer once it is paid – and therefore has no impact on the quantity the
consumer will purchase once the fee is paid.
The only question for the consumer is then whether she wants to pay the fixed fee in order to
be able to purchase from the monopolist. Since she expects a consumer surplus of CS n when she
faces a per-unit price of pc in the absence of a fixed fee, she will be willing to pay any fixed fee that
is less than or equal to CS n . The monopolist can therefore set a two-part tariff, with the overall
payment P n charged to consumer n equal to
P n (x) = CS n + pc x.
(23.17)
Under this two-part tariff, the monopolist has set a price policy for consumer n that will leave the
consumer with no surplus but results in the efficient level of consumption by consumer n. The fixed
portion of the price policy is different for each type of consumer, which implies the monopolist must
know each consumer’s type in order to implement the first-degree price discrimination if consumers
have different demands.
Exercise 23B.7 Illustrate graphically the two different parts of the two-part tariff in equation (23.17).
23B.2.2
Third Degree Price Discrimination
Suppose now that the monopolist is selling to two different distinct markets but is limited to
charging per-unit prices in each market (and thus cannot implement a two-part tariff of the type in
equation (23.17)). With knowledge of the two inverse demand functions p1 (x) and p2 (x) for the two
markets, the monopolist will then try to maximize her profit across the two markets by choosing
how much to produce in each market (and thus also how much to charge in each market); i.e. the
monopolist will solve the problem
max π = p1 (x1 )x1 + p2 (x2 )x2 − c(x1 + x2 ),
x1 ,x2
(23.18)
where c is the firm’s total cost function. Taking first order conditions, we get
∂π
= p1 (x1 ) +
∂x1
∂π
= p2 (x2 ) +
∂x2
which can simply be rewritten as
dp1 1
x −
dx1
dp2 2
x −
dx2
dc
= 0,
dx
dc
= 0,
dx
(23.19)
23B. The Mathematics of Monopoly
877
M R1 = M C = M R2 ,
(23.20)
where M Ri is the marginal revenue function derived from the ith market’s inverse demand
function. Since we know from equation (23.10) how to write M R functions in price elasticity terms,
we can write this as
p
1
1+
1
ǫD 1
1
= MC = p 1 +
ǫD 2
2
(23.21)
which simply extends equation (23.15) to two separate markets, with the “mark-up” in each
market reflecting the price elasticity in each market. This then implies
(ǫD2 + 1)ǫD1
p1
=
.
p2
(ǫD1 + 1)ǫD2
(23.22)
Put into words, regardless of what the M C of production is, the price charged in one market
relative to that charged in the other market depends only on the price elasticities of demand in the
two markets when M C is constant.
Suppose, for instance, that a monopoly faces constant marginal cost equal to c and that the
demand functions in two different markets are x1 = A − αp and x2 = B − βp. These demand
functions give rise to inverse demand functions
p1 =
B − x2
A − x1
and p2 =
,
α
β
(23.23)
and the first order conditions requiring marginal revenue to be equal to marginal cost in both
markets imply
x1 =
A − αc
B − βc
and x2 =
2
2
(23.24)
p1 =
A + αc
B + βc
and p2 =
.
2α
2β
(23.25)
and
Exercise 23B.8 Verify that equation (23.22) holds for this example. (Be sure to evaluate elasticities at
the optimal output levels.)
Exercise 23B.9 * True or False: The higher priced market under (third degree) price discrimination is
more price inelastic.
As we noted in our Section A discussion of third degree price discrimination, the welfare effect of
eliminating such discrimination is ambiguous and requires an analysis of the gains by low elasticity
consumers relative to the losses by high elasticity consumers (and the monopolist).
878
Chapter 23. Monopoly
2-Part Tariff Restrictions for Different Forms of Price Discrimination
None 1st Degree 3rd Degree 2-part tariff
2nd Degree
F1
=0
=0
F2
=0
=0
= F1
1
p
p2 = p1
= p1
Table 23.1: F n = type n’s fixed charge; pn = type n’s per-unit price
23B.3
Discrimination when Consumer Types are Not Observable
First and third degree price discrimination are relatively straightforward since firms are assumed
to know the types of consumers they face. When they do not know the consumer types but
are only aware of the fraction of the population that falls into each category, the monopolist’s
problem becomes more difficult and involves more strategic considerations. In particular, since
the monopolist has no external signal about the consumer types she is facing, she must design her
pricing policy in such a way that consumers themselves choose to reveal what type they are through
the types of purchases they make. As you may have noticed already in Section A, all the various
ways of thinking about monopoly pricing involve the firm choosing two-part tariffs of the form
P n (x) = F n + pn x for n = 1, 2.
(23.26)
In other words, we can express each of the pricing strategies as separate two-part tariffs aimed
at the two types of consumers. The difference in all these strategies is that in some cases we are
restricting fixed charges F n to be zero and in some cases we are restricting the monopolist to
only a single pricing schedule. Table 23.1 illustrates this for the forms of price discrimination we
have treated above and those we are about to discuss below. For instance, we began the chapter
in Section 23B.1 with a monopolist who was restricted to charging a single per-unit price to all
consumers, effectively assuming F 1 = F 2 = 0 and p2 = p1 as in the first column of the table. Under
first degree price discrimination, on the other hand, we make no restrictions on the fixed and perunit prices that the monopolist can use. Under third degree price discrimination, no fixed fees are
permitted (i.e. F 1 = F 2 = 0) but no restrictions are placed on the per-unit prices the monopolist
can charge. Below we will revisit the case where no restrictions are placed on fixed fees or perunit prices (as in first degree price discrimination) but under the informational constraint that the
firm cannot observe consumer type prior to consumers making their purchasing decisions. This is
second degree price discrimination, represented in the last column of Table 23.1. But we will build
up to this full second-degree price discrimination by first considering the case where a firm does
not observe consumer type and is restricted to posting a single two-part tariff (rather than separate
two-part tariffs aimed at different consumer types). This is represented in the second-to-last column
in Table 23.1.
To simplify the analysis to its essentials, we will also allow a single preference parameter to
differentiate the different consumer types in this section.4 In particular, suppose that consumer n
has tastes for the monopoly good x that can be represented by the utility function
4 Previously, we have allowed different consumer types to differ in both the intercept and the slope of their demand
curves.
23B. The Mathematics of Monopoly
879
U n = θn u(x) − P (x),
(23.27)
5
where P (x) is the total charge for consuming quantity x. Differences in consumer tastes are
then captured by differences in the value of θn . Note that this is not the typical type of utility
function we have worked with given that it is defined over only a single good. However, as we
demonstrate in a short appendix, this type of “reduced form” utility function can be justified as
arising from preferences that are separable (between other consumption and the good x) when the
overall spending on the good x represents only a small portion of the consumer’s income. In fact,
we demonstrate in the appendix that we can assume identical underlying (separable) preferences
where consumers differ only in their income, and that the differences in the value of θn in the
reduced form utility function above are then simply related to underlying differences in consumer
income.
23B.3.1
Second Degree Price Discrimination with a single Two-Part Tariff
As already mentioned, we begin our consideration of second degree price discrimination with a
restricted version that we did not discuss in Section A of the chapter: a version in which the
monopolist is limited to using a single two-part tariff for both consumer types (rather than different
two-part tariffs aimed at the two different types). If the monopolist is so constrained, P (x) has to
take the form
P (x) = F + px,
(23.28)
where F is the fixed charge and p the per-unit price – with neither being superscripted by n
(since the same price schedule applies to both types). Maximizing consumer utility from equation
(23.27) given the two part tariff from equation (23.28) entails the simple optimization problem
max θn u(x) − F − px
x
(23.29)
and gives us the first order condition
du(x)
= p.
(23.30)
dx
The analysis becomes particularly clean if we assume the following functional form for u(x):
θn
1 − (1 − x)2
(23.31)
2
which has a first derivative with respect to x that is just (1 − x). Plugging this into equation
(23.30) and solving for x, we then get the consumer’s demand function as
u(x) =
xn (p) =
θn − p
.
