Transcript 21_Monopoly

Chapter 21
Monopoly
1. Auctions and Monopoly
2. Prices and Quantities
3. Segmenting the Market
1. Auctions and Monopoly
We begin this chapter by putting auctions in a more
general context to highlight the similarities and
differences between auctions and monopolies. In this
spirit we investigate the sale of multiple units by
auction, to see when the selling mechanism affects
the outcome, and how.
Within the context of a multiple unit auction we derive
our first result in finance, the efficient markets
hypothesis, that in its simplest form, states prices of
stocks follow a random walk.
Are auctions just like monopolies?
Monopoly is defined by the phrase “single
seller”, but that would seem to characterize
an auctioneer too.
Is there a difference, or can we apply
everything we know about a monopolist to
an auctioneer, and vice versa?
We now begin to make the transition
between auctions and markets by noting the
similarities and differences.
Two main differences between
most auction and monopoly models
The two main differences distinguishing models of
monopoly from a auction models are related to the
quantity of the good sold:
1. Monopolists typically sell multiple units, but most
auction models analyze the sale of a single unit. In
practice, though, auctioneers often sell multiple
units of the same item.
2. Monopolists choose the quantity to supply, but most
models of auctions focus on the sale of a fixed
number of units. But in reality the use of reservation
prices in auctions endogenously determines the
number the units sold.
Other differences between
most auction and monopoly models
1. Monopolists price discriminate through market
segmentation, but auction rules do not make the winner’s
payment depend on his type. However holding auctions
with multiple rounds (for example restricting entry to
qualified bidders in certain auctions) segments the
market and thus enables price discrimination.
2. A firm with a monopoly in two or more markets can
sometimes increase its value by bundling goods together
rather than selling each one individually. While auction
models do not typically explore these effects, auctioneers
also bundle goods together into lots to be sold as
indivisible units.
An agenda for the first portion of
our work on monopoly
We will focus on two issues:
1. How does a multiunit auction differ from a single
unit auction?
2. What can we learn about market behavior from
multiunit auctions?
Auctioning multiple units
to single unit demanders
Suppose there are exactly Q identical units of a good
up for auction, all of which must be sold.
As before we shall suppose there are N bidders or
potential demanders of the product and that N > Q.
Also following previous notation, denote their
valuations by v1 through vN.
We begin by considering situations where each buyer
wishes to purchase at most one unit of the good.
Decisions for the seller to make
in multiunit auctions
The seller must decide whether to sell the
objects separately in multiple auctions or
jointly in a single auction.
The seller must choose among different
auction formats.
Open auctions for selling identical
units
Descending Dutch auction:
Suppose the auctioneer has five units for
sale. As the price falls, the first five bidders
to submit market orders purchase a unit of
the good at the price the auctioneer offered
to them.
Ascending Japanese auction:
The auctioneer holds an ascending auction
and awards the objects to the five highest
bidders at the price the sixth bidder drop
out.
Multiunit Japanese auction
In a Japanese auction, bidders drops out
until there are only as many remaining
bidders in the auction as there are items.
The winning bidders pay the price at
which the last bidder dropped out of the
auction.
In this auction it is easy to see that the
bidders with the highest valuation win the
auction.
Multiunit sealed bid auctions
Sealed bid auctions for multiple units can be
conducted by inviting bidders to submit limit order
offers, and allocating the available units to the
highest bidders.
In discriminatory auctions the winning bidders pay
different prices. For example they might pay at the
respective prices they posted.
In a uniform price auction the winners pay the same
price, such as a kth price auction (where k could
range from 1 to N.)
Revenue equivalence revisited
Suppose each bidder:
- knows her own valuation
- only want one of the identical items up for auction
- is risk neutral
Consider two auctions which both award the auctioned
items to the highest valuation bidders in equilibrium.
Then the revenue equivalence theorem applies, implying
that the mechanism chosen for trading is immaterial
(unless the auctioneer is concerned about entry
deterrence or collusive behavior).
Prices follow a random walk
In repeated auctions that satisfy the revenue
equivalence theorem, we can show that the price
of successive units follows a random walk.
Intuitively, each bidder is estimating the bid he
must make to beat the demander with (Q+1)st
highest valuation, that is conditional on his own
valuation being one of the Q highest.
