The Limits of Price Discrimination
Dirk Bergemanny
Benjamin Brooksz
Stephen Morrisx
First Version: August 2012
Current Version: March 9, 2013
Preliminary and incomplete version prepared for March 13 NYU seminar
Abstract
We analyze the welfare consequences of a monopolist having additional information, beyond the prior distribution, about consumers’ tastes; the additional information can be used to charge di¤erent prices to di¤erent
segments of the market, i.e., carry out "third degree price discrimination".
We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that 1) consumer surplus is non-negative, 2) producer surplus is
at least as high as uniform monopoly pro…ts, and 3) total surplus does not exceed the gains from trade.
We also examine the limits of how quantities and prices can change under market segmentation and the limits
of quantity and quality (second degree) discrimination and thus screening problems for generally.
Keywords: Third Degree Price Discrimination, Private Information, Bayes Correlated Equilibrium.
JEL Classification: C72, D82, D83.
We gratefully acknowledge …nancial support from NSF SES 0851200 and ICES 1215808. We would like to thank Ricky Vohra for
an informative discussion and Alex Smolin for excellent research assistance.
y
Department of Economics, Yale University, New Haven, U.S.A., dirk.bergemann@yale.edu.
z
Department of Economics, Princeton University, Princeton, U.S.A., babrooks@princeton.edu.
x
Department of Economics, Princeton University, Princeton, U.S.A., smorris@princeton.edu.
1
The Limits of Price Discrimination March 9, 2013
1
2
Introduction
A classic and central issue in the economic analysis of monopoly is the impact of discriminatory pricing on consumer
and producer surplus. A monopolist engages in third degree price discrimination if he uses additional information
about characteristics to o¤er di¤erent prices to di¤erent segments of the aggregate market. A large and classical
literature (reviewed below) examines the impact of particular segmentations on consumer and producer surplus,
as well as on output and prices.
In this paper, we characterize what could happen to consumer and producer surplus for all possible segmentations. We know at least two points that will be attained. If there is no segmentation and the producer charges the
uniform monopoly price, the producer will get the associated monopoly pro…t, which incidentally forms the lower
bound of the producer surplus and the consumer will get a positive consumer surplus, the standard information
rent. This is marked in Figure 1 by point A.
Figure 1: The Surplus Triangle of Price Discrimination
This corresponds to the situation in which the monopolist has zero information (beyond the prior distribution
of the valuations). Second, if the market is completely segmented, we have perfect price discrimination, or …rst
degree price discrimination, and the allocation will be e¢ cient, but the producer will receive the entire feasible
surplus and the consumer will get zero surplus.
This is marked in Figure 1 by point B. We can also identify
some elementary bounds on consumer and producer surplus. First, the consumer must get non-negative surplus
as a consequence of the participation constraint. Second, the producer must get at least the surplus that he could
get if there was no segmentation and he charged the uniform monopoly price. Third, the sum of consumer and
producer surplus cannot exceed the total value to the consumer of always getting the good when it is e¢ cient to
do so. The right angled triangle in Figure 1 illustrates these bounds.
The Limits of Price Discrimination March 9, 2013
3
Our main result is that every point in the right angled triangle is attainable by some market segmentation.
The point marked C is one where consumer surplus is maximized; in particular, the producer is held down to his
uniform monopoly pro…ts, but the consumer gets the rest of the social surplus from an e¢ cient allocation. And
the point marked D is one where total surplus is minimized by holding both consumer and producer down to their
minimum surplus (i.e., zero for the consumer and the uniform monopoly pro…ts for the producer).
We can explain these results most easily in the case where there are only a …nite set of possible consumer
valuations and zero cost of production (a normalization we will maintain throughout the paper).
explain how consumer surplus is maximized, i.e., point C is realized.
Let us …rst
The set of market prices will consist of
every valuation less than or equal to the uniform monopoly price. Suppose that we could divide the market into
segments corresponding to each of these prices in such a way that (i) in each segment, the consumer’s valuation was
always greater than or equal to the price for that segment; and (ii) in each segment, the producer was indi¤erent
to charging the price for that segment and the uniform monopoly price. Then, since the producer is indi¤erent
between charging the uniform monopoly price to everybody, his pro…ts must be equal to his uniform monopoly
pro…ts. But since the allocation is e¢ cient, the consumer must obtain the rest of the surplus above the uniform
monopoly pro…ts.
So creating a market segmentation satisfying (i) and (ii) is a su¢ cient condition for attaining consumer surplus
maximization. We now describe a way of constructing such a market segmentation iteratively. Start with a "lowest
price segment" where the lowest price will be charged. Put everyone with that valuation into that segment. For
each higher valuation, put the same proportion (to be determined) of consumers with that valuation into the
lowest price segment. Choose that proportion between zero and one such that the producer will be indi¤erent
between charging that minimum price and the uniform monopoly price. We know this must be possible, because
if the proportion were equal to one, the uniform monopoly price would be pro…t maximizing for the producer (by
de…nition); if the proportion were equal to zero - so only lowest valuation consumers were in the market - the lowest
price would be pro…t maximizing; and, by keeping the relative proportions above the lowest valuation constant,
there is no possibility of a price other than these two becoming optimal. Now we have created one market segment
satisfying properties (i) and (ii) above. But notice that the remaining consumers, not put into the lowest price
segment, are in the same relative proportions as they were in the original population. Now we can construct a
segment who will be charged the second lowest valuation in the same way: put all the remaining consumers with
the second lowest valuation into this market; for higher valuations, put a …xed proportion of remaining consumers
into that segment; choose the proportion so that the producer is indi¤erent between charging the second highest
valuation and the uniform monopoly price. This construction iterates, and we have attained point C.
A similar argument shows how to minimize total surplus, i.e., attain point D. Now let the set of market prices
consist of every valuation more than or equal to the uniform monopoly price. Suppose that we could divide the
market into segments corresponding to each of these prices in such a way that (i) in each segment, the consumer’s
The Limits of Price Discrimination March 9, 2013
4
valuation was less than or equal to price for that segment; and (ii) in each segment, the producer was indi¤erent
to charging the price for that market and the uniform monopoly price.
Now consumer surplus would be zero,
since consumers are always charged a price greater than or equal to their valuations; and producer surplus would
equal uniform monopoly pro…ts, since the producer is always indi¤erent to charging the uniform monopoly price.
A similar, but slightly more complex, iterative construction will be used to show that this su¢ cient condition for
attaining minimum total surplus is satis…ed.
We can thus establish that points B, C and D can be attained. But every point in their convex hull, i.e., the
right angled triangle in Figure 1, can also be attained simply by averaging the segmentations that work for each
extreme point. Thus we have a complete characterization of possible welfare outcomes.
While we focus on welfare implications, we can also completely characterize possible output levels and derive
implications for prices. An upper bound on output is the e¢ cient quantity and this is realized by any consumer
surplus maximizing segmentation. In such segmentations, prices are always (weakly) below the uniform monopoly
price. A lower bound on output can be constructed as follows. The monopolist must receive at least his uniform
monopoly pro…ts. So surplus equal to the uniform monopoly pro…ts must be created. Say that a segmentation
is conditionally e¢ cient if, conditional on the amount of output sold, it is sold to higher valuation consumers.
The output minimizing way of setting total surplus equal to uniform monopoly pro…ts is use a conditionally
e¢ cient allocation. We show that we can attain this bound. In particular, we construct a segmentation which
is simultaneously surplus minimizing and output minimizing.
In this segmentation, prices are always (weakly)
higher than the uniform monopoly price.
Our result holds with both discrete and continuous distributions over valuations. In the continuum version
of the results, a continuum of prices are charged and there are mass points of consumers with valuations equal to
the price being charged, with valuations above (for consumer surplus maximization) and below (for total surplus
minimization) distributed according to densities.
to di¤erential equations.
The analysis is constructive involving closed form solutions
In both the discrete and the continuum versions of the results, there are multiple
segmentations that attain welfare bounds. In the continuum case, we give an example attaining consumer surplus
maximization where the producer is indi¤erent between all prices between the lowest valuation in each segment
and the uniform monopoly price.
Our paper contributes to a large literature on third degree price discrimination, starting with Pigou (1920).
This literature examines what happens to price, quantities, consumer surplus, producer surplus and total welfare
as the market is segmented into (usually, two) pieces. Thus Pigou (1920) considered the case of segments with
linear demand where both segments are served in equilibrium. He showed that (in this special case) output did not
change under price discrimination. Since di¤erent prices were charged in the two segments, this means that some
higher valuation consumers must be being replaced by lower value consumers, and thus total welfare (the sum of
consumer and producer surplus) must decrease. We can visualize the results of Pigou (1920) and other authors in
The Limits of Price Discrimination March 9, 2013
5
Figure 1. Pigou (1920) showed that this particular segmentation resulted in a west north west move (i.e., move
from point B corresponding to uniform price monopoly to a point below the negative 450 line going through B).