θn
(23.32)
Notice that we therefore have specified underlying preferences in such a way as to once again
have linear demand curves of the form x(p) = A − αp where A = 1 and α = 1/θn .
5 This exposition draws on similar exposition in Tirole, J. (2001), The Theory of Industrial Organization, Cambridge, MA: The MIT Press. For the interested student, this text is an excellent reference for matters related to
market power, but it is quite advanced.
880
Chapter 23. Monopoly
Exercise 23B.10 Intuitively, why does the fixed charge F from the two-part tariff not show up in the
demand function?
Exercise 23B.11 Derive the price charged to consumer n by a third-degree price discriminating monopolist
with constant marginal cost c.
In Graph 23.7 below, we depict the inverse demand curve for this demand function and illustrate
the consumer surplus triangle CS n that, for a particular per-unit price p with F = 0, is of size
CS n (p) =
(θn − p)2
(θn − p)xn (p)
.
=
2
2θn
(23.33)
Graph 23.7: Consumer n’s Inverse Demand Curve
Now suppose that a monopolist faces two types of consumers, type 1 and 2 with preference
parameters θ1 and θ2 respectively and with θ1 < θ2 . Suppose further that the monopolist knows
that a fraction γ < 1 of the consumers are of type 1, with the remaining fraction (1 − γ) made
up of consumers of type 2. Finally, suppose the monopolist faces a constant marginal cost of c.
Whatever per-unit price the monopolist chooses, she then has to respect the constraint that the
lower demand consumer 1 will not choose to consume any of the good if the fixed charge F is set
above CS 1 (p) = (θ1 − p)2 /2θ1 .6 Thus, for a given per-unit price p, the monopolist’s optimal fixed
charge is CS 1 (p).
Knowing this, the monopolist needs to determine the optimal per-unit charge in the two-part
tariff. One way to think of this is as a process in which the monopoly maximizes the expected profit
from each encounter with a consumer – knowing the fractions of the consumer pool that fall into
one type or the other. This expected profit takes the form
E(π) = CS 1 (p) + γ(p − c)x1 (p) + (1 − γ)(p − c)x2 (p).
(23.34)
The CS 1 (p) term is simply the fixed charge that we have concluded the firm will set in its
two part tariff, a charge that will be paid by both types of consumers. Thus, the firm receives
that for sure each time a customer shows up. With probability γ, the firm faces a consumer of
6 This
constraint is often referred to as the individual rationality constraint.
23B. The Mathematics of Monopoly
881
type 1 who will purchase x1 (p) at price p. When multiplied by the difference between price p and
marginal cost c, we get the expected additional profit from facing this type of consumer. Similarly,
with probability (1 − γ) the firm will face a consumer of type 2 and with it an additional profit of
(p − c)x2 (p).
Substituting in for what we derived for CS 1 (p), x1 (p) and x2 (p), and rearranging terms, the
expected profit can then be expressed as
(1 − γ)
γ
(θ1 − p)2
p
(23.35)
+ (p − c) 1 −
+
E(π) =
2θ1
θ1
θ2
Exercise 23B.12 Verify that this equation is correct.
The only choice variable for the monopolist in this expected profit equation is p. Thus, maximizing the expected profit subject to the implicit constraint that only a two-part tariff can be
employed is simply maximizing E(π) by choosing p. Solving the first order condition from this
maximization problem for p, we get the optimal per-unit price p∗
p∗ =
Exercise 23B.13
**
c(γθ2 + (1 − γ)θ1 )
2(γθ2 + (1 − γ)θ1 ) − θ2
(23.36)
Verify that this equation is correct.
In panel (a) of Graph 23.8, the line CS 1 (p∗ ) + p∗ x represents the two part tariff P (x) that
indicates, for any quantity x, the total price charged to consumers. What makes this a two-part
tariff is that the line has a vertical intercept – which puts in place a fixed cost to the consumer for
purchasing from the firm. Were the line to go through the origin, we would have a simple per-unit
price.
Graph 23.8: Second Degree Price Discrimination with Two-Part Tariffs
In panel (b) of the graph, we illustrate the shape of indifference curves for the two types of
consumers, with the blue indifference curves representing type 1 and the magenta indifference
curves representing type 2. Consumers prefer to have more of x and less of P – and thus become
better off as they move toward indifference curves to the south-east of the graph.
882
Chapter 23. Monopoly
Exercise 23B.14 Are these preferences convex?
Exercise 23B.15 Note that each set of blue and magenta indifference curves cross once, with the magenta
indifference curve having a steeper slope at that point than the blue indifference curve. Can you give an
intuitive explanation for this?
Finally, in panel (c) of the graph, we put indifference curves and the two-part tariff-induced
constraint into a single graph to illustrate the consumers’ optimal choices, with type 1 consumers
optimizing at point A and type 2 consumers optimizing at point B. Note that the optimal blue
indifference curve for type 1 crosses the origin, which implies that type 1 consumers are as well off
at point A as they are at point (0, 0) where they consume no x and pay no price. Put differently,
consumers of type 1 attain zero consumer surplus at point A under the two-part tariff that has
been set by the firm.
Exercise 23B.16 Given what you know of how the firm constructed the two-part tariff, can you give an
intuitive explanation for this?
While the firm that is implementing the two-part tariff does not know what type of consumer
she faces prior to a consumption decision, the graph illustrates that the two-part tariff allows the
firm to know what type of consumer she faced after the decision has been made. Put differently and
in the language of Chapter 22, the firm has induced a separating equilibrium, with the consumer
types signaling their type through their consumption choices.
23B.3.2
Second Degree Price Discrimination more Generally
In our definition of second degree price discrimination in Section A, we did not limit the monopolist
to using a single two-part tariff but allowed her to create price/quantity packages that in effect
allowed her to charge different fixed fees and different per-unit prices. In order to reconcile our
treatment here with the graphs we drew in Section A, particularly Graph 23.5, we can again
consider the problem using demand curves rather than indifference curves. Panel (a) of Graph 23.9
then illustrates the inverse demand curves for type 1 (blue) and type 2 (magenta) as well as the
per-unit price p∗ in the single two-part tariff that we just derived.
Exercise 23B.17 Explain why, for the preferences we have been working with, the two demand curves have
the same horizontal intercept.
Since we know the monopolist sets the fixed charge in the two-part tariff equal to the consumer
surplus type 1 would get under only the per-unit price, the shaded blue area is equal to the fixed
charge F . This implies zero consumer surplus for type 1 consumers and consumer surplus equal to
the magenta area for type 2 consumers.
We began our exploration of second degree price discrimination in Section A by proposing that
the firm set a per-unit price at M C = c and then charge the highest possible fixed fees to each
consumer type such that the consumers would in fact choose different bundles. We replicate this in
panel (b) of Graph 23.9 for the demand curves we are working with, taking the liberty of drawing
these in a particular way so as to minimize the number of areas we have to keep track of. Here,
the firm sets its per-unit price at M C and then charges a fee F 1 = (a + b + c) to type 1 (thereby
capturing all of type 1’s consumer surplus) and a fee F 2 = (a + b + c + d) to type 2 consumers.
The expected profit from a consumer of unknown type is then (a + b + c + (1 − γ)d) under this
pricing policy, while it is (a + b + (1 − γ)(c + e)) under the single two-part tariff we calculated in
the previous section.
23B. The Mathematics of Monopoly
883
Graph 23.9: Two-Part Tariff illustrated with Demand Curves
Exercise 23B.18 Why is F 2 = (a + b + c + d) the highest possible fixed fee the firm can charge to type 2
consumers given that it sets per unit prices at M C and charges type 1 F 1 = (a + b + c)?
Exercise 23B.19 Why is the expected profit from the single two-part tariff (a + b + (1 − γ)(c + e))?
It is then easy to see in this example that charging the proposed different fixed fees might in fact
result in more profit for the monopolist. Suppose, for instance, that γ = 0.5. Then the expected
profit from a given consumer of unknown type under different fixed fees and marginal cost pricing
is (a + b + c + 0.5d) while it is (a + b + 0.5(c + e)) under the single two-part tariff with per-unit
price p∗ . Since areas c and e are equal to each other, the profit from the two-part tariff can also
be written as (a + b + c), which is lower than the profit from charging two different fixed fees and
pricing at marginal cost.