If the expected price from the qs+1 item exceeds
that of the qs item before either is auctioned, then
we would expect this to cause more (less)
aggressive bidding for qs item (qs+1 item) to get
a better deal, thus driving up (down) its price.
Multiunit Dutch auction
To conduct a Dutch auction the auctioneer
successively posts limit orders, reducing the limit
order price of the good until all the units have been
bought by bidders making market orders.
Note that in a descending auction, objects for sale
might not be identical. The bidder willing to pay the
highest price chooses the object he ranks most
highly, and the price continues to fall until all the
objects are sold.
Clusters of trades
As the price falls in a Dutch auction for Q units, no one
adjusts her reservation bid, until it reaches the highest bid.
At that point the chance of winning one of the remaining
units falls. Players left in the auction reduce the amount of
surplus they would obtain in the event of a win, and
increase their reservation bids.
Consequently the remaining successful bids are clustered
(and trading is brisk) relative to the empirical probability
distribution of the valuations themselves.
Hence the Nash equilibrium solution to this auction creates
the impression of a frenzied grab for the asset, as herd like
instincts prevail.
Why the Dutch auction
does not satisfy the conditions
for revenue equivalence
We found that the revenue equivalence theorem applies
to multiunit auctions if each bidder only wants one item,
providing the mechanism ensures the items are sold to
the bidders who have the highest valuations.
In contrast to a single unit auction, the multiunit Dutch
auction does not meet the conditions for revenue
equivalence, because of the possibility of “rational
herding”.
If there is herding we cannot guarantee the highest
valuation bidders will be auction winners.
Multiunit demanders
By a multiunit demander we mean that each bidder
might desire (and bid on) all Q units for himself. We
now drop the assumption that N > Q.
Relaxing the assumption that each bidder demands
one unit at most seriously compromises the
applicability of the Revenue Equivalence theorem.
Typically auctions will not yield the same resource
allocation even if the usual conditions are met
(private valuations, risk neutrality, lowest feasible
expects no rent from participation).
Example: Two unit demanders
in a third price sealed bid auction
Consider a third price sealed bid auction for two units
where there are two bidders, each of whom wants
two units. Thus N = Q = 2. Each bidder submits two
prices.
We suppose the first bidder has a valuation of v11 for
his first unit and v12 for for his second, where v11 > v12
say. Similarly the valuations of the second bidder are
v21 and v22 respectively, where v21 > v22.
Example continued
The arguments given for single unit second price
sealed bid auctions apply to the highest price of each
bidder. One of his prices is highest valuation.
There is some probability that each bidder will win
one unit, and in this case the price paid by one of the
bidders will be determined by his second highest bid.
Recognizing this in advance, he shades his valuation
on his second highest bid.
Vickery auctions defined
A Vickery auction is a sealed bid auction, and units are
assigned according to the highest bids (as usual).
Each bidder pays for the (sum of the) price(s) for the
losing bid(s) his own bids displaced. By definition the
losing bids he displaced would have been included within
the winning set of bids if the bidder had not participated
in the auction, and everybody else had submitted the
same bids. In a single unit auction this corresponds to the
second highest bidder.
The total price a bidder pays in a Vickery auction for all
the units he has won is the sum of the bids on the units
he displaced.
Vickery auctions are efficient
A Vickery auction is the multiunit analogue to a
second price auction, in that the unique solution
(derived from weak dominance) is for each bidder to
truthfully report his valuations.
This implies that a Vickery auction allocates units
efficiently, in contrast to many multiunit auction
mechanisms.
Summary
This session compared auctions with monopoly, and
thus established the close connections between them.
We found the revenue equivalence theorem applies to
multiunit auctions if each bidder only wants one item.
Prices in first and second price sealed bid repeated
multiunit auctions follow a random walk.
When bidders demand more than one unit each, the
revenue equivalence theorem breaks down.
The Vickery auction is efficient, in contrast to many
other auction mechanisms.
2. Prices and Quantities
This section of the chapter analyzes how the
determination of quantity impacts on the
monopolist’s optimization problem. We begin
with a discussion of the reservation price in
an auction, before moving on to monopoly
supply. Although traditional arguments
suggest that monopolists are inefficient, we
argue the monopolist has an incentive to be
as efficient as a competitive industry.