A literature since then has focussed on identifying su¢ cient conditions on the shape of demand in two segments
for total welfare to increase or decrease with price discrimination. A recent paper of Aguirre, Cowan, and Vickers
(2010) uni…es and extends this literature1 and, in particular, identi…es su¢ cient conditions for price discrimination
to either increase or decrease total welfare (i.e., move above or below the negative 450 line through B). Restricting
attention to market segments that have concave pro…t functions and an additional property ("increasing ratio
condition") that they argue is commonly met, they show that welfare is lower if the direct demand in the higher
priced market is at least as convex as that in lower priced market; welfare is higher if prices are not too far apart
and the inverse demand function in the lower priced market is locally more convex than that in the higher priced
market. They note how their result ties in with an intuition of Robinson (1933): concave demand means that
price changes have a small impact on quantities and thus welfare, while convex demand means that prices have a
large impact on quantities and thus welfare.
Our paper also gives su¢ cient conditions for di¤erent welfare impacts of segmentation. However, unlike most
of the literature, we allow for segments with non-concave pro…t functions. Indeed, the segmentations giving rise to
extreme points in welfare space (i.e., consumer surplus maximization at point C and total surplus minimization at
point D) always rely on non-concave pro…t functions. This ensures that the type of local conditions highlighted in
the existing literature will not be relevant. Our non-local results suggest some very di¤erent intuitions. Consumer
surplus is of course always increased if prices drop in all markets. We show that for any demand curves, pooling
low valuation consumers with the right number of high valuation consumers will always give the producer an
incentive to o¤er prices below the monopoly price.
If the producer does not have a strong incentive to o¤er
discounted prices to these groups, then the producer surplus cannot be increasing much and the consumer must
be extracting the e¢ ciency gain.
The literature also has results on the impact of segmentation on output and prices. On output, the focus is
on identifying when increasing output is su¢ cient for an increase in welfare.
Although we do not analyze the
question in detail in this paper, most output levels are associated with many di¤erent levels of producer, consumer
and total welfare, so there is no necessary relationship between output and welfare.
However, we do identify
the highest and lowest possible output in any market segmentation. On prices, Nahata, Ostaszewski, and Sahoo
(1990) o¤er examples with non-concave pro…t functions where third degree price discrimination may lead either
prices in all market segments to decrease, or prices in all market segments to increase.
We show that one can
create such segmentations independent of the original demand curve. In other words, in constructing our critical
market segmentations, we incidentally show that it is always possible to have all prices fall or all prices rise (with
non-concave pro…t functions in the segments remaining a necessary condition, as shown by Nahata, Ostaszewski,
1
Key intervening work includes Robinson (1933), Schmalensee (1981) and Varian (1985).
The Limits of Price Discrimination March 9, 2013
6
and Sahoo (1990)).
If market segmentation is exogenous, one might argue that the segmentations that deliver extremal surpluses
are special and might be seen as atypical. But to the extent that market segmentation is endogenous, our results
can be used to o¤er predictions about what segmentation might arise endogenously. For example, consider internet
companies that have large amounts of data about the valuations of large numbers of consumers. They then have
an incentive to sell that information to producers who could use it to price discriminate.
A recent literature -
see, for example, Taylor (2004) and Acquisti and Varian (2005) - examines the dynamic impact of this possibility,
highlighting how consumers might strategically conceal information anticipating that it might be sold in this way.
But suppose that, for regulatory and/or legal reasons, internet companies who had collected this market data were
not able to sell the information to producers. Our results show that internet companies with complete information
about their consumers could, by selectively revealing information (for free) to producers, extract all the e¢ ciency
gains of perfect price discrimination, but pass all the welfare gains on to consumers. In this case, the consumers
would have an incentive to reveal their valuations in a dynamic setting. In this scenario, the internet companies
would be endogenously deciding how to segment the market. A subtlety of this story, however, is that this could
only be done by randomly allocating consumers with same valuation to di¤erent segments with di¤erent prices.
We also consider the extension of our results to the case where the quality (or quantity) of the good can vary,
and thus there is second degree price discrimination. In doing so, we provide an analysis of optimal segmentation
in screening problems more generally.
We provide a complete analysis of the case where each consumer has
(continuous) demand for multiple units and the producer can screen with using di¤erent qualities, i.e., the problem
originally analyzed by Mussa and Rosen (1978). We derive a closed form characterization of the set of attainable
consumer and producer surplus pairs.
The extreme result that either consumer surplus can be maximized or
total surplus minimized consistent with the producer being held down to his pooling (i.e., uniform pricing) surplus
no longer holds.
But there continues to be a very large set of feasible consumer surplus and producer surplus
pairs, and thus large scope for changes in segmentations to be Pareto-improving or Pareto-worsening. Allowing
non-linear cost, increasing marginal cost) also relaxes the extreme benchmark result.
Our work has a methodological connection to two strands of literature.
Kamenica and Gentzkow (2011)’s
study of "Bayesian persuasion" considers how sender would choose to transmit information to a receiver, if he
could commit to such an information revelation strategy before observing his private information. They provide
a characterization of such optimal communication strategies as well as applications. Our results in this paper are
another application. Suppose that a representative consumer, who did not yet know his valuation of the good,
could decide on what noisy signal to send a producer about his valuation. Our construction of the segmentation
that achieves consumer surplus maximization is an optimal solution to this problem (as noted earlier, there are
other solutions, but we also provide partial characterizations of all such segmentations).
In the two type case,
Kamenica and Gentzkow (2011) show that if one plots the utility of the "sender" as a function of the proportions of
The Limits of Price Discrimination March 9, 2013
7
each his two types, his highest attainable utility can be read o¤ from the concavi…cation of that function. Setting
the sender’s utility equal to the arbitrary weighted sums of consumer and producer surplus, we use this argument
to map out the attainable consumer and producer surplus pairs in a two type version of our main result, as well
as the extension to the second degree price discrimination problem of Mussa and Rosen (1978).
Bergemann and Morris (2013) examine the general question, in strategic (many player) settings, of what
behavior could arise in an incomplete information game if players might observe additional information not known
to the analyst. We provide a characterization of what behavior might arise, which is equivalent to an incomplete
information version of correlated equilibrium that we label "Bayes correlated equilibrium". In Bergemann and
Morris (2013), we highlight the connection a one player version of our analysis and its connection to the work of
Kamenica and Gentzkow (2011) and others. In this language, this paper considers a one player game, i.e., decision
problem, for a producer choosing prices. We characterize what could happen for any information structure that
he might observe. Thus we identify possible payo¤s of the one player (the producer) and an unmodelled player
(the consumer) in all Bayes correlated equilibria of the one player price setting game.
In Bergemann, Brooks,
and Morris (2012), we are examining properties of Bayes correlated equilibria of a …rst price auction, which can
be seen as a many player version of the one player game studied here.
Thus we provide substantive new results about the limits of third degree price discrimination and a striking
illustration of the methodology of Kamenica and Gentzkow (2011) and Bergemann and Morris (2013).
We present our main result in the case of discrete values in Section 2. In Section 3, we develop a continuum
analogue of the result. The discrete and continuum analyses are complementarity: while they lead to the same
substantive conclusions and economic insights, the arguments and mathematical formulations look very di¤erent,
so we …nd it useful to treat both cases independently. In Section 4, we re-visit our analysis of the discrete type
case in the special case of two types. In this case, we illustrate how we could have proved our results by using
the concavi…cation argument from Kamenica and Gentzkow (2011), and analyze the two type Mussa and Rosen
(1978) extension. In Section 5, we discuss the increasing marginal cost case and conclude.
2
Discrete Values
There a continuum of consumers, each of whom has unit demand for a good. There is a monopoly producer. We
normalize the constant marginal cost of the good to zero. In this Section, we assume that there are K possible
values that consumers might have,
0 < v1 < ::: < vK .
Let qj > 0 be the proportion of buyers with valuation vj , so
q1 + :::: + qK = 1.
The Limits of Price Discrimination March 9, 2013
8
We normalize the size of the population to 1. In this setting, an alternative and equivalent interpretation is that
there is a single consumer with unknown valuation, and qj is the probability that the consumer’s valuation is vj .
2.1
Uniform Monopoly Price
Let us introduce some key variables in this simple setting. Feasible surplus is
w ,
K
X
qj vj .
(1)
j=1
The price will always be set equal to one of the K possible values. If the monopolist cannot price discriminate,
he will set the price equal to the uniform monopoly price p = vi where
0
1
0
1
K
K
X
X
@
@
qj A vi
qj A vi
j=i
(2)
j=i
for all i = 1; ::; K.
Uniform monopoly pro…ts are then
0
,@
Consumer surplus is
1
K
X
qj A vi =
j=i
u ,
K
X
max
i2f1;:::;N g
qj (vj
0
1
K
X
@
qj A vi .