Exercise 23B.20 Can you think of alternative scenarios under which the single two-part tariff yields more
profit?
But then we also illustrated in Graph 23.5 that, when allowed to design fixed fee and per unit
pricing packages that differ in both dimensions, the monopolist can do better by raising the per-unit
price on the low demand consumer and thus increasing the fixed fee for the high demand consumer.
Complete freedom in designing pricing when faced with different consumer types then results in
high demand consumers purchasing the socially optimal quantity but paying a higher fixed fee, and
the lower demand consumers purchasing sub-optimal quantities and paying a lower fixed fee.
A potentially optimal level of second degree price discrimination (analogous to what we derived
in Section A) is pictured once again in Graph 23.10. It can be viewed as consisting of two separate
two-part tariffs, with consumers free to choose which one to select. The two-part tariff targeted
at low-demand consumers consists of a per unit price p accompanied by a fixed fee equal to that
884
Chapter 23. Monopoly
Graph 23.10: Optimal Second Degree Price Discrimination using Two-Part Tariffs
consumer type’s consumer surplus CS 1 (p) under the per unit price p. Under this two-part tariff,
type 1 consumers will choose x1 (p) and pay a total tariff
P 1 = CS 1 (p) + px1 (p).
(23.37)
which is equal to the shaded blue area in the graph plus the rectangle cx1 (p) underneath the
shaded blue area. The tariff aimed at high demand consumers, on the other hand, consists of a
per-unit price c equal to marginal cost and the highest possible fixed fee that will keep type 2
consumers from taking the two-part tariff aimed at type 1 consumers. This will result in type 2
consumers purchasing the quantity x2 (c), leaving them with consumer surplus equal to the shaded
blue, green and magenta areas in the absence of a fixed fee. Since type 2 consumers can obtain
consumer surplus equal to the shaded green area by accepting the two-part tariff aimed at low
demand consumers, the most that the firm can then charge in a fixed fee is equal to the shaded
blue plus the shaded magenta areas. The resulting two-part tariff P 2 aimed at type 2 consumers is
then given by
2
P =
(
1
CS (p) + (p − c)x1 (p) +
"
p2 (x1 (p)) − c
2
#)
x2 (c) − x1 (p)
+ cx2 (c),
(23.38)
where the first bracketed term represents the shaded blue area and the second bracketed term
represents the shaded magenta area – which together compose the fixed fee charged to type 2
consumers.
This implies that the firm can expect profit of
π 1 (p) = CS 1 (p) + (p − c)x1 (p)
from type 1 consumers and
(23.39)
23B. The Mathematics of Monopoly
885
π 2 (p) = CS 1 (p) + (p − c)x1 (p) +
"
p2 (x1 (p) − c
2
#
x2 (c) − x1 (p)
.
(23.40)
from type 2 consumers. The expected profit from encountering a consumer of unknown type is
then E(π) = γπ 1 (p) + (1 − γ)π 2 (p) or
1
1
E(π) = CS (p) + (p − c)x (p) + (1 − γ)
"
p2 (x1 (p) − c
2
#
x2 (c) − x1 (p)
(23.41)
The only variable in the expression for E(π) that is under the control of the monopolist is the
price p – because the setting of p determines the fixed charges that can be levied on the two types
of consumers and we already know that the per-unit price for type 2 consumers is c. Thus, the
monopolist’s problem is to choose p to maximize E(π) and then to define the two-part tariffs for
the two consumer types accordingly.
For the preferences we have used in this section, we can substitute in for the various functions
in E(π) and write the firm’s problem as
(θ2 − c) (θ1 − p)
(θ1 − p)2
(θ1 − p) (1 − γ) θ2 p
+ (p − c)
+
−c
−
max E(π) =
p
2θ1
θ1
2
θ1
θ2
θ1
(23.42)
With a bit of careful math, the first order condition for this maximization problem can then be
solved for p to yield
p=
θ1 γ
θ1 − (1 − γ)θ2
c
(23.43)
from which the two-part tariffs P 1 and P 2 can be derived.
We have then derived full second-degree price discrimination in the form of two separate twopart tariffs, with different fixed fees and different per-unit prices targeted at the two consumer
types in such a way as to get each consumer type to utilize the two-part tariff intended for her
while maximizing the monopolist’s profit (conditional on the monopolist not being able to a priori
identify the consumer types).
There is one final caveat for the monopolist who is contemplating this pricing policy: If there are
sufficiently many high demand consumers (i.e. if γ is sufficiently low) or if the high demanders have
sufficiently greater demand than low demanders (i.e. θ2 is sufficiently above θ1 ), it may be better
for the monopolist to write off the type 1 market and simply set a single two-part tariff intended to
extract the most possible surplus from type 2 consumers. You can see this clearly in Graph 23.10.
Suppose, for instance, that γ = 0.5 – implying an equal number of type 1 and type 2 consumers.
By choosing second degree price discrimination, the monopolist chooses to forego capturing the
shaded green area in type 2’s consumer surplus in exchange for instead getting the shaded blue
area of type 1’s consumer surplus. The alternative is for the firm to capture the green area of type
2’s surplus and not offer anything that type 1 consumers would choose – thus foregoing the shaded
blue area. Note that, in our graph, the green area is larger than the blue area. Thus, with γ = 0.5,
the monopolist is better off engaging in first degree price discrimination with respect to type 2
consumers (and not sell to type 1 consumers) than to engage in second degree price discrimination.
886
Chapter 23. Monopoly
F1
F2
p1
p2
x1
x2
CS 1
CS 2
E(π)
TS
Different Forms of Monopoly Price Discrimination
None 1st Degree 3rd Degree 2-part tariff 2nd Degree
$0
$28.13
$0
$23.63
$12.50
$0
$52.08
$0
$23.63
$33.33
$72.50
$25.00
$62.50
$31.25
$50.00
$72.50
$25.00
$87.50
$31.25
$25.00
0.2750
0.7500
0.3750
0.6875
0.5000
0.5167
0.8333
0.4167
0.7917
0.8333
3.7813
0
7.0313
0
0
20.0208
0
13.0208
23.3724
18.7500
18.8021
40.1042
20.0521
28.2552
29.1667
30.7031
40.1042
30.0781
39.9414
38.5417
Table 23.2: θ1 = 100, θ2 = 150, γ = 0.5, c = 25
Exercise 23B.21 If the monopolist is restricted to offering a single two-part tariff (rather than two separate tariffs intended for the two consumer types), is she more or less likely to forego second degree price
discrimination in favor of first degree price discrimination with respect to the high demand type?
23B.3.3
Comparing Different Monopoly Pricing: An Example
We noted at the beginning of our discussion of second-degree price discrimination that we can think
of each of the pricing strategies we have covered as different personalized two-part tariffs of the form
P n (x) = F n + pn x. Under some strategies we assume fixed charges F n to be zero; under others
we require them to be equal for the different consumer types (as summarized in Table 23.1). This
then gives us a convenient way of comparing the different forms of price discrimination.
Table 23.2 undertakes this comparison for a particular example in which θ1 = 100, θ2 = 150,
γ = 0.5 and the marginal cost c = 25. The first column begins by presenting the outcome of
monopoly behavior when no price discrimination takes place, with the next two columns presenting
the outcome for first and third degree price discrimination where the firm knows each consumer’s
type and the final two columns presenting the outcome when the firm does not know each consumer’s
type and is at first restricted to using a single two-part tariff and then permitted to employ separate
two-part tariffs aimed at the two consumer types. In each case, we begin with the fixed fees and
the per-unit prices charged to the two consumer types and then report the consumption levels,
consumer surpluses and the firm’s expected profit per consumer. The final row of the table then
sums the consumer surpluses and the firm’s profit to arrive at the total surplus.