Choosing quantity
When analyzing monopoly, an important issue is
the quantity the monopolist chooses to supply and
sell.
Regulators argue that compared to a competitively
organized industry where there are many firms
supplying the product, a monopolist restricts the
supply of the good and charges higher prices to
high valuation demanders in order to make rents
out of his position of sole source.
Is this true in practice?
Reservation prices for auctions
One reason for an auctioneer to set a reservation
price is because of the value of the auctioned item
to him if it is not sold. This value represents the
opportunity cost of auctioning the item. For
example he might sell it at another auction at some
later time, and maybe use the item in the
meantime.
Should the auctioneer set a reservation above its
opportunity cost?
A related question is whether the auctioneer has
the power to commit himself to setting a
reservation price above its opportunity cost.
Auction Revenue
What is the optimal reservation price in a private
value, second price sealed bid auction, where
bidders are risk neutral and their valuations are
drawn from the same probability distribution
function?
Let r denote the reservation price, let v0 denote the
opportunity cost, let F(v) denote the distribution of
private values and N the number of bidders. Then
the revenue from the auction is:

Frv0 NFr 
1 Fr
r NN 1 vFvN2
1 Fv
F vdv
r
N
N1
Solving for the optimal reservation price
Differentiating with respect to r, we obtain the first
order condition for optimality below, where r0 denotes
the optimal reservation price.
Note that the optimal reservation price does not
depend on N.
Intuitively the marginal cost of the top valuation
falling below r, so that the auction only nets v0 instead
of r0, equals the marginal benefit from extracting a
little more from the top bidder when he is the only
one to bidder to beat the reservation price.

ro v 0 
F 
ro 1 F
ro 
The uniform distribution
When the valuations are distributed uniformly with:
F
v 
v v 
/v v 
then:
r v v 0 
/2
o
Designing a monopoly game
with a quantity choice
In the game below, the valuations of buyers are
uniformly distributed between $10 and $20 for one
unit, and have no desire to purchase multiple units.
Each buyer is endowed with $20.
The monopolist’s production capacity is 100 units of
the good. The marginal cost of producing each unit
up to capacity is constant at $10.
What is the equilibrium quantity bought and sold?
Eleven buyers and one seller
20
19
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10
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MC=10
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10 11
q
Demand schedule
In this example the marginal cost is $10.
Price
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Quantity Revenue Marginal Total costs Profit
revenue
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38
54
68
80
90
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108
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110
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0
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30
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50
60
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80
90
100
110
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18
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28
30
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28
24
18
10
0
Static Solution to game
There are two outputs that yield the
maximum profit, which is $30.
If the monopolist offers 6 units for sale,
the market will clear at a price of $15.
If the monopolist offers 5 units for sale,
the market will clear at a price of $16.
A differential approach
The traditional argument can be framed as follows.
Let c denote the cost per unit produced, and
suppose consumers demand quantity q(p) when
the price is p.
Assume q(p) is differentiable and declining in p,
and write p(q) as its inverse function. That is:
q(p(q)) = q.
The monopolist chooses q to maximize:
(p(q) – c) q
Marginal revenue equals marginal cost
Let qm denote the profit maximizing quantity supplied
by the monopolist. Then qm satisfies the first order
condition for the optimization problem, which is:
p(qm) + p’(qm) qm = c
The two terms on the left side of the equation
comprise the marginal revenue from increasing the
quantity sold. When an additional unit is sold it fetches
p(q) if we ignore any downward pressure on prices.
The traditional argument is that the monopolist will
only produce sell an extra unit if the marginal revenue
from doing so exceeds the marginal cost.
Uniform distribution
In the uniform distribution example. if there is a large
number of potential customers with mass of one unit
q(p) = 20 – p (if 10 < p < 20)
so:
p(q) = 20 – 20q (if 0 < q < 1)
and marginal revenue is:
20 – 40q
Setting marginal revenue equal to marginal cost yields
the equation:
20 – 40q = 10q
and solving we obtain: q = ¼ and p = 15.
Intermediaries with market power
We typically think of monopolies owning the
property rights to a unique resource. Yet the
institutional arrangements for trade may also be
the source of monopoly power.
If brokers could actively mediate all trades
between buyers and sellers, then they could
extract more of the gains from trade.
How should a broker set the spread between the
buy and sell price? A small spread encourages
greater trading volume, but a larger spread nets
him a higher profit per transaction.
Real estate agents
Suppose real estate agencies jointly determined the fees
paid by home owners selling their real estate to buyers.
How should the cartel set a uniform price that
maximizes the net revenue for intermediating between
buyers and sellers?
We denote the inverse supply curve for houses by fs(q)
and the inverse demand curve for houses by fd(q).
Writing price p = fs(q) means that if the price were p
then suppliers would be willing to sell q houses. Similarly
if p = fd(q), then at price p demanders would be willing
to purchase q houses.
Optimization by a real estate cartel
By convention the seller is nominally responsible for the
real estate fees. Let t denote real estate fees and q the
quantity of housing stock traded. The cartel maximizes tq
subject to the constraint that t = fd(q) - fs(q), or chooses q
to maximize:
[fd(q) - fs(q)]q
The interior first order condition is:
[fd(q) + f’d(q)q] = [fs(q) + f’s(q)q]
The marginal revenue from a real estate agency selling
another unit (selling more houses at a lower price) is
equated with the marginal cost of acquiring another house
(and thus driving up the price of all houses being sold).
NYSE dealers
In the NYSE dealers see the orders entering
their own books, in contrast to the brokers
and investors who place limit orders.
The exchange forbids dealers from
intervening in the market by not respecting
the timing priorities of the orders from
brokers and investors as they arrive.
However dealers are expected to use their
informational advantage make the market by
placing a limit order in the limit order books
if it is empty.
The gains from more information
If dealers do not mediate trades, but merely place their
own market orders, their ability to make rents is
severely curtailed, but not eliminated. The trading
game is characterized by differential information.
1. The order flow is uncertain, everyone sees past
transaction prices and volume but only the dealer sees
the existing limit orders, so the dealer is in a stronger
position than brokers to forecast future transaction
prices.
2. If valuations are affiliated then the broker is also more
informed about the valuations of investors and brokers
placing future orders.
Perfect price discrimination
Suppose the monopolist knows the valuation each
consumer places on a unit of the item or service and
there is no possibility of re-trade amongst consumers.
In that case, legal issues aside, the monopolist should
offer the item to each consumer who values it at more
than the marginal production cost, at his or her
valuation (or for a few cents less).
The monopolist’s profit is then the integral of demand
up to the point where the demand crosses the marginal
cost curve, less total costs, which clearly exceeds the
profit from charging a uniform price.
Comparison with competitive equilibrium
Note that the and the production level of a perfectly
discriminating monopolist is the competitive equilibrium
level, where price equals marginal cost.
The basic difference is that a price discriminating
monopolist extracts all the gains from trade, whereas a
in a competitive equilibrium, all the gains from trade go
to the consumers in the case where marginal costs are
constant.
In the example with 11 consumers, the perfectly
discriminating monopolist garners profits of 55, a
uniform price monopolist 30, and a competitively
organized industry nothing.
Laws against price discrimination
The 1936 Robinson-Patman Act of updated the earlier
1914 Clayton Act instituting laws against price
discrimination. The Federal Trade Commission (FTC) is
charged with the oversight of these laws.
The fact that different consumers pay different prices is
not sufficient to prove illegal price discrimination has
occurred.
A firm cannot be found guilty of engaging in illegal
price discrimination unless there are ill effects on
competition, meaning competition is reduced, or a
monopoly is sustained, or a monopoly is created.
How important are these legal issues?
Economists are skeptical about how much
competition has been fostered by laws against
price discrimination.
More than half the firms prosecuted for breaking
price discrimination laws are relatively small (local)
monopolies.
Perhaps the most important reason we observe less
price discrimination than the simple static model
analysis predicts, is that the monopolist typically
does not know how each consumer values his
goods and services.
Summary
Monopolists are said to create inefficiencies, restricting
supply by trading off higher prices with less demand.