(3)
j=i
vi )
j=i
and deadweight loss is
w
u
=
iX1
qj vj .
j=1
We will use a simple example to illustrate results in this section. Suppose that there are …ve possible valuations,
1; 2; 3; 4 and 5, each with equal proportions.
Thus K = 5; and vj = j and qj =
simple calculations show that feasible total surplus is w =
price is p = 3, so i = 3 and vi = 3.
u =
1
5
(3
1
5
3) + (4
1
5
3) + (5
3) =
3
5.
1
5
1
5
for each j.
(1 + 2 + 3 + 4 + 5) = 3.
Uniform monopoly pro…ts are
Deadweight loss is 3
9
5
3
5
=
=
3
5.
3
53
=
In this case,
The uniform monopoly
9
5.
Consumer surplus is
See Figure 2 for a plot of consumer
and producer surplus based on these numbers.
2.2
Segmentations
A segmentation is - in general - a division of consumers into separate markets.
producer will always set the price equal to one of the K possible values.
However, we know that the
And if there were multiple segments
where the same price were charged, we might as well treat them as the same segment.
Thus we will restrict
The Limits of Price Discrimination March 9, 2013
9
Figure 2: The Surplus Triangle of Price Discrimination: A Numerical Example
attention to segmentations with at most K possible segments.
Now a segmentation is a K
K non-negative
matrix X with (i; j) th element xij equal to the proportion of consumers who are in a market segment facing price
vi and have valuation vj ; the total proportions with each valuation must match the original distribution, so that
K
X
xij = qj
(4)
i=1
for each j = 1; ::; K.
Segmentation X is an equilibrium segmentation if it is optimal for the monopolist to charge price vi to segment
i, so that
0
1
K
X
@
xij A vi
0
@
j=i
for all i; k = 1; ::; K.
K
X
j=k
1
xij A vk
The inequalities (5) are the incentive constraints for this problem.
(5)
In particular, the
monopolist must not have an incentive to deviate to the uniform monopoly price in any segment, so setting k = i
in the above inequality, we have
0
1
K
X
@
xij A vi
j=i
for all i = 1; ::; K.
0
@
K
X
j=i
1
xij A vi
(6)
Now we can associate with any market segmentation X its producer surplus,
and its consumer surplus
e (X) =
u
e (X) =
K X
K
X
xij vi
(7)
i=1 j=i
K
K X
X
i=1 j=i
xij (vj
vi ) .
(8)
The Limits of Price Discrimination March 9, 2013
10
An important equilibrium segmentation is the uniform price monopoly where
8
< q , if i = i
j
xij =
.
: 0, otherwise
In the example, this corresponds to the matrix
0
and gives producer surplus
=
9
5
0
B
B 0
B
B
B 1
B 5
B
B 0
@
0
0
0
0
0
0
0
1
5
1
5
1
5
0
0
0
0
0
0
0
1
C
0 C
C
C
1 C
5 C
C
0 C
A
0
and consumer surplus u = 35 .
Another important equilibrium segmentation is perfect price discrimination, where
8
< q , if i = j
j
xij =
: 0, otherwise
In the example, this corresponds to the matrix
0
1
5
B
B 0
B
B
B 0
B
B
B 0
@
0
and gives pro…ts
0
0
0
1
5
0
0
0
1
5
0
0
0
1
5
0
0
0
0
1
C
0 C
C
C
0 C
C
C
0 C
A
1
5
= 3 and consumer surplus 0. We plot these (consumer surplus, producer surplus) pairs in
Figure 2 for the example.
We want to characterize the (consumer surplus, producer surplus) pairs that can arise in an equilibrium
segmentation, i.e.,
8
<
V = (u; ) 2 R2+
:
9
there exists an equilibrium segmentation X =
;
satisfying u
e (X) = u and e (X) =
Now consumer surplus u
e (X) must be non-negative: this follows from the de…nition of consumer surplus, (8). The
The Limits of Price Discrimination March 9, 2013
11
sum of consumer surplus and producer surplus, u
e (X) + e (X), cannot exceed the feasible surplus, w :
u
e (X) + e (X) =
=
K X
K
X
xij (vj
vi ) +
i=1 j=i
K X
K
X
K X
K
X
xij vi , by (7) and (8)
i=1 j=i
xij vj
i=1 j=i
K X
K
X
xij vj
i=1 j=1
=
K
X
qj vj , by (4)
j=1
= w , by (1).
And producer surplus e (X) must be at least uniform monopoly pro…ts
e (X) =
K X
K
X
:
xij vi
i=1 j=i
K X
K
X
xij vi , by (6)
i=1 j=i
=
K
X
vi
j=i
=
K
X
K
X
xij
i=1
vi qj , by (4)
j=i
=
, by (3)
These three inequalities provide an upper bound (in the set order) on the set of feasible (producer surplus, consumer
surplus) pairs:
V
8
>
>
<
(u; ) 2 R2+
>
>
:
u
e (X)
e (X)
0
u
e (X) + e (X)
w
9
>
>
=
>
>
;
.
(9)
The set on the right hand side for our example is plotted as the shaded right angled triangle in Figure 2:
2.3
Consumer Surplus (and Output) Maximizing Segmentations
To give a partial characterization of consumer surplus maximizing segmentations, let us identify some key properties
of segmentations. Segmentation X attains the consumer surplus upper bound if
u
e (X) = w
The Limits of Price Discrimination March 9, 2013
12
and thus
e (X) =
.
Segmentation X is e¢ cient if the price is always less than the consumer’s valuation, i.e.,
xij = 0 if i > j.
If a segmentation is e¢ cient, then all consumers buy the good and thus it is also said to be output maximizing.
Segmentation X is price reducing if the price is always less than or equal to the uniform monopoly price, i.e.,
xij = 0 if i > i .
Segmentation X is uniform price indi¤ erent if the producer is always indi¤erent between charging the segment
price and the uniform monopoly price, i.e., in addition to all the incentive compatibility inequalities (5), we have
that (6) holds with equality:
0
1
0
1
K
K
X
X
@
xij A vi = @
xij A vi
j=i
(10)
j=i
for all i = 1; ::; K. Now we have the following partial characterization of consumer surplus maximizing segmentations:
Lemma 1 (Necessary Conditions for Consumer Surplus Maximization)
1. If an equilibrium segmentation attains the consumer surplus upper bound, then it is e¢ cient, output maximizing, price reducing and uniform price indi¤ erent.
2. If an equilibrium segmentation is e¢ cient and uniform price indi¤ erent, then it is output maximizing and
attains the consumer surplus upper bound.
For (1), observe that the producer can guarantee himself surplus
, any segmentation attaining the consumer
surplus upper bound must have total surplus is w , which requires e¢ ciency. E¢ ciency implies output maximizing.
If the segmentation attains the consumer surplus upper bound, and is e¢ cient, but is not price reducing, then the
producer would be able to guarantee himself surplus strictly greater than
by raising all prices strictly below
the uniform monopoly price to the uniform monopoly price, a contradiction. Thus any segmentation attaining
the consumer surplus upper bound must be price reducing. Finally, the producer surplus is equal to that from
always charging the uniform monopoly price. If he obtained surplus greater than what he would get charging the
uniform monopoly price in any segment in a consumer surplus maximizing segmentation, then he could raise his
total pro…t above
by charging the uniform monopoly price in all other segments.
For (2), observe that if a segmentation is e¢ cient, the total surplus is w . If the segmentation is uniform price
indi¤erent, then the producer surplus must be
and the consumer surplus must be w
.
The Limits of Price Discrimination March 9, 2013
13
We prove the existence of a consumer surplus maximizing equilibrium segmentation by constructing one (which
is necessarily e¢ cient, price reducing and uniform price indi¤erent). Note that e¢ ciency and price reducing imply
- in our example - that the segmentation has support on the price - valuation pairs with crosses in the following
matrix:
0
B
B 0
B
B
B 0
B
B
B 0
@
0
1
0
0
0
0
0
0
0
C
C
C
C
C.