We know from our work above that first degree price discrimination results in full efficiency,
with the entire surplus accruing to the firm. It is therefore not surprising to see that the firm’s
profit and the total surplus are the largest under first degree price discrimination, nor is it surprising
that this is the least preferred outcome for consumers whose entire surplus is taken in fixed fees by
the monopolist. It should also not be surprising that the firm’s profit is the lowest when it is not
permitted to engage in any price discrimination. After all, we can see from Table 23.1 that the firm
is most restricted in its pricing policy in that case, with no possibility of charging a fixed fee and
no possibility of differentiating between the consumer types in terms of the per-unit price charged.
These restrictions are lifted partially under third degree price discrimination, resulting in higher
23B. The Mathematics of Monopoly
887
firm profit, and fully lifted under first degree price discrimination. It is therefore natural to expect
the firm’s profit from third degree price discrimination to fall in between the no-discrimination and
full (first degree) discrimination scenarios.
In the case where firm’s can discriminate but do not know the consumer types (represented
in the last two columns), it is again not surprising that the firm makes more profit than it dues
in the no-discrimination case, nor should it be surprising that firm profit is higher when the firm
can charge two separate two-part tariffs (in the last column) than when it is restricted to a single
two-part tariff (in the second-to-last column). The only case that is theoretically ambiguous with
respect to firm profit is the comparison between third degree price discrimination and the two forms
of second degree price discrimination in the last two columns. For our particular example, it turns
out that both forms of second degree price discrimination result in greater profit than third degree
price discrimination, but for other examples the reverse could be true.
Exercise 23B.22 From looking at Table 23.1, it seems that the firm is unambiguously less restricted in
its pricing under second degree price discrimination than under third degree price discrimination. So how
could it theoretically be the case that profit is higher under third degree price discrimination?
We can summarize these implications in two sets of equations, with
π(None) ≤ π(2-part tariff) ≤ π(2nd Degree) ≤ π(1st degree)
(23.44)
comparing profit under the second degree price discrimination scenarios to the extremes of no
discrimination and perfect discrimination, and with
π(None) ≤ π(3rd Degree) ≤ π(1st degree)
(23.45)
comparing third degree price discrimination to these same extremes.
Exercise 23B.23 * Can you think of a scenario under which all the inequalities turn to equalities in the
two equations above? (Hint: Think of goods for which consumers demand only 1 unit.)
Turning from profit to consumer surplus, we can derive the following implications for the low
demand consumers:
0 = CS 1 (1st Degree) = CS 1 (2-part tariff) = CS 1 (2nd Degree)
≤ CS 1 (None) ≤ CS 1 (3rd Degree).
(23.46)
Exercise 23B.24 Can you give an intuitive explanation for why this has to hold?
For the high demand consumers, however, the implications for consumer surplus are not nearly
as unambiguous. We can definitively conclude that
0 = CS 2 (1st Degree) ≤ CS 2 (3rd Degree) ≤ CS 2 (None)
(23.47)
CS 2 (1st Degree) ≤ CS 2 (2nd Degree) ≤ CS 2 (2-part tariff),
(23.48)
and
but we again cannot be certain about how consumer surplus for the high demand type under no
and third degree price discrimination compares to consumer surplus under the two forms of second
888
Chapter 23. Monopoly
degree price discrimination. In our example, third degree price discrimination happens to be worse
for high demand consumers than either of the forms of second degree price discrimination but no
discrimination is better than second degree price discrimination.
The theoretical ambiguities with respect to profit and consumer surplus of high demand consumers then create theoretical uncertainty about the overall efficiency (or total surplus) under
different monopoly behavior. The only conclusions that hold regardless of the types of demand
are that total surplus is largest under first degree price discrimination. For instance, by simply
changing γ in our example from 0.5 to 0.4, the ranking of total surplus changes from one in which
second degree price discrimination is more efficient than no discrimination which is more efficient
than third degree price discrimination (as illustrated in Table 23.2) to one where no discrimination
is more efficient than third degree price discrimination which is more efficient than second degree
price discrimination. It is therefore important from an efficiency-focused policy perspective to know
as much as possible about underlying demands before intervening in monopoly pricing behavior.
Furthermore, it may be that policy makers are less concerned about monopoly profit and more
concerned about consumer welfare, in which case overall surplus is not the relevant outcome to
consider.
Exercise 23B.25 Can you think of any definitive policy implications if the goal of policy is to maximize
consumer welfare (with no regard to firm profit)?
Exercise 23B.26 Explain all the zeros in Table 23.2.
Exercise 23B.27 In Table 23.1 we note that there are no restrictions on per-unit prices for the two consumer types under either first or second degree price discrimination, with firms being able to tell consumer
types apart in the former case but not the latter. Yet in Table 23.2, the firm appears to be charging exactly
the same per unit prices to the two consumers under first degree price discrimination when it can tell the
consumers apart and different per-unit prices under second-degree price discrimination when the firm cannot
tell the consumer types apart. Explain this intuitively.
23B.4
Barriers to Entry and Natural Monopoly
In Section A, we concluded with a discussion of barriers to entry that create monopolies and
particularly focused on the case of natural monopolies that are characterized by downward sloping
average cost curves. The mathematical treatment of such monopolies is relatively straightforward,
and we therefore leave its development to end-of-chapter exercise 23.8. We will also return to the
role of barriers to entry in creating market power in Chapters 25 and 26.
Conclusion
In this chapter, we have begun exploring market power by focusing on the extreme case in which a
single firm controls the entire market for a particular good and thus faces the market demand curve
rather than the perfectly elastic demand curve that arises for a firm’s output under perfect competition. We noted at the beginning that “market power” is a relative concept that is closely linked
to the price elasticity of demand that the firm is facing, with infinite price elasticity representing
the extreme case of no market power. We then illustrated how monopolies can take advantage of
market power to increase profit, whether it is by charging a single per-unit price to all consumers or
by price discriminating in various ways that depend on which pricing strategies are available to the
23B. The Mathematics of Monopoly
889
firm, whether it is possible to prevent re-sale and how much information regarding consumer types
the firm has. Unless a firm is able to perfectly price discriminate, we concluded that monopoly
behavior results in deadweight loss because monopolies will strategically restrict output in order
to raise price. This deadweight loss might be even higher in cases where firms engage in socially
wasteful activities in order to attain or maintain monopoly power. At the same time, we noted that
our models probably over-predict the size of deadweight losses in many circumstances in which a
single firm might in fact control the market for a particular good but in which its monopoly power
is disciplined by fear of the possible entry of future competitors. In the case of government-induced
monopolies, however, our models may under-estimate the deadweight loss if monopolists expend
resources to lobby for government protection.
The emergence of deadweight loss from the existence of market power raises the possibility
that government intervention in markets characterized by market power might result in efficiency
enhancements. But whether such intervention is possible and will in fact lead to increased efficiency
depends on the precise nature of the monopoly and the information available to policy makers. In
some cases, monopolies might exist for good reasons – such as in the case of natural monopolies
that have cost curves which make the presence of multiple firms in the market inherently inefficient.
Government intervention in such cases might require information about cost curves that is not
readily available to regulators, with the added problem that firms have an explicit incentive to
misrepresent their true costs and a possible incentive to not innovate if regulation simply guarantees
a “fair market return.” At the same time, we discussed market-based interventions, such as the
public provision of the fixed cost infrastructure that might open up the possibility of multi-firm
competition along the infrastructure that would otherwise result in a natural monopoly.
Often, monopolies exist because governments create market power. Governments might, as
we will see more clearly in Chapter 26, offer market power in the form of copyrights and patents
in order to provide powerful incentives for innovations that might otherwise not occur, and the
surplus from such innovation may well outweigh the deadweight losses from underproduction that
arises due to the granted market power. At the same time, governments might grant market power
as a result of lobbying efforts by firms that seek profit, thus bestowing “concentrated benefits”
on owners of the firm while creating “diffuse costs” that nevertheless exceed the benefits. In
such circumstances, efficiency and consumer welfare would clearly be enhanced by the removal of
such market power. Finally, when faced with a monopoly exercising its market power through
price discrimination, we found that it is not always obvious whether the mere tempering of price
discrimination through government intervention will necessarily increase social welfare. In such
circumstances, much depends on the underlying specifics of the case. As a result, courts that are
asked to adjudicate in anti-trust law suits that challenge monopoly pricing will typically need to
take great care to understand the specifics of the case at hand.