Intermediaries can also sometimes exploit their
monopolistic position by creating a wedge between
their buy and sell prices.
If monopolists price discriminate they produce where
the lowest price consumer pays the marginal cost of
production, an efficient outcome.
Laws against price discrimination are directed against
anticompetitive practices that limit entry, and are not
primarily concerned with how trading surplus is divided
between consumers and producers.
3. Segmenting the Market
Perfect price discrimination is often hard
to impose directly. However quantity
discounting, product bundling and
dynamic pricing strategies sometimes
provide the means for achieving its
objective of value maximization.
Segmenting the market
To profitably engage in explicit price
discrimination, the monopolist must be able to
1.
Identify the individual reservation prices
by his clientele for his goods
2.
Prevent resale from customers with low
reservation prices to potential customers
with high reservation prices.
3.
Be free of incrimination from laws of
price discrimination.
When the monopolist knows the distribution of
demand but not the characteristics of individual
demanders, or alternatively is subject to laws
against price discrimination, it can sometimes
segment the market to increase its profits.
Quantity discounting
We first consider a geographically isolated retail
market monopolized by a firm selling kitchen and
laundry detergents or bathroom toiletries to two
types of consumers, large volume commercial
buyers and small volume households.
The commercial demanders are willing to search
over a wider area for suppliers, and consider a
greater range of close substitutes (paper towels
versus blow dry).
Households have less incentive to search for these
low cost items, rarely consider substitute products,
and limited space to store these items; household
rental rates for inventory storage are typically
greater commercial property rates (per cubic foot).
A parameterization
Suppose the reservation value of a commercial
demander is vc and the reservation price of a
household is vh where vc < vh.
We also assume a commercial demander would
buy k units if the price is less than its reservation
value, whereas a household would only buy one
unit.
Commercial and household demanders are
distributed in proportion p and (1 – p) respectively
throughout the local market catchment area.
Unit (wholesale) costs for the monopolist are c,
where c < vc.
Solution to the parameterization
If the firm adopts a uniform pricing policy, then the
maximum monopoly profits are found by charging a high
price and only serving households, or charging a low
price to capture all the local demand:
max{p(vc – c) + (1 - p)k(vc – p), p(vh – c) }
If the firm charges a high price for single units and a
discount price for bulk orders of k units then the
maximum monopoly profits are
p(vh – c) + (1 - p)k(vc – p)
Comparing the net profits of the two, we see that
discounting bulk orders is profitable.
When can perfect price discrimination be
achieved through quantity discounting?
Here perfect price discrimination is achieved without resort
to charging households and commercial demanders
different prices!
Note that if vc > vh then segmenting the market in this
way cannot be achieved unless the monopolist can restrict
the number of individual units purchased separately (which
is typically infeasible).
This result on segmentation can be extended to monopoly
markets with several consumer types. We only assume
that the consumer types demanding more units have lower
reservation values. The same logic applies.
Product bundling
Consider now another related method for
segmenting market demand to extract greater
economic rent.
The firm exploits the idea that customers who
demand several of the firm’s products might
exhibit more elastic demands (be more price
sensitive) than customers who only wish to
purchase a smaller subset of the firm’s products.
Indeed the monopoly offer a bundle of goods and
services that includes its monopoly product as
well as a product that is available separately at a
competitive price elsewhere.
Ski resort
Enthusiastic skiers bring their own equipment to the
resort, while casual skiers rent. Enthusiastic skiers are
willing to pay up to ve for a ski ticket, but casual skiers
are only willing to pay vc where ve > ve.
Resort employees at the ticket booth cannot distinguish
between a casual skier versus an enthusiastic skier,
because enthusiastic skiers have lots of experience
watching and listening to casual skiers.
There is, however, a competitive market for rental skis.
The price of renting skis, poles and boots is p, and this
reflects the cost of running a rental firm.
How does the resort maximize its value?
Solution to the ski resort’s problem
If the resort charges ve for ski tickets, and does not offer
any other services, only the enthusiastic skiers will visit.