C
C
0 C
A
0
We describe an algorithm for constructing a segmentation which is e¢ cient, price reducing and uniform price
indi¤erent and thus consumer surplus maximizing, formalizing the construction discussed in the introduction. We
…rst set
x11 = q1
and, letting
1
2 [0; 1] solve
set
0
0
11
K
X
@q 1 + 1 @
qj AA v1 =
1
j=2
x1j =
0
@
K
X
j=i
1
qj A vi ,
1 qj
for each j = 2; :::; K. This leaves the producer indi¤erent between setting the price equal to v1 or vi in the …rst
segment. The remaining population has valuation at least v2 and the relative mass is the same as the original
demand curve. Then we set
x21 = 0
x22 = (1
and, letting
set
2
2 [0; 1] solve
0
@q2 +
2
0
@
K
X
j=3
1 ) q2
11
qj AA v2 =
x2j = (1
1)
2
0
@
K
X
j=i
1
qj A vi ,
2 qj
for each j = 3; :::; K. Again, this leaves the producer indi¤erent between setting the price equal to v2 or vi in
the second segment. Now we iterate this construction. Thus each segment i
xij
= 0 for all j < i
xii =
i 1
Y
k=1
(1
!
k)
qi
i is given by
The Limits of Price Discrimination March 9, 2013
and, letting
i
2 [0; 1] solve
0
@qi +
set
i
0
@
14
K
X
j=i+1
11
qj AA vi =
i 1
Y
xij =
(1
i
0
@
!
k)
k=1
1
K
X
qj A vi ,
j=i
i qj
for each j = i + 1; :::; K. Thus, for each i = 1; :::; i ,
=
i
K
P
qj
j=i
Observe that
i
!
qi vi
K
P
vi
qj
j=i+1
!
vi
= 1 by construction, so the i th segment will contain all the remaining consumers. Observe that
(2) implies that for each i = 1; :::; i
1,
0
@
and so
qi vi
K
X
j=i
0
@
0
i
0
1
K
X
@
qj A vi
qj A vi
K
X
j=i
and thus
1
j=i
1
0
qj A vi
@
K
X
j=i+1
qj A vi
qi vi
=
K
P
K
P
qk vi
k=i
1
qk
k=i+1
!
1.
vi
We can fully describe the segmentation as:
8
>
0;
if j < i and i
>
>
>
>
iQ1
>
>
>
(1
if j = i and i
<
k ) qi ;
k=1
xij =
iQ1
>
>
>
(1
k)
i qj ; if j > i and i
>
>
>
k=1
>
>
:
0;
if
i < i:
For our example,
1
= 51 ,
2
=
2
3
i ;
i ;
(11)
i ;
and thus
0
1
5
B
B 0
B
B
B 0
B
B
B 0
@
0
1
25
4
25
0
1
25
8
75
4
75
1
25
8
75
4
75
0
0
0
0
0
0
1
25
8
75
4
75
1
C
C
C
C
C.
C
C
0 C
A
0
(12)
By construction, the segmentation (11) satis…es e¢ ciency and uniform price indi¤erence. Thus we have by Lemma
1.
The Limits of Price Discrimination March 9, 2013
15
Theorem 1 (Consumer Surplus Maximizing Segmentation)
The segmentation (11) is an equilibrium segmentation that attains the upper bound on consumer surplus, maximizes
output and is price reducing.
We describe one particular segmentation with these properties, but there are many segmentations satisfying the
necessary and su¢ cient conditions identi…ed in Lemma 1. For example, there is a consumer surplus maximizing
market segmentation where no valuation 2 consumers buy the good:
1
0
1
1
1
1
0
30
30
30
C
B 5
B 0 1 2
2
2 C
B
5
15
15
15 C
C
B
1
1 C;
B 0 0 1
B
30
30
30 C
C
B
B 0 0 0 0 0 C
A
@
0 0 0 0 0
(13)
and there is also a consumer surplus maximizing market segmentation where the monopolist is indi¤erent between
prices 1, 2 and 3:
0
1
5
B
B 0
B
B
B 0
B
B
B 0
@
0
1
15
1
10
0
2
45
4
45
1
15
2
45
4
45
1
15
0
0
0
0
0
0
2
45
4
45
1
15
1
C
C
C
C
C.
C
C
0 C
A
0
(14)
More generally, if we …x the value of x12 , i.e., the mass of consumers with valuation 2 who are sold the good
at price 1, at y, then uniform price indi¤erence will pin down the proportion of buyers with valuations greater
than or equal to 3, the uniform monopoly price, who pay each price. Thus the following segmentation is also a
market segmentation that attains the upper bound on consumer surplus as long as y
1
15 ,
required to ensure that prices 1 and 3 give as high pro…ts as price 2 in the …rst segment:
0
1
1
1
1
1
1
1
1
y
5
30 + 6 y 30 + 6 y 30 + 6 y
C
B
B 0 1 y 2
2
2
2
2
2 C
y
y
y
B
5
15
3
15
3
15
3 C
C
B
1
1
1
1
1
1 C.
B 0
0
+
y
+
y
+
y
B
30
2
30
2
30
2 C
B
C
B 0
C
0
0
0
0
@
A
0
0
0
0
which is the condition
(15)
0
As y increases, prices are spread from 2 to 1 and 3. Thus when y = 0 - i.e., segmentation (13) - the distribution on
3 3 1
10 ; 5 ; 10
2 2 1
5; 5; 5 .
prices (1; 2; 3) is
on prices is
and when y takes its maximum value of
1
15 ,
i.e., in segmentation (14), the distribution
In Section ??, we examine in more detail the multiplicity of consumer surplus maximizing segmentations
in a continuum example.
In particular, we will see a "greedy" algorithm for constructing a consumer surplus
The Limits of Price Discrimination March 9, 2013
16
maximizing segmentation, like that in (14), where the monopolist is indi¤erent between the segment price, the
uniform monopoly price, and a continuum of price in between.
This segmentation generates a more disbursed
distribution of prices, while holding welfare constant.
2.4
Total Surplus (and Output) Minimizing Segmentations
Segmentation X attains the total surplus lower bound if
u
e (X) + e (X) =
.
Segmentation X has zero consumer surplus if the price is always greater than or equal to the consumer’s valuation,
i.e., u
e (X) = 0 and thus
xij = 0 if i < j.
(16)
Segmentation X is conditionally e¢ cient if, conditional on the total output that is sold, it is sold to those with
the highest valuation, i.e., there exists b
j such that
j > b
j ) xij = 0 for all i > j
j < b
j ) xij = 0 for all i
Total surplus must be at least the uniform monopoly pro…ts
.
j
The lowest output consistent with attaining
this surplus is to choose a conditionally e¢ cient allocation which generates total surplus
b 2 (0; 1] uniquely solve
then
qbi vbi +
K
X
j=bi+1
Q = qbi +
qj vj =
K
X
. Thus if we let bi and
,
(17)
qj
j=bi+1
is a lower bound. Segmentation attains the output lower bound if
K X
K
X
xij = Q.
i=1 j=i
Segmentation X is price increasing if the price is always greater than or equal to the uniform monopoly price, i.e.,
xij = 0 if i < i .
Now we have the following partial characterization of total surplus minimizing segmentations:
Lemma 2 (Necessary Conditions for Surplus Minimizing Segmentation)
The Limits of Price Discrimination March 9, 2013
17
1. If an equilibrium segmentation attains the total surplus lower bound, then it has zero consumer surplus and
is price increasing.
2. If an equilibrium segmentation has zero consumer surplus and is uniform price indi¤ erent, then it attains the
total surplus lower bound. If, addition, it is conditionally e¢ cient, then it attains the output lower bound.
For (1), the monopolist can guarantee himself the total surplus lower bound, so if the bound is attained, the
consumer must have zero surplus.
Suppose that there was a positive mass who bought the good at a price vk
strictly below the uniform monopoly price.
Since there is zero consumer surplus, there must be no consumers
with a valuation above vk . So the monopolist could make pro…ts above the uniform monopoly pro…ts by charging
vi in all segments other than the price vk segment.
For (2), zero consumer surplus and uniform monopoly pro…ts together imply that the total surplus lower bound
is attained. Zero consumer surplus and conditional e¢ ciency imply that the output lower bound is attained.
Note that zero consumer surplus, conditionally e¢ cient and price increasing imply - in our example - that the
segmentation has support on the price - valuation
0
0
B
B 0
B
B
B 0
B
B
B
@
pairs with crosses in the following matrix:
1
0 0 0 0
C
0 0 0 0 C
C
C
0 0 0 0 C
C
C
0 C
A
0
Theorem 2 (Surplus Minimizing Segmentation)
There exists an equilibrium segmentation which attains the total surplus lower bound, the output lower bound and
is price increasing.
We prove the result in the appendix. Again, there will be many ways of doing this. The algorithm used to
prove the Theorem in the Appendix works by …rst …xing mass along the diagonal (i.e., where price equals valuation)
according to the unique conditionally e¢ cient allocation that attains the total surplus lower bound. Assign zero
mass to prices below vbi , the lowest valuation allocated the object. For prices vi greater than or equal to vbi , assign
zero mass to (i) valuations strictly above vi (to ensure zero consumer surplus); and to (ii) valuations strictly above
vbi and strictly below vi (to ensure conditional e¢ ciency).
Finally, iterating downwards for valuations from vbi
to v1 , allocate consumers to prices in such a way as to maintain incentive compatibility. One can show that it
is always possible to do so, exploiting the fact that bi is chosen such that producer surplus in the zero consumer
surplus conditionally e¢ cient segmentation equals the uniform monopoly pro…ts which exceed the surplus from
charging any other price in the unsegmented market. This algorithm leads to the following segmentation in our
example:
The Limits of Price Discrimination March 9, 2013
0
18
0
B
B 0
B
B
B 0
B
B 4
B
@ 45
1
9
2.5
0
0
0
0
0
0
0
0
0
4
45
1
9
1
15
2
15
1
5
0
1
C
0 C
C
C
0 C
C.