Our focus in this chapter has been exclusively on the ways in which monopolists can use pricing
to exercise market power and generate profit. There are, however, a variety of other ways in
which monopolies might exercise market power. These include differentiating the quality of its
output across different consumer types and strategically bundling different goods so as to extend
monopoly power from one market to another. An entire course can easily be taught on such topics,
and probably is taught in your department under the heading of antitrust economics or industrial
organization??. If this chapter has been interesting to you, you might want to consider taking such
a course in your future studies.
We will proceed in Chapters 25 and 26 by investigating market structures that lie in between
the extremes of perfect competition and monopoly. Before doing so, however, we need to develop
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Chapter 23. Monopoly
some concepts that assist economists in thinking about strategic behavior – concepts that come
under the heading of game theory. It turns out that we have implicitly begun to use some of these
concepts in this chapter as we thought through the strategic choices made by monopolists under
different pricing strategies (as we illustrate in end-of-chapter exercise 24.11 in the next chapter).
We will now formalize these and other concepts – and then return to the topic of market power and
its impact on efficiency in a wider array of settings.
Appendix: Deriving a “Reduced Form” Utility Function from
Separable Preferences
In Section 23B.3 we introduced what we called a “reduced form” utility function representing
preferences for the monopoly good x that took the form
U n = θn u(x) − P (x)
(23.49)
where θn became our preference parameter that distinguished consumer types and P (x) was the
total charge to the consumer for consuming the quantity x of the monopoly good. We indicated
at the time that this way of representing preferences for a single good can be derived from a more
typical utility function over x and a composite good y. We furthermore indicated that one can
assume that consumers in fact have identical underlying preferences and that the parameter θn is
simply a measure of consumer income, with consumer demands therefore differing solely because
of underlying income differences. We will now illustrate this more fully.
Suppose that consumers have underlying preferences that can be represented by the utility
function
U(x, y) = u(x) + v(y).
(23.50)
If spending on the monopoly good x represents a relatively small fraction of the consumer’s
income I, we can approximate this utility function by writing it as
U (x, I) ≈ u(x) + v(I) − P (x)
dv(I)
.
dI
(23.51)
When we then solve the optimization problem maxx U (x, I), the terms v(I) plays no role in the
first order conditions, leaving only the portion (u(x)−P (x)dv(I)/dI) as relevant for the optimization
problem. We can then define θ = 1/(dv(I)/dI) and multiply this relevant portion of the utility
function by θ to get
e θ) = θu(x) − P (x) with θ =
U(x,
1
.
dv(I)/dI
(23.52)
The term θ is then simply the inverse of the “marginal utility of income”. It is common to
assume that marginal utility of income declines in income – i.e. dv(I)dI < 0. Since θ is the inverse
of marginal utility of income, this implies that θ is increasing in incomce – i.e. dθ/dI > 0.
Suppose, then, that we have two consumers with identical preferences that can be represented
by the separable utility function in equation (23.50) but their incomes are I1 < I2 . Then we can
represent their preferences for purposes of determining demand for the monopoly good x by the
equation
23B. The Mathematics of Monopoly
U (x) = θn u(x) − P (x) with θ1 < θ2 .
891
(23.53)
Thus, low demand consumers will be those with less income than high demand consumers.
This then implies that, for instance, under full second degree price discrimination, lower income
consumers purchase the monopoly good at a higher per-unit price but are charged a lower fixed fee
than high demand consumers.
End of Chapter Exercises
23.1 Suppose that the demand curve for a product x provided by a monopolist is given by p = 90 − x and suppose
further that the monopolist’s marginal cost curve is given by M C = x.
A: In this part, we will focus on a graphical analysis – which we ask you to revisit with some simple math in
part B. (It is not essential that you have done Section B of the chapter in order to do (a) through (d) of part
B of this question.)
(a) Draw a graph with the demand and marginal cost curves.
(b) Assuming that the monopolist can only charge a single per-unit price for x, where does the marginal
revenue curve lie in your graph?
(c) Illustrate the monopolist’s profit maximizing “supply point”.
(d) In the absence of any recurring fixed costs, what area in your graph represents the monopolist’s profit.
(There are actually two areas that can be used to represent profit – can you find both?)
(e) Assuming that the demand curve is also the marginal wilingness to pay curve, illustrate consumer surplus
and deadweight loss.
(f) Suppose that the monopolist has recurring fixed costs of an amount that causes her actual profit to be
zero. Where in your graph would the average cost curve lie? In particular, how does this average cost
curve relate to the demand curve?
(g) In a new graph, illustrate again the demand, M R and M C curves. Then illustrate the monopolist’s
average cost curve assuming the recurring fixed costs are half of what they were in part (f).
(h) In your graph, illustrate where profit lies. True or False: Recurring fixed costs only determine whether a
monopolist produces – not how much she produces.
B: Consider again the demand curve and M C curve as specified at the beginning of this exercise.
(a) Derive the equation for the marginal revenue curve.
(b) What is the profit maximizing output level xM ? What is the profit maximizing price pM (assuming that
the monopolist can only charge a single per-unit price to all consumers)?
(c) In the absence of recurring fixed costs, what is the monopolist’s profit?
(d) What is consumer surplus and deadweight loss (assuming that demand is equal to marginal willingness to
pay).
(e) What is the cost function if recurring fixed costs are sufficiently high to cause the monopolist’s profit to
be zero?
(f) Use this cost function to set up the monopolist’s optimization problem and verify your answers to (b).
(g) Does the average cost curve relate to the demand curve as you concluded in part A(f)?
(h) How does the profit maximization problem change if the recurring fixed costs are half of what we assumed
in part (e)? Does the solution to the problem change?
23.2 Everyday and Business Application: Diamonds are a Girl’s Best Friend: Historically, most of the diamond
mines in the world have been controlled by a few companies and governments. Through clever marketing by diamond
producers, many consumers have furthermore become convinced that “diamonds are a girl’s best friend” because
“diamonds are forever.” In fact, the claim is that the only way to show true love is to give a diamond engagement
ring that costs the equivalent of 3 months of salary. (We will refer to this throughout the exercise as “the claim.”)
A: For purposes of this question, assume that diamonds are only used for engagement rings, that there is no
secondary market for engagement rings and that the diamond industry acts as a single monopoly.
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Chapter 23. Monopoly
(a) Let x be the size of diamonds (in karats). Draw a demand curve for x (with the price per karat on
the vertical axis) – and make the shape of this demand curve roughly consistent with the claim at the
beginning of the question.
(b) If this claim is true, what is the price elasticity of demand for diamonds?
(c) What price per karat would be consistent with the diamond monopoly maximizing its revenues (assuming
the claim accurately characterizes demand)?
(d) What price is consistent with profit maximization?
(e) How large would the diamonds in engagement rings be if the marketing campaign to convince us of the
claim at the beginning of the question was fully successful and if the diamond industry really has monopoly
power?
(f) True or False: By observing the actual size of diamonds in engagement rings, we can conclude that either
the market campaign has not yet fully succeeded or the diamond industry is not really a monopoly.
B: Suppose that demand for diamond size is x = (A/p)(1/(1−β)) .
(a) What value must β take in order for the claim to be correct?
(b) How much revenue will the diamond monopoly earn if the claim holds? Does this depend on what price
it sets?
(c) Derive the marginal revenue function (assuming the claim holds). Assuming M C > 0, does M R every
cross M C?
(d) If M C = 0, how large a diamond size per engagement ring is consistent with profit maximization (assuming
the claim holds)?
(e) Suppose the diamond monopoly has recurring fixed costs that are sufficiently high to cause it’s profits to
be zero. If marginal costs were zero, what would be the relationship between the demand curve and the
average cost curve?
(f) Suppose β = 0.5 and M C = x. What is the profit maximizing diamond size now?
(g) What if instead β = −1?