If the resort only charges vc for ski tickets, then not all
the rent is extracted from enthusiastic skiers
Suppose the ski resort sells its tickets for ve but offer its
rentals for:
p – (ve - vc)
In that case enthusiastic skiers pay their reservation
price for skiing, while casual skiers pay their reservation
price for the package of skiing and renting, and after
cross subsidization from the ticket office, the resort
breaks even on its rental operation.
Principles for product bundling
More generally the solution to this problem is
found by identifying a product that the lower
valuation customers demand but the high
valuation customers do not want, and offering a
package deal on the bundle.
The package is typically marketed as a bargain.
Note that we have said nothing about the costs of
doing business. If the ski resort has high fixed
costs from running its lifts or preparing its runs,
then it might not be profitable to operate unless it
can engage in this form of price discrimination.
Other examples
1. Firms sell assembled goods such as cars to new car
buyers, and also meet demand from previous
buyers for plus replacement parts arising from
collision damage or wear and tear.
2. Restaurants sometimes offer complete dinners with
a limited range of items on the menu, and also
offer portions a la carte to those willing to spend
more.
3. Travel agencies offer all inclusive vacation packages
for travel and lodging as well as sell tickets for
individual items.
The static solution revisited
Price in dollars
20
inverse demand curve
Uniform price
solution
unit cost
9
marginal revenue curve
quantity
0
Uniform quantity
solution
Residual demand
Price
20
New vertical axis for origin of
residual inverse demand
Uniform price
solution
Unit cost
9
New marginal revenue curve
Quantity
0
A dynamic inconsistency?
After selling the original demanders the item at price
p(qm) the monopolist would have an incentive to sell the
item to the remaining consumers at a lower price.
If the original consumers knew that the product would go
on sale later they might delay their purchase. Does that
undermine our prediction that qm will be bought if the
price is p(qm)?
One possibility is that the monopolist commit to a uniform
price policy by promising everyone the lowest price he
offers to anyone.
These issues cannot be fully resolved within the context
of a static model.
Dynamic considerations
There are several ways to model the dynamics of price
setting and the service flow from the good over time.
If all trading must occur before customers take delivery
of their purchases, we can separate considerations of
price dynamics from those of the service flow.
Another approach is that the game lasts a fixed amount
of time and that consumers receive service flows from
the good as soon as they buy it. This approach
provides a natural way of modeling durable goods.
In both case we assume that agreements to trade
occur instantaneously, meaning transactions can be
conducted in infinitesimal amounts of time.
Dynamic pricing policy
in a closed time interval
Suppose that all trading must take place in a
closed interval of time, say [0,1], and customers
receive the good after trading closes.
This corresponds to a situation where the market
closes at a give fixed time.
At time t = 1 consumers recognize that the
monopolist will solve the static problem. Therefore
no consumer will buy above that price.
By a backwards induction argument we conclude
that the monopolist cannot charge more than the
uniform price in that case.
Dynamic pricing policy
in an open time interval
Suppose that all trading must take place in a (half)
open interval of time, say [0,1), and as before
customers receive the good at time t =1.
This corresponds to the case where the monopolist is
open ended about when trading will end.
Suppose the monopolist refuses to lower his price
until everyone with a higher reservation price than the
current price has purchased his product. In that case
consumers are sequentially presented with all or
nothing offers that are subgame perfect.
The monopolist reaps the full benefits of price
discrimination.
Durable goods
Now consider what happens in a closed trading interval
[0,1] when the good yields a service flow over the portion
of the interval that a consumer owns it.
For example if the consumer buys the good at t = ½ then
she receives a service flow between times t = ½ and 1,
and her total benefit is half her valuation.
In this case there are no consumer benefits from trading
at t =1.
A simple adaptation of our arguments in the open interval
case proves that the monopolist can extract all the
benefits of discriminating by sequentially reducing prices
at the beginning of the game from highest reservation
value to the lowest.
Summary
In the traditional view, monopolists maximize
their value by setting price where marginal
revenue equals marginal cost and restricting
trade, that is compared to competitive equilibrium
where price equals marginal cost.
We showed that the monopolist has an incentive
to price discriminate, extracting more of the gains
from trade, and raising output to the efficient
outcome achieved in competitive equilibrium.
When the conditions for perfect price
discrimination are absent, quantity discounting,
product bundling and dynamic pricing policies
may provide the means to the same end.