C
0 C
A
1
5
0
Summary
The set V of attainable (producer surplus, consumer surplus) pairs attainable in market segmentations is convex.
Thus we have as an immediate Corollary of Theorems 1 and 2 that:
8
>
>
<
V = (u; ) 2 R2+
>
>
:
u
e (X)
0
e (X)
u
e (X) + e (X)
w
9
>
>
=
>
>
;
.
(18)
Our arguments in this section were constructive. There might be more abstract linear algebraic arguments
that could be used to establish the result. In Section 4.1, we will re-prove the results in this Section in the special
case of only two values with an insightful "concavi…cation" argument.
The proof is insightful in this two type
case, and can be used to extend the results to more general screening problems, although it is not clear that a
many type version of it o¤ers any insight over the constructive arguments in this Section.
3
Continuum of Values
We now extend the argument to a setting with a continuum of valuations, and construct segmented markets that
mirror those in the environment with a …nite number of valuations. Suppose then that a continuum of buyers,
identi…ed by their valuation, are distributed on the interval [v; v]
R+ according to a smooth distribution function
F with a density f . The total surplus in the market is then given by
w ,
Zv
vf (v) dv,
v
and let the uniform monopoly price p be de…ned as:
p , arg max fp (1
F (p))g ;
p
where we assume without loss of generality that there is a unique maximizer of the above pro…t maximization
problem. The monopoly pro…ts under the uniform price p are:
, p (1
F (p )) ,
The Limits of Price Discrimination March 9, 2013
19
and consumer surplus under uniform price monopoly is
u ,
Zv
(v
p ) f (v) dv,
p
and deadweight loss is
w
u
=
Zp
vf (v) dv.
v
In this Section, we focus on the consumer surplus maximizing market segmentation. However, the algorithm
that we presented earlier for the total surplus and quantity minimizing allocation can be constructed in a manner
analogous to the construction of the consumer surplus maximizing segmentation.
We now construct a market segmentation where the consumer surplus is w
and output is maximal and
equal to one. The segmentation will be analogous to that we constructed in the discrete case. For every price
p 2 [v; p ], there will be a market segment associated with that price p. Each segment will consist of a point mass
of consumers with valuation equal to the price of that segment.
In addition, there will be positive probability
of valuations strictly above the segment price p. These will be distributed proportional to the density f of the
aggregate market restricted to the interval (p; v]. The distribution of valuations in the segment p, denoted by
Fp (v) is therefore described by two parameters, ;
0:
8
>
0;
if v v < p;
>
<
Fp (v) =
;
if
v = p;
>
>
:
+ (F (v) F (p)) , if p < v v:
(19)
The mass of probability on p will be chosen so that the monopolist is indi¤erent between charging price p or the
uniform monopoly price p on that segment. Thus we require that the revenue for o¤ering price p and price p is
the same:
p = p (1
(F (p )
F (p)))
(20)
and that the segment has mass 1 :
+ (1
F (p)) = 1:
(21)
The unique solution to the conditions (20) and (21) is
=
p (1
F (p )) p (1 F (p))
;
p (1 F (p ))
=
p
p (1 F (p ))
and hence the distribution in the segment p is given by:
8
>
0;
if v v < p;
>
<
p(1 F (p))
Fp (v) =
1 p (1 F (p )) ; if
v = p;
>
>
:
F (v))
1 pp(1
v:
(1 F (p )) ; if p < v
(22)
The Limits of Price Discrimination March 9, 2013
20
The distribution over valuations, conditional on the received price recommendation p, Fp (v), thus has (i) a
mass point at p, (ii) has zero probability below p, and (iii) replicates (is proportional to) the prior distribution
above p. The mass point at p is just large enough to support the indi¤erence condition between p and p . It
follows that if the optimal price p is the unique optimum, then for every recommend price p, exactly two global
optima exist given the segment distribution Fp (v), and no further incentive constraints are binding.
To complete the description of the market segmentation, we need to specify the distribution of the buyers
across the price segments. We write H for the distribution (and h for the corresponding density) over the prices
[v; p ] associated with the segments. Now, given the upper triangular structure of the segments, and the fact that
in each segment the density of valuations is proportional to the original density, it is su¢ cient to insist that for
all v 2 [v; p ] ; we have
Z
v
v
p
f (v) h (p) dp + 1
p (1 F (p ))
v (1
p (1
F (v))
F (p ))
h (v) = f (v) .
(23)
In other words, the density f (v) of every valuation v in the aggregate is recovered by integrating over the continuous
parts of the segmented markets, as v is present in every segmented market p with p < v, and the discrete part,
which is due to the presence of the valuation v in the segmented market p with p = v. where the parts are given
by (23). In particular, the solution to h (v) for all v 2 [v; v ] will automatically guarantee that for all v 2 (v ; v]:
Z v
p
f (v) h (p) dp = f (v) .
(24)
F (p ))
v p (1
One could imagine that a construction of the distribution H might fail, or might fail to be representable by a
density.
In particular, suppose that we tried to construct H (p) from below, and thus attempted to have the
residual density of valuation v being absorbed by the segment p = v. Then it could be that we either run out of
probability to complement the probability of p = v with higher valuations necessary to construct (22), or we could
arrive at p = v and still have a positive residual probability Pr (v
v ) to allocate, which again would not allow
us to establish the speci…c segment p = p , Fp (v). However, as indicated already by the …nite valuation result
in Theorem 1, we now show that the condition (23) implicitly de…nes a separable ordinary di¤erential equation
whose unique solution, given the boundary condition H (p ) = 1, is given by:
h (p) =
(1
(1 F (p )) f (p) p
e
F (p )) p
(1 F (p)) p
Zp
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
,
(25)
and the associated distribution function:
H (p) = 1
e
Zp
s=0
sf (s)
ds
(1 F (p ))p
(1 F (s))s
.
(26)
Thus there exists an equilibrium segmentation that attains the upper bound on the consumer surplus in the model
with a continuum of valuations that mirrors the earlier result for the case of a …nite number of valuations.
The Limits of Price Discrimination March 9, 2013
21
Theorem 3 (Consumer Surplus Maximization)
There exists an equilibrium segmentation, represented by segments Fp (v) and a distribution over segments H (p),
given by (22) and (25) respectively, that attains the upper bound on consumer surplus and maximizes output among
all market segmentations.
The proof is in the Appendix. We can obtain the density and distribution function, given by (25) and (26)
directly from the di¤erential equation implicit in (23). Interestingly, the resulting di¤erential equation is solved
most directly by the method of substitution, which suggests a speci…c algorithm to implement the equilibrium
segmentation.
Consider the following information structure for the pricing decision of the monopolist. Suppose that nature
independently draws a tentative price recommendation r on the interval [v; p ] according to distribution function
G with associated density g, which is smooth except that there may be a mass point at p . If the tentative price
recommendation is below the buyer’s valuation, r
v, then the …nal price recommendation p equals the tentative
price recommendation r, i.e., p = r. If the tentative price recommendation r exceeds the buyer’s valuation, r > v,
then the …nal price recommendation p is set equal to the buyer’s valuation v, i.e., p = v.
We can then identify conditions on the distribution of G such that this information structure, which by construction, delivers an e¢ cient allocation and holds the monopolist down to his uniform price monopoly pro…ts.
We want a monopolist following price recommendation p 2 [v; p ) to be indi¤erent between the price recommendation p and the uniform monopolist price p , and to prefer either to any other price.
If the monopolist gets
price recommendation p, this happens either because the tentative price recommendation was p and the buyer’s
valuation exceeded it or because the buyer’s valuation was p and the tentative price recommendation was above
p. Thus the seller is indi¤erent to p and p if
[f (p) (1
G (p)) + g (p) (1
F (p))] p = g (p) (1
F (p )) p
(27)
This can be re-written as
f (p) (1
G (p)) p = g (p) [(1
F (p )) p
(1
F (p)) p]
or
g (p)
=
1 G (p)
(1
pf (p)
F (p )) p
(1
F (p)) p
:
(28)
Now, the …nal price distribution, or distribution over price segments, can be identi…ed with:
h (p) = [f (p) (1
G (p)) + g (p) (1
F (p))] ,
and using (28), it follows that we have the following simple relationship:
h (p) = g (p) (1
F (p ))
p
.
p
(29)
The Limits of Price Discrimination March 9, 2013
22
Now, remarkably if we substitute the expression on the right hand side of (29), then the sweeping up condition
on h (p), given by (23), exactly equals the indi¤erence condition (28) from which we can obtain as solution to the
di¤erential equation (28), the distribution function of recommended bids, G (p), and the distribution function of
market segments, H (p).
We observe that the distribution H (p) implements a particular set of segmented markets which achieve the
e¢ cient allocation with the largest possible consumer surplus. Clearly, it is not the unique market segmentation
which allows to maximize the consumer surplus and maintain the e¢ cient allocation. This observation follows
immediately from the fact that for any market segment p only two, distant prices, namely p and p , formed local
and global optima. In particular, we could add further local optimal between p and p (and hence di¤erent market
segmentations) without upsetting the e¢ ciency result or the resulting consumer surplus.
The Uniform Distribution
We illustrate the nature and variety of consumer surplus maximizing segmentation
when the distribution of the valuations in the aggregate market are given by the uniform distribution on the unit
interval [0; 1]. In this case, the monopoly price is p =
1
2.
We …rst consider the consumer surplus maximizing
segmentation we construct for the general case. The density of …nal recommended prices on 0; 12 is
e
h (p) =
(1
2p
1 2p
2p)3
;
(30)
and the associated distribution function is
H (p) = 1
1 p
e
1 2p
2p
1 2p
.
The distribution of valuations in the price p segment is now given by
8
>
0,
if v < p;
>
<
Fp (v) =
(1 2p)2
if v = p;
>
>
:
1 4p (1 v) if v > p:
(31)
(32)
Figure 3 displays the segmentation associated with the price p = 1=4; Fp (v). Thus, every segmented market
preserves the density and distribution of the aggregate market, but is truncated from below at v = p. At v = p,
the segmented market has a mass point which is su¢ ciently large to make the seller indi¤erent between selling at
the price p = v and at the optimal price p = p of the aggregate market.
Given the distribution of prices, we can then compute the interim expected utility of each type in the segmented
markets, and compare it with the interim expected utility in the aggregate market. To obtain these values, we need
to compute the conditional distribution of prices of the …nal recommended prices, given by H (p). Alternatively,
we can directly use the distribution of the initial recommendation, where we get
Z v
E [p jp v ] =
pg (p) dp + (1 G (v)) v.
0
The Limits of Price Discrimination March 9, 2013
F(v/p)
23
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
v
Figure 3: Segmented Market Fp (v) with p = 1=4.
The interim expected utility is then given by:
E [v
p jp
v] = v
Z
v
pg (p) dp + (1
G (v)) v:
0
We show in Figure 4 the expected prices paid by the buyer with valuation v, E [p jp
v ]. The expected is increasing,
rather than constant as it would be under the uniform monopoly price. In addition, the expected price is always
below the uniform price of the monopolist in the absence of segmentation.Similarly, we illustrate in Figure 5, the
E[p/p<v]
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
v
Figure 4: Expected prices paid by buyer with valuation v.
interim utility, conditional on purchase, E [v
p jp
v ], which is also everywhere larger than in the uniform price
monopoly. In the uniform price monopoly, the buyer with valuation v = p = 1=2 would receive a zero interim
The Limits of Price Discrimination March 9, 2013
24
utility, whereas in this surplus maximizing allocation it is strictly above 0:
v
0.25
0.20
0.15
0.10
0.05
0.00
0.0
0.1
0.2
0.3
0.4
0.5
E[v-p/p<v]
Figure 5: Expected interim utility of buyer with valuation v
The Uniform Distribution: A Di¤erent Market Segmentation
The information structure discussed above
guaranteed that the incentive compatibility is binding only at two price points, the monopoly price p for the
aggregate market and the recommended price p of the segmented market. By contrast, all the other prices are
strictly inferior in the assessment of the seller. Alternatively, we might seek to achieve the e¢ cient allocation by a
greedy like algorithm. Suppose we were to match a low valuation v (which equals the price of the segment p) with
as many high values as we can. This would lead to a distribution function over valuations in market segment p,
Fp (v):
v (1
Fp (v)) = p , Fp (v) = 1
p
,
v
and an associated density function:
fp (v) =
p
:
v2
(33)
Such a market segmentation would have the property that the seller is indi¤erent between o¤ering any price p0
between p and p . But in fact, the seller would be indi¤erent between p and any p > p . In other words, the above
distribution has unbounded support, namely v 2 [p; 1). However, let us now replace the (33) by the uniform
density above v
p and thus weaken the uniform incentive constraints for all values above p > p . Thus we have
at p = 21 , two segments of a distribution, the linear and the quadratic component:
8
<
1 vp
if v p
Fp (v) =
.
: (1 2p) + v 1 if v > p
2
The Limits of Price Discrimination March 9, 2013
25
We need to guarantee that they sum to one, after introducing the respective weights, with v = 1 :
(1
and hence
Fp (v) =
and associated density
2p) +
8
<
: 1
1
2
=1,
1
p
v
fp (v) =
if v
4p (1
8
>
>
< 0
>
>
:
p
,
v) if v > p
if
p
v2
= 4p,
0
v<p
if p
4p if
v
.
p
(34)
v>p
We can verify that is incentive compatibility to o¤er the price p of the segment, and he veri…cation is supported
by the virtual utility, which is given
1
v
8
0
if 0
Fp (v) <
=
: 2v 1 if p
fp (v)
v<p
v
.
(35)
1
We can now construct the segmented markets with weights w (v) with the property that the weights sum up
the density of the prior density:
Z
p
v
dv = 1;
p2
w (v)
0
(36)
and that the weights pick up the residual distribution as well, i.e. is su¢ ciently large, and hence the inequality:
Z 1
2
w (v) (v) dv 1:
0
From (36), we …nd that we need:
p
w (p) 2
p
2
Z
0
p
v
w (v) 3 dv = 0 , w (p) p2
p
2
Z
and hence we can set w (v) = 2 for all v, then we satisfy (36) for all p
Z p
v
2dv = 1.
2
0 p
p
w (v) vdv = 0,
0
1=2 :
Thus the greedy algorithm has a uniform prior distribution over all prices p = v between 0; 12 . Conditional on
selecting p the distribution is given by the density function (34) which we illustrate below for p = 1=4 :
The expected price to be paid is
1
Ew [v] = ;
4
(37)
as in the earlier construction, but just as before, higher types have to pay higher prices on average. The interim
utility of type v is computed by E [p jp
v] :
E [p jp
v] =
Z
0
v
p
p
2
2dp = v
2
v
3
The Limits of Price Discrimination 4March 9, 2013
26
f(v/p)
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
v
Figure 6: Conditional Density f (v jp ) with the greedy algorithm
and hence the interim utility is
v
1
1
2
v = v for all v < .
3
3
2
For all v > 12 , we have
E [p jp < v ] =
We can then plot the interim utility and get
u (v) =
and
w (v) =
8
<
: v
0
1
3v
: v
8
<
Z
1
3
0
1
2
1
2
1
p4p2dp = .
3
if 0 < v
1
2
1
2
v
1
if 0 < v
1
2
1
2
1
if
if
v
(38)
(39)
We compare the nature of the segmentation in the market with local incentive constraints and with global
incentive constraints. In Figure 7, the distribution of prices displays more variance in the segmented market with
global incentive constraints, where the density generated by the local incentive constraints (the humped and red
line) is more concentrated around the mean, and the density generated by the global incentive constraints (the
‡at and blue line) is uniformly distributed between [0; p ]. The expected prices paid by a buyer with valuation
v is then also di¤erent in the local and global incentives constraints. Correspondingly, we …nd that the expected
prices are initially higher for the local incentive constraints (red line) to be eventually overtaken by the global
incentive constraints.
We can …nally plot and compare the interim utility with the greedy algorithm, the locally segmented market
and the aggregate market in Figure 9. We then have the interim utility of the buyer with type v for the aggregate
market (black line), the segmented market with local indi¤erence (blue) and the segmented market with global
indi¤erence (red).
Thus the segmented market with local indi¤erence generates a interim utility with greater
The Limits of Price Discrimination March 9, 2013
h(p), k(p)
27
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
p
Figure 7: Density of O¤ered Prices under Local and Global Incentive Constraints
variance than the segmented market with global indi¤erence. In turn, the local incentive compatibility condition
increase the information rent less than the greedy algorithm, which leaves the agent with a constant proportion of
their value. But eventually, the local conditions give the agents more information rent, and so the high valuations
of the buyer receive larger information rent. Thus the global incentive compatibility leads to a larger variance in
the prices, but a smaller variance in the interim utilities. The ex ante expected price and the ex ante expected
net utility however are the same across the two di¤erent segments, and each one leads to the e¢ cient allocation.
This illustrates, that there is no unique solution to the consumer maximizing surplus, rather there are a number
of segmentations which all leads to the same ex ante utilities and revenue, but with very di¤erent implications for
the interim utilities and the distribution of prices.
4
Discrimination and Segmentation: A Second Approach
In the previous Sections, we derived the extremal segmentations by direct constructive arguments. These direct
arguments exploited the linearity of the allocated quantity, both in terms of the utility of the consumer, and in
terms of the revenue and the cost of the seller. We now present a di¤erent argument and route to establish these
results which does not rely at all on the linearity of the payo¤ functions.
This line of argument can then be
used to analyze and obtain explicit results for the quality discrimination model of Mussa and Rosen (1978), where
the cost of providing higher quality is convex. This second argument considers the complete information payo¤
functions, and then concavi…es the payo¤ functions. This argument explicitly identi…es when and how to construct
the segments. This concavi…cation argument is prominently used in Kamenica and Gentzkow (2011) to construct
optimal information structures from the point of view of a speci…c player. Here, we construct the entire concavi…ed
The Limits of Price Discrimination March 9, 2013
E[p/p<v]
28
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
v
Figure 8: Expected prices paid by buyer with type v
frontier by looking at the weighted sum of the payo¤s with positive and negative weights can be negative. Thus
we can construct the entire frontier of the social surplus, the e¢ cient, but also the ine¢ cient parts of the frontier.
While the concavi…cation argument can be established for the case of …nitely many types, here we shall present
the analysis for two types, in part as it will allows to geometrically represent the argument.
4.1
Third Degree Price Discrimination
We begin the analysis with the earlier model of a single unit demand, but now specialize to the case of binary
values, with 0 < vl
by
vh . We assume a constant marginal cost and, as before, normalize this to zero. We denote
be the proportion of low valuation consumers in the market. With binary values, the outcome of the uniform
price monopoly leads to an extreme allocation in terms of consumer surplus. If
price outcome minimizes the consumer surplus, and conversely, if
>1
<1
vl =vh , then the uniform
vl =vh , then the uniform price outcome
maximizes the consumer surplus. Finally, for the critical value of the prior, namely
b,1
vl
,
vh
any pricing policy that randomizes between vl and vh achieves the same pro…t level, but the consumer surplus
is increasing in the probability by which the low price is charged. Consequently, the producer surplus in the
aggregate market with a uniform price is given by:
8
< (1
) vh , if
( ),
:
vl ,
if
<1
1
vl
vh ;
vl
vh ;
The Limits of Price Discrimination 0.7
March 9, 2013
29
E[v-p/p<v]
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
v
Figure 9: Interim Utility of Buyer with Type v
and the consumer surplus is given by:
u( ) ,
8
<
0,
: (1
) (vh
if
<1
vl ) , if
1
vl
vh ;
vl
vh :
The dependence of the producer surplus and the consumer surplus on the share
of low valuation consumers is
illustrated in Figure 10.
We will consider the weighted sum of the consumer and the producer surplus given by, where we attach the
weights
u
and
, to the consumer and producer surplus respectively:
uu (
)+
( );
(40)
Now, in order to describe the entire boundary of the set of feasible distribution of the social surplus, it is necessary
to consider both positive as well as negative weights of
u
and
. For our formal analysis in this section, we
shall restrict attention to the case where the weight on consumer surplus,
u,
is strictly positive. However, the
analysis extends straightforwardly to zero or negative weight on consumer surplus and we use that analysis in our
illustrations.
With this restriction to
u
u
> 0, it is convenient to normalize the weight of the consumer surplus to be one:
, 1, and vary the weight of the producer surplus, setting
by:
w ( ) , u( ) +
( )=
We can then identify how the relative weight
8
<
:
(1
vl + (1
,
2 R+ , and thus the weighted sum is described
) vh ,
) (vh
if
<1
vl ) , if
1
vl
vh ;
vl
vh :
(41)
of the producer surplus leads to an optimal segmentation of the
market, as well as identify the the nature of the segmentation (Proposition 1), and in the process associate with
The Limits of Price Discrimination March 9, 2013
Figure 10: The producer surplus
30
( ) and the consumer surplus u ( )
every weight a locus on the border of the social welfare triangle (Proposition 2).
The weighted objective function given by (41) is linear and decreasing from 0 until it reaches the critical
fraction b , jumps up at b , and is linear and decreasing from b to 1.
weights, namely
For
= 1,
> 1 and 0 <
It is illustrated below for three distinct
< 1, in Figure 11.
> 1, the concavi…cation of w will be a straight line joining (0; vh ) and (1; vh ), so the optimal
segmentation will take the form of full information and perfect price segmentation. For
of w will be a straight line joining (0; vh ) and (b ; vl + (1
(b ; vl + (1
b ) (vh
b ) (vh
< 1, the concavi…cation
vl )), and then a straight line joining
vl )) and (1; vh ). In this case, the optimal segmentation will involve with segments with
proportions 0 and b or segments with proportions b and 1. More generally, we denote a segment with a fraction
of low valuation buyers by s , we have:
Proposition 1 (Welfare Weights and Segmentation)
The weighted welfare sum w ( ) = u ( ) +
1. for
( ) is maximized by:
> 1, pure segmentation or perfect price discrimination; the population is divided into segments s with
2 f0; 1g;
2. for
= 1, any segmentation using segments s with
3. for
< 1, mixed segmentation: if
2 f0g [ [b ; 1] ;
b , the population is divided into segments s with
b , the population is divided into segments s with
2 fb ; 1g.
2 f0; b g; if
We learned earlier that the social surplus triangle consists of three line segments, namely the e¢ cient frontier
The Limits of Price Discrimination March 9, 2013
31
Figure 11: Weighted welfare for
= 1,
> 1, and
< 1.
= w and
w
line W :
W , (u; ) 2 R2+ ju +
the minimal producer surplus line
;
:
, (u; ) 2 R2+ j =
and 0
u
w
;
and the minimal consumer surplus line U :
U , (u; ) 2 R2+ ju = 0 and
w
:
We can relate the entire frontier of the social surplus triangle to the weights in the welfare maximization problem
and recover the entire frontier by varying the weights.
Proposition 2 (Welfare Weights and Social Surplus)
A pair (u; ) is at the frontier of the social surplus triangle if and only if it is the solution to weighted welfare
maximization problem (41) for some weight . In particular, the maximizers consist of
1. W \ U = f(0; w )g if and only if
2. W if and only if
= 1;
3. W \
;
= f(w
> 1;
)g if and only if
< 1.
We gave an explicit characterization of the segmentation which solves the welfare maximization, as achieved by
the concavi…cation in Proposition 1, only for the case of strictly positive weight on consumer surplus but similar
analysis for all welfare weights (
u;
) give solutions that also only appeal to the use of three segments, namely
The Limits of Price Discrimination March 9, 2013
32
s0 ; s b ; s1 , that already appeared in Proposition 1. Moreover, all pairs (u; ) on the social surplus frontier, that are
not on the e¢ cient frontier, can be achieved by the convex combination of just two pairs, namely either fs0 ; s b g
or fs b ; s1 g. Thus, the solution for any set of weights given by (
u;
) is obtained by a linear combination of
just three segments, where importantly the composition of each segment is independent of the particular pair of
weights (
4.2
u;
) under consideration.
Second Degree Price Discrimination
We now generalize to a screening problem.
We will completely solve a two type problem originally studied by
Mussa and Rosen (1978). Suppose now that goods can produced at a variety of qualities q 2 R+ . Proportion
of consumers derive utility vl q from consuming a good of quality q, while proportion 1
where 0 < vl < vh .
The constant marginal cost of producing a good with quality q is
derive utility vh q,
1 2
2q .
Note that if the
consumer has valuation v, the e¢ cient quality to produce is v giving a total surplus of 12 v 2 to be divided between
the producer and the consumer.
An equivalent interpretation is that there is a single consumer who has valuation vl with probability
valuation vh with probability 1
.
and
Under this single consumer interpretation, we could also interpret q as
quantity and 12 q 2 as the cost of producing quantity q for the single consumer. The quantity interpretation is not
valid in the continuum interpretation, however, as now allocations to di¤erent consumers are not longer separable
(the case was studied in Maskin and Riley (1984)).
As before, we are interested in identifying all combinations of consumer and producer surplus that could arise.
With full information, the producer extracts all the surplus and gets the full gains from trade:
v( ) =
1
2
) vh2 ,
vl2 + (1
as the socially e¢ cient allocation has ql = vl and qh = vh .
If the producer knows nothing about the type of the consumer, then the well known optimal screening contract
has the producer "exclude" the low valuation type if the probability of the low type is su¢ ciently low, in particular
if
vl
,
vh
^,1
that is not to sell at all to low valuation consumers, and sell the e¢ cient quantity vh and full extract the surplus
of high valuation type, so that producer surplus is
1
(1
2
If the probability of the low type is higher, i.e.,
) vh2 .
^ =1
vl
vh ,
then the high type is again sold the e¢ cient
quantity, the surplus of the low type is set equal to zero and the high type is given enough rent to prevent her
The Limits of Price Discrimination March 9, 2013
33
mimicking the low type. In particular, the low type consumer receives quality
1
ql ( ) , vl
(vh
vl )
and pays
tl ( ) ,
1
vl
(vh
vl ) vl
while the high quality consumer receives the e¢ cient quality qh ( ) , vh and pays
th ( ) , vh2
(vh
1
vl ) vl
(vh
vl )
=
1
(vh
vl )2 + vh vl :
Thus the producer surplus is
( )=
and consumer surplus is
8
<
:
u( ) =
Now for any given
8
<
:
1
2
1
2
(vh
) vh2 ,
(1
vl )2
vh (vh
2vl )
0,
1
(vh
vl ) (vl
(1
1
if
1
if
1
) vh ) if
1
vl
vh ;
vl
vh ;
(42)
vl
vh ;
vl
vh :
(43)
, we know that payo¤s must be contained in a triangle as before: total surplus is not more
v ( ), consumer surplus is at least 0 and producer surplus is at least
when
if
( ). Figure 12 plots the feasible payo¤s
= 0:58, vl = 1 and vh = 2, which are described by the triangle.
By contrast, we will see the set of
equilibrium payo¤s that can arise in some form of segmentation are given by the shaded area, and hence by a
strict subset of the surplus triangle. The distribution of the surplus among buyers and the seller in the aggregate
market are denoted as the Bayes Nash equilibrium point. We describe the construction of the equilibrium surplus
set next.
First, suppose we want to identify the maximum possible consumer surplus across segmentations and the
segmentation that attains it.
Suppose that there were K segments, with proportion
k
of the population in
segment k, and proportion rk of segment k have the low valuation. We require that
K
X
k rk
=
(44)
k=1
and consumer surplus is
K
X
k u (rk )
(45)
k=1
Thus our problem is choose a segmentation to maximize (45) subject to (44).
But as noted by Kamenica and
Gentzkow (2011), the solution to this problem is found by considering the concavi…cation of u ( ), denoted by
u ( ).
The Limits of Price Discrimination March 9, 2013
34
Figure 12: Quality discrimination and an inclusive prior: the high end market.
Lemma 3 (Concavi…cation of Consumer Surplus)
The concavi…cation of u ( ) is given by
8
<
u ( ),
:
1
4
1
(vh
vl2
vl ) (vl
(1
if
;
) vh ) , if
;
(46)
with
,
2vh
2vh
2vl
.
vl
Proof. The payo¤ function u ( ) achieves its maximum at
r
vh
0
u ( )=0, =
vl
vh
;
and as
u00 ( ) =
for all
2
(vh
3
vl )2 < 0;
^ , it follows that the strictly positive segment of u ( ) is a concave function, and hence the concavi…cation
of u ( ) is achieved by the line segment
such that
satis…es:
, min f 2 R+ j
u ( ) ; 8 2 [0; 1] g ,
which is determined by:
= u( );
= u0 ( ) ,
(47)
and thus we get
=
,
2vh
2vh
2vl
and
vl
1
= vl2 ,
2
(48)
The Limits of Price Discrimination March 9, 2013
35
which establishes (46).
Figure 4.2 plots
( ) and u ( ), as well as their concavi…ed versions, namely
( ) and u ( ) with vl = 1 and
vh = 2.
pi
u
2.0
0.15
1.5
0.10
1.0
0.05
0.5
0.0
0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
alpha
The pro…t function
concavi…cation
alpha
( ) and its
The consumer surplus u ( ) and its
( )
concavi…cation u ( ).
;then it is optimal for the consumer to not segment the market and the consumer surplus
It follows, that if
is maximized in the aggregate market. But if
, then it is optimal for the consumer to have two segments,
of low types. The size of each segment must then be chosen to
one with no low types and one with proportion
match the total proportion.
We can generalize this argument to the problem of maximizing any weighted sum of consumer surplus plus
producer surplus.
uu (
)+
( );
(49)
As before, we will do our formal analysis for the case of strictly positive weight on consumer surplus and normalize
the objective to
w ( ) , u( ) +
so that
w ( )=
8
<
:
1
2
2
(vh
vl )2
vh (vh
(1
2vl ) +
( ),
) vh2 ,
1
(vh
vl ) (vl
(1
if
1
) vh ) if
1
vl
vh
vl
vh
(50)
The results extend straightforwardly to zero and strictly negative weights on consumer surplus, and we will use
this extension in our illustrations.
The concavi…cation of the weighted sum w ( ), denoted by w ( ), is therefore given by a linear segment that
connects w (0) with an interior point of the function w ( ), and the linear function has the form
l( ) ,
1 2
v +
2 h
,
The Limits of Price Discrimination March 9, 2013
36
and w equal to its concavi…cation:
l ( ) = w ( ) ; l0 ( ) = w 0 ( ) ,
which uniquely determines
and
as follows
,
((2
) (vh
(vh vl ) + (1
and
,
1 vl2
2
(2
2
vl ))
) vh
) vh2
(51)
:
(52)
Now, we can verify that the contact by the linear concavi…cation occurs in the right range:
((2
) (vh
(vh vl ) + (1
vl ))
) vh
1
vl
vh
=
vl
vh (vh
vh vl
vl ) + (1
) vh
>0
and that the concavi…cation spans the entire unit interval as soon as:
((2
) (vh
(vh vl ) + (1
vl ))
=1,
) vh
= 1:
Proposition 3 (Segmentation and Second Degree Price Discrimination)
The weighted welfare sum w ( ) = u ( ) +
1. for
2. for
( ) is maximized by :
> 1, pure segmentation; the population is divided into segments with s with
1, mixed segmentation; if
2 f0; 1g;
, the population is divided into segments s with
, there is no segmentation and the population is pooled in a segment s .
2 f0; b g; if
Using this characterization (and an omitted analysis of what happens when there is negative or zero weight on
consumer surplus), we can calculate the set of equilibrium consumer and producer surplus pairs. We plot these
for the case where vl = 1 and vh = 2 for di¤erent values of
. We shall focus in our discussion on the consumer
surplus maximizing segmentation and the comparison with the equilibrium surplus in the aggregate market. If
there are few buyers with a low valuation, then in the aggregate market, the seller will not o¤er a product to the
low valuation agents, we refer to this as the case of the exclusive prior. Hence in equilibrium the seller extracts
all the surplus from the high valuation buyers, and in consequence the equilibrium is socially highly ine¢ cient.
An important consequence of the exclusive prior is that any segmentation beyond the aggregate market will now
increase social surplus, and hence strictly increase the revenue of the seller, and weakly increase the surplus of the
buyers.
In turn, their surplus is maximized with the unique binary segmentation which uses s0 and s , where
we determined
in Lemma 3. But importantly, in the case of quality (or quantity) discrimination, any attempt
to increase the surplus of the buyers, and hence their information rent, leads to an ine¢ cient allocative decision
by the seller. In consequence, the e¢ cient frontier can only be reached with perfect segmentation s 2 fs0 ; s1 g,
The Limits of Price Discrimination March 9, 2013
37
Figure 13: Quality discrimination: the case of the exclusive prior
whereas any other segmentation, even if it increases the consumer surplus will lead away from the e¢ cient frontier.
This is illustrated for the case of
=
1
3
in Figure 13.
If we increase the proportion of low value buyers,
, beyond the critical value b at which the seller starts
o¤ering a low quality version of the product to the low value buyers, but remain below , then the equilibrium in
the aggregate market is not as ine¢ cient as before in the sense that the low value buyers are now longer excluded.
We refer to this as the case of the inclusive prior. Nonetheless, there are segmentations of the aggregate market,
in particular those involving s0 and s , that can increase both the revenue of the …rm and the surplus of the
buyers. But in contrast to the case of the exclusive prior, there now do exist segmentations which strictly improve
the revenue of the seller while lowering the consumer surplus. As before the e¢ cient frontier can be attained only
through perfect segmentation. This was already illustrated for
Eventually, as the proportion of low value buyers,
= 0:58 in Figure 12.
, increases, and in particular increases above
, i.e.
> , then the equilibrium in the aggregate market leads to the largest possible consumer surplus. In fact, any
segmentation now increases the revenue of the seller and strictly decreases the surplus of the buyers. We have
thus arrived at an environment where segmentation and hence additional information by the seller unambiguously
increases his revenue and decreases the consumer surplus. This is illustrated for
= 0:9 in Figure 14.
The Limits of Price Discrimination March 9, 2013
38
Figure 14: Quality discrimination with an inclusive prior: the middle end market.
5
Conclusion
Constant vs Increasing Marginal Cost
In our analysis of the discrete case in Section 2, the producer surplus
from segmentation X was given by
K X
K
X
e (X) =
xij (vj
c),
i=1 j=i
where we normalized the constant marginal cost c to zero. More generally, we could allow a more general cost
function so that C (q) was the cost of producing a total of q units. In this case, producer surplus would be
0
1
K X
K
K X
K
X
X
e (X) =
xij vj C @
xij A .
i=1 j=i
i=1 j=i
A non-linear cost function introduces externalities between price changes in di¤erent segments, since altering prices
in one segment alters the marginal cost of production in all segments. For this reason, the incentive constraints
must allow for multi-segment deviations, of which there are many. It continues to be the case, though, that there
is a uniform monopoly optimal price vi solving
0
1
0
1
K
K
X
X
@
qj A vi
C@
qj A
j=i
for all k, giving uