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Chapter 21 Monopoly. Auctions and Monopoly Prices and Quantities Segmenting the Market. 1. Auctions and Monopoly.

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chapter 21 monopoly

Chapter 21Monopoly

  • Auctions and Monopoly
  • Prices and Quantities
  • Segmenting the Market
1 auctions and monopoly

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
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:

  • 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.
  • 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.

Two main differences between most auction and monopoly models

other differences between most auction and monopoly models

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.

  • 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.

Other differences between most auction and monopoly models

an agenda for the first portion of our work on monopoly
An agenda for the first portion ofour work on monopoly
  • We will focus on two issues:
  • How does a multiunit auction differ from a single unit auction?
  • What can we learn about market behavior from multiunit auctions?
auctioning multiple units to single unit demanders
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
Decisions for the seller to makein 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
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
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
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
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
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
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
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
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
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
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
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
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
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.

  • 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
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
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
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
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:
solving for the optimal reservation price
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.
the uniform distribution
The uniform distribution
  • When the valuations are distributed uniformly with:
  • then:
designing a monopoly game with a quantity choice
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
Eleven buyers and one seller
























| | | | | | | | | | |


1 2 3 4 5 6 7 8 9 10 11

demand schedule
Demand schedule

In this example the marginal cost is $10.











static solution to game
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
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
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
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
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
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
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
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
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.
  • 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.
  • If valuations are affiliated then the broker is also more informed about the valuations of investors and brokers placing future orders.
perfect price discrimination
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
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
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
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.
  • 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
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
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
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
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
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
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
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
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
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
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
Other examples
  • 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.
  • 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.
  • Travel agencies offer all inclusive vacation packages for travel and lodging as well as sell tickets for individual items.
the static solution revisited
The static solution revisited

Price in dollars


inverse demand curve

Uniform price solution

unit cost


marginal revenue curve



Uniform quantity solution

residual demand
Residual demand


New vertical axis for origin of

residual inverse demand


Uniform price solution

Unit cost


New marginal revenue curve



a dynamic inconsistency
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
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
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
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
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.
  • 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.