23.3 Business and Policy Application: Monopoly Pricing in Health Insurance Markets: In Chapter 22, we worked
with models in which high and low cost customers compete for insurance. Consider the level x of health insurance
that consumers might choose to buy, with higher levels of x indicating more comprehensive insurance coverage.
A: Suppose that there are relatively unhealthy type 1 consumers and relatively healthy type 2 consumers. The
marginal cost of providing additional insurance coverage is then M C 1 and M C 2 , with M C 1 > M C 2 . Unless
otherwise stated, assume that d1 = d2 – i.e. the individual demand curves for x are the same for the two types.
Also, suppose that the number of type 1 and type 2 consumers is the same, and some portion of each demand
curve lies above M C 1 .
(a) Begin by drawing a graph with the individual demands for the two types, d1 and d2 , as well as the marginal
costs. Indicate the efficient levels of health insurance x1∗ and x2∗ for the two types.
(b) Suppose the monopolist cannot tell consumers apart and can only charge a single price to both types.
What price will it be and what level of insurance will each type purchase?
(c) How does your answer change if the monopolist can first-degree price discriminate?
(d) What if he can third-degree price discriminate?
(e) Suppose you worked for the Justice Department’s anti-trust division and you only cared about efficiency.
Would you prosecute a first-degree price discriminating monopolist in the health insurance market? What
if you cared only about consumer welfare?
(f) In the text we suggested that it is generally not possible without knowing the specifics of a case whether
third degree price discrimination is more or less efficient than no price discrimination by a monopolist.
For the specifics in this case, can you tell whether type 1 consumers are better off without this pricediscrimination? What about consumer type 2?
(g) Would it improve average consumer surplus to prohibit the monopolist from third-degree price discriminating? Would it be more efficient?
B: Suppose next that we normalize the units of health insurance coverage such that the demand function is
xn (p) = (θn − p)/θn for type n. You can interpret x = 0 as no insurance and x = 1 as full insurance. Let
θ1 = 20 and θ2 = 10 for the two types of consumers, and let M C 1 = 8 and M C 2 = 6.
23B. The Mathematics of Monopoly
893
(a) Determine the efficient level of insurance for each consumer type.
(b) If a monopolist cannot tell who is what type and can only charge a single per-unit price for insurance,
what will she do assuming there are γ type 1 consumers and (1 − γ) type 2 consumers, with γ < 0.5?
(Hint: Define the monopolist’s expected profit and maximize it.)
(c) What would the monopoly price be if γ = 0? What if γ = 2/7? What is the highest that γ can be and
still result in type 2 consumers buying insurance?
(d) Suppose that the monopolist first-degree price discriminates. How much insurance will each consumer
type purchase? How much will each type pay for her coverage?
(e) How do your answers to (d) change if the monopolist third-degree price discriminates?
(f) Let the payment that individual n makes to the monopolist be given by P n = F n + pn xn . Express your
answers to (c), (d) and (e) in terms of F 1 , F 2 , p1 and p2 .
(g) Suppose γ = 0.5 – i.e. half of the population is type 1 and half is type 2. Can you rank the three scenarios
in (c), (d) and (e) from most efficient to least efficient?
(h) Can you rank them in terms of their impact on consumer welfare for each type? What about in terms of
population weighted average consumer welfare?
* Business and Policy Application: Second-Degree Price Discrimination in Health Insurance Markets: In exercise 23.3, we analyzed the case of a monopoly health insurance provider. We now extend the analysis to second-degree
price discrimination, with x again denoting the degree of health insurance coverage.
A: Consider the same set-up as in part A of exercise 23.3 and assume there is an equal number of type 1 and
type 2 consumers.
(a) Begin again by drawing a graph with the individual demands for the two types, d1 and d2 , as well as the
marginal costs. Indicate the efficient levels of health insurance x1∗ and x2∗ for the two types.
23.4
(b) Under second degree price discrimination, the monopolist does not know who is what type. What two
packages of insurance level x and price P (that can have a per-unit price plus a fixed charge) will the
monopolist offer? (Hint: You can assume that, if consumers are indifferent between two packages, they
each buy the one intended for them.)
(c) Is the outcome efficient? Are consumers likely to prefer it to other monopoly pricing strategies?
(d) Suppose next that the demand from type 1 consumers is greater than the demand from type 2 consumers,
with d1 intersecting M C 1 to the right of where d2 intersects M C 2 . Would anything fundamental change
for a first-degree or third-degree price discriminating monopolist?
(e) Illustrate how a second-degree price discriminating monopolist would now structure the two health insurance packages to maximize profit. Might relatively healthy individuals no longer be offered health
insurance?
(f) True or False: Under second-degree price discrimination, the most likely to not buy any health insurance
are the relatively healthy and the relatively young.
B: ** Consider again the set-up in part B of exercise 23.3. Suppose that a fraction γ of the population is of type
1, with the remainder (1 − γ) of type 2. In analyzing second degree price discrimination, let the total payment
P n made by type n be in the form of a two-part tariff pn = F n + pn xn .
(a) Begin by assuming that the monopolist will set p2 = p and p1 = M C 1 = 8. Express the level of insurance
x2 for type 2 consumers as a function of p. Then express consumer surplus for type 2 consumers as a
function of p and denote it CS 2 (p).
(b) Why would a second-degree price discriminating monopolist set F 2 equal to CS 2 (p) once she has figured
out what p should be? What would the payment P 2 (p) made by type 2 consumers to the monopolist be
under p and F 2 (p)?
(c) Suppose M C 2 < p < M C 1 . For p in that range, what is the largest possible F 1 that the monopolist
can charge to type 1 consumers if she sets p1 = M C 1 = 8. (Hint: Draw the graph with the two demand
curves – and then ask how much consumer surplus type 1 consumers could get by simply pretending to
be type 2 consumers and accepting the package designed for type 2 consumers.)
(d) Suppose instead that M C 1 < p < 10. What would now be the largest possible F 1 that is consistent
with type 1 consumers not buying the type 2 insurance (assuming still that p1 = M C 1 = 8? (Hint: Use
another graph as you did in the previous part to determine the answer.)
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Chapter 23. Monopoly
(e) Given that the fraction of type 1 consumers is γ (and the fraction of type 2 consumers is (1 − γ)), what
is the expected profit E(π(p)) per customer from setting p2 = p when M C 2 < p < M C 1 ? What if
M C 1 < p < 10?
(f) For both cases – i.e. for M C 2 < p < M C 1 and when M C 1 < p < 10 – set up the optimization problem
the second degree price discriminating monopolist solves to determine p. Then solve for p in terms of γ.
(Hint: You should get the same answer for both cases.)
(g) Determine the value for p when γ = 0. Does your answer make intuitive sense? What about when
γ = 0.1, when γ = 0.2, when γ = 0.25? True or False: As the fraction of type 1 consumers increases,
health insurance coverage for type 2 consumers falls.
(h) At what value for γ will type 2 consumers no longer buy insurance? If we interpret the difference in
types as a difference in incomes (as outlined in the appendix), can you determine which form of price
discrimination is best for low income consumers?
23.5 Business and Policy Application: Labor Unions Exercising Market Power: Federal anti-trust laws prohibit
many forms of collusion in price setting between firms. Labor unions, however, are exempt from anti-trust laws and
are allowed to use market power to raise wages for their members.
A: Consider a competitive industry in which workers have organized into a union that is now renegotiating the
wages of its members with all the firms in the industry.
(a) To keep the exercise reasonably simple, suppose that each firm produces output by relying solely on labor
input. How does each firm’s labor demand curve emerge from its desire to maximize profit? Illustrate a
single firm’s labor demand curve (with the number of workers on the horizontal axis). (Note: Since these
are competitive firms, this part has nothing to do with market power.)
(b) On a graph next to the one you just drew, illustrate the labor demand and supply curves for the industry
as a whole prior to unionization.
(c) Label the competitive wage w ∗ and use it to indicate in your first graph how many workers an individual
firm hired before unionization.
(d)
* Suppose that the union that is negotiating with the firm in your graph is exercising its market power with
an aim toward maximizing the overall gain for its members. Suppose further that the union is sufficiently
strong to be able to dictate an outcome. Explain how the union would go about choosing the wage in this
firm and the size of its membership that will be employed by this firm. (Hint: The union here is assumed
to have monopoly power – and the marginal cost of a member is that member’s competitive wage w ∗ .)
(e) If all firms in the industry are becoming unionized, what impact will this have on employment in this
industry? Illustrate this in your market graph.
(f) Suppose that those workers not chosen to be part of the union migrate to a non-unionized industry. What
will be the impact on wages in the non-unionized sector?
B: * Suppose that each firm in the industry has the same technology described by the production function
f (ℓ) = Aℓα with α < 1, and suppose that there is some fixed cost to operating in this industry.
(a) Derive the labor demand curve for each firm.
(b) Suppose that the competitive wage for workers of the skill level in this industry is w ∗ . Define the optimization problem that the labor union must solve if it wants to arrive at its optimal membership size and
the optimal wage according to the objective defined in A(d). (It may be more straightforward to set this
up as a maximization problem with w rather than ℓ as the choice variable.)
(c) Solve for the union wage w U that emerges if the union is able to use its market power to dictate the wage.
What happens to employment in the firm?
(d) Can you verify your answer by instead finding M R and M C from the perspective of the union – and then
setting these equal to one another?
(e) Given the fixed cost to operating in the industry, would you expect the number of firms in the industry
to go up or down?
23.6
*
Business and Policy Application: Monopsony: A Single Buyer in the Labor Market: The text treated extensively the case where market power is concentrated on the supply side – but it could equally well be concentrated
on the demand side. When a buyer has such market power, he is called a monopsonist. Suppose, for instance, the
labor market in a modest-sized town is dominated by a single employer (like a large factory or a major university).
In such a setting, the dominant employer has the power to influence the wage just like a typical monopolist has the
power to influence output prices.
23B. The Mathematics of Monopoly
895
A: Suppose that there is a single employer for some type of labor, and to simplify the analysis, suppose that
the employer only uses labor in production. Assume throughout that the firm has to pay the same wage to all
workers.
(a) Begin by drawing linear labor demand and supply curves (assuming upward sloping labor supply). Indicate
the wage w ∗ that would be set if this were a competitive market and the efficient amount of labor ℓ∗ that
would be employed.
(b) Explain how we can interpret the labor demand curve as a marginal revenue curve for the firm. (Hint:
Remember that the labor demand curve is the marginal revenue product curve.)
(c) How much does the first unit of labor cost? Where would you find the cost of hiring a second unit of labor
if the firm could pay the second unit of labor more than the first?
(d) We are assuming that the firm has to pay all its workers the same wage – i.e. it cannot wage discriminate.
Does that imply that the marginal cost of hiring the second unit of labor is greater or less than it was in
part (c)?
(e) How does the monopsony power of this firm in the labor market create a divergence between labor supply
and the firm’s marginal cost of labor – just as the monopoly power of a firm causes a divergence between
the output demand curve and the firm’s marginal revenue curve?
(f) Profit is maximized where M R = M C. Illustrate in your graph where marginal revenue crosses marginal
cost. Will the firm hire more or fewer workers than a competitive market would (if it had the same demand
for labor as the monopsonist here)?
(g) After a monopolist decides how much to produce, he prices the output at the highest possible level at
which all the product can be sold. Similarly, after a monopsonist decides how much to buy, he will pay
the lowest possible price that will permit him to buy this quantity. Can you illustrate in your graph the
wage w M that our dominant firm will pay workers?
(h) Suppose the government sets a minimum wage of w ∗ (as defined in (a)). Will this be efficiency enhancing?
(i) We gave the example of a modest-sized town with a dominant employer as a motivation for thinking about
monoposonist firms in the labor market. As it becomes easier to move across cities, do you think it is
more or less likely that the monopsony behavior we have identified is of significance in the real world?
(j) Labor unions allow workers to create market power on the supply side of the labor market. Is there a
potential efficiency case for the existence of labor unions in the presence of monopsony power by firms in
the labor market? Would increased mobility of workers across cities strengthen or weaken this efficiency
argument?
B: Suppose that the firm’s production function is given by f (ℓ) = Aℓα (with α < 1) and the labor supply curve
is given by ws (ℓ) = βℓ.
(a) What is the efficient labor employment level ℓ∗ ? (Hint: You should first calculate the marginal revenue
product curve.)
(b) At what wage w ∗ would this efficient labor supply occur?
(c) Define the firm’s profit maximization problem – keeping in mind that the wage the firm must pay depends
on ℓ.
(d) Take the first order condition of the profit maximization problem. Can you interpret this in terms of
marginal revenue and marginal cost?
(e) How much labor ℓM does the monopsonist firm hire – and how does it compare to ℓ∗ ?
(f) What wage w M does the firm pay – and how does it compare to w ∗ ?
(g) Consider the more general case of a monopsonist firm with production function f (ℓ) facing a labor supply
curve of w(ℓ). Derive the M R = M C condition (which is the same as the condition that the marginal
revenue product equals M C) from the profit maximization problem.
(h) Can you write the M C side of the equation in terms of the wage elasticity of labor supply?
(i) True or False: As the wage elasticity of labor supply increases, the monopsonist’s decision approaches
what we would expect under perfect competition.
23.7 Business and Policy Application: Taxing Monopoly Output: Under perfect competition, we found that the
economic incidence of a tax – i.e. who ends up paying a tax – had nothing to do with statutory incidence – i.e. who
the law said should pay the tax.
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Chapter 23. Monopoly
A: Suppose the government wants to tax the good x which is exclusively produced by a monopoly with upward
sloping marginal cost.
(a) Begin by drawing the demand, marginal revenue and marginal cost curves. On your graph, indicate the
profit maximizing supply point (xM , pM ) chosen by the monopolist in the absence of any taxes.
(b) Suppose the government imposes a per-unit tax of t on the production of x – thus raising the marginal
cost by t. Illustrate how this changes the profit maximizing supply point for the monopolist.
(c) What happens to the price paid by consumers? What happens to the price that monopolists get to keep
(given that they have to pay the tax)?
(d) Draw a new graph as in (a). Now suppose that the government instead imposes a per-unit tax t on
consumption. Which curves in your graph are affected by this?
(e) In your graph, illustrate the new marginal revenue curve – and the impact of the consumption tax for the
monopolist’s profit maximizing output level.
(f) What happens to the price paid by consumers (including the tax)? What happens to the price received
by monopolists?
(g) In terms of who pays the tax, does it matter which way the government imposes the per-unit tax on x?
(h) By how much does deadweight loss increase as a result of the tax? (Assume that demand is equal to
marginal willingness to pay.)
(i) Why can’t monopolists just use their market power to pass the entire tax onto the consumers?
B: Suppose the monopoly has marginal costs M C = x and faces the demand curve p = 90 − x as in exercise
23.1.
(a) If you have not already done so, calculate the profit maximizing supply point (xM , pM ) in the absence of
a tax.
(b) Suppose the government introduces the tax described in A(b). What is the new profit maximizing output
level? How much will monopolists charge?
(c) Suppose the government instead imposed the tax described in A(d). Set up the monopolist’s profit
maximization problem and solve it.
(d) Compare your answers to (b) and (c). Is the economic incidence of the tax affected by the statutory
incidence?
(e) What fraction of the tax do monopolists pass onto consumers when monopolists are statutorily taxed?
What fraction of the tax do consumers pass onto monopolists when consumers are statutorily taxed?
23.8 Business and Policy Application: Two Natural Monopolies: Microsoft versus Utility Companies: We suggested
in the text that there may be technological reasons for the barriers to entry required for the existence of a monopoly.
In this exercise, we consider two examples.
A: Microsoft and your local utilities company have one thing in common: They both have high fixed costs and
low variable costs. In the case of Microsoft, the fixed cost involves producing software which, once produced,
can be reproduced cheaply. In the case of your local utility company, the fixed cost involves maintaining the
infrastructure that distributes electricity to homes, with the actual delivery of that electricity costing relatively
little if the infrastructure is in good shape.
(a) Let’s begin with Microsoft. Draw a graph with low constant marginal costs and a downward sloping
demand curve. Add Microsoft’s marginal revenue curve and indicate which point on the demand curve
Microsoft will choose (assuming, until later chapters, that it is not worried about potential competitors).
Then draw a second and similar graph for your local utilities company.
(b) There is one stark difference between Microsoft and your local utilities company: Microsoft has not asked
the government for help to allow it to operate but has instead been under strict scrutiny by governments
around the world for potential abuse of its market power. Utility companies, on the other hand, have
often asked for government aid in regulating prices in such a way that the companies can earn a reasonable
profit. What is missing from your two graphs that can explain this difference?
(c) Put into words the “problem” in the two cases from a government’s perspective (assuming the government
cares about efficiency)?
(d) In the case of Microsoft, how can the granting of a copyright on the software explain the existence of “the
problem”? How much is Microsoft willing to pay for this copyright?
23B. The Mathematics of Monopoly
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(e) Now consider the “problem” in the utilities industry. How would setting a two-part tariff allow the utilities
company to produce at zero profit? If properly structured, might its output level be efficient?
(f) Explain how the alternative of having the government lay and maintain the infrastructure on which
electricity is delivered could address the same “problem”.
(g) What would be the analogous government intervention in the software industry – and why might you
think that this was not a very good idea there? (Hint: Think about innovation.) Could you think of a
way to offer a similar criticism regarding the proposal of having the government provide the infrastructure
for electricity delivery?
B: We did not develop the basic mathematics of natural monopolies in the text and therefore use the remainder
of this exercise to do so. Suppose demand for x is characterized by the demand curve p(x) = A − αx. Suppose
further that x is produced by a monopolist whose cost function is c(x) = B + βx.
(a) Derive the monopolist’s profit maximizing supply point – i.e. the price and quantity (pM , xM ) under the
implicit assumption of no price discrimination.
(b) At the output level xM , what is the average cost paid by the monopolist?
(c) How high can fixed costs be and still permit the monopolist to make non-negative profit by choosing the
supply point you calculated in (a)?
(d) How much is Microsoft willing to pay its lawyers to get copyright protection?
(e) Suppose both Microsoft and your local utility company share the same demand function. They also share
the same cost function except for the fixed cost B. Given our description of the “problem” faced by
Microsoft versus your utility company, whose B is higher?
(f) Suppose B for the utility company is such that it cannot make a profit by behaving as you derived in (a)
and suppose there are N households. Suggest a two-part tariff that will allow the utility company to earn
a zero profit while getting it to produce the efficient amount of electricity.
(g) Suppose the government were to build and maintain the infrastructure needed to deliver electricity to
people’s homes. It furthermore allows any electricity firm to use the infrastructure for a fee δ (per unit
of electricity that is shipped). Can the electricity industry be competitive in this case? What has to be
true about the fee for using the infrastructure in order for this industry to produce the efficient level of
electricity?
23.9 Policy Application: Some Possible “Remedies” to the Monopoly Problem: At least when our focus is on
efficiency, the core problem with monopolies emanates from the monopolist’s strategic under-production of output
– not from the fact that monopolists make profits. But policy prescriptions to deal with monopolies are often based
on the presumption that the problem is that monopolies make excessive profits.
A: Suppose the monopoly has marginal costs M C = x and faces the demand curve p = 90 − x as in exercise
23.1. Unless otherwise stated, assume there are no recurring fixed costs. In each of the policy proposals below,
indicate the impact the policy would have on consumer welfare and deadweight loss.
(a) The government imposes a 50% tax on all economic profits.
(b) The government imposes a per-unit tax t on x. (In problem 23.7, you should have concluded that it does
not matter whether the tax is levied on production or consumption.)
(c) The government sets a price ceiling equal to the intersection of M C and demand. (Hint: How does this
change the marginal revenue curve?)
(d) The government subsidizes production of the monopoly good by s per unit.
(e) The government allows firms to engage in first-degree price discrimination.
(f) Which of the analyses above might change if the firm also has recurring fixed costs.
(g) True or False: In the presence of distortions from market power, price distorting policies can be efficient.
B: Suppose demand and marginal costs are as specified in part A. Unless otherwise stated, assume no recurring
fixed costs.
(a) Determine the monopolist’s optimal supply point (assuming no price discrimination). Does it change
when the government imposes a 50% tax on economic profits?
(b) Suppose the government imposes a $6 per unit tax on the production of x. Solve for the new profit
maximizing supply point.
(c) Is there a price ceiling at which the monopolist will produce the efficient output level?
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Chapter 23. Monopoly
(d) For what range of recurring fixed costs would the monopolist produce prior to the introduction of the
policies in (a), (b) and (c) but not after their introduction?
(e) What is the profit maximizing output level if the monopolist can perfectly price discriminate?
(f) How high a per-unit subsidy would the government have to introduce in order for the monopolist to
produce the efficient output level?
(g) For what range of recurring fixed costs does the monopolist not produce in the absence of a subsidy from
part (f) but produces in the presence of the subsidy? If recurring fixed costs are in this range, will the
monopolist produce the efficient quantity under the subsidy?
23.10 Policy Application: Pollution and Monopolies: In Chapter 21, we discussed the externality from pollutionproducing industries within a competitive market.
A: Suppose now that the polluting firm is a monopolist.
(a) Begin by illustrating a linear (downward sloping) demand curve and an upward sloping M C curve for the
monopolist. Indicate the efficient level of production in the absence of any externalities.
(b) Draw the marginal revenue curve and illustrate the monopolist’s profit maximizing “supply point”.
(c) Suppose that the monopolist pollutes in the process of producing, with the social marginal cost curve
SM C therefore lying above the monopolist’s marginal cost curve. Does this change anything in terms of
the monopolist’s profit maximizing decision?
(d) Illustrate a SM C curve with sufficient pollution costs such that the monopoly’s output choice becomes
efficient.
(e) True or False: In the presence of negative production externalities, the per-unit tax that would cause the
monopolist to behave efficiently might be positive or negative (i.e. it might take the form of a tax or a
subsidy).
(f) Suppose that the production externality were positive instead of negative. True or False: In this case,
the monopolist’s output level will be inefficiently low.
B: Suppose a monopolist faces the cost function c(x) = βx2 , but production of each unit of x causes pollution
damage B.
(a) What is the marginal cost function for the monopolist? What is the social marginal cost function?
(b) Suppose demand curve is equal to p(x) = A − αx. Determine the monopolist’s output level xM (assuming
no price discrimination).
(c) What is the monopoly price?
(d) For what level of B is the monopolist’s output choice efficient?
23.11 Policy Application: Regulating Market Power in the Commons: In exercises 21.9 and 21.10, we investigated
the case of many firms emitting pollution into a lake. We assumed the only impact of this pollution was to raise the
marginal costs for all firms that produce on the lake.
A: Revisit part A(g) of exercise 21.10.
(a) How does a merging of all firms around the lake (into one single firm) solve the externality problem
regardless of how large the pollution externality is?
(b) Suppose you are an anti-trust regulator who cares about efficiency. You are asked to review the proposal
that all the firms around this lake merge into a single firm. What would you decide if you found that,
despite being the only firm that produces output x on this lake, there are still plenty of other producers
of x such that the output market remains competitive.
(c) Suppose instead that, by merging all the firms on the lake, the newly emerged firm will have obtained a
monopoly in the output market for x. How would you now think about whether this merger is a good
idea?
(d) How would your answers to (c) and (d) change if the externality emitted by firms on the lake lowered
rather than raised everyone’s marginal costs?
B: Suppose, as in exercise 21.9 and 21.10, that each of the many firms around the lake has a cost function
c(x) = βx2 + δX where x is the firm’s output level and X is the total output by all firms around the lake.
(a) In exercise 21.10B(a), we discussed how a social planner’s cost function for each firm would differ from
that of each individual firm. Review this logic. How does this apply when all the firms merge into a single
company that owns all the production facilities around the lake?
(b) Will the single company make decisions different from that of the social planner in exercise 21.10? What
does your answer depend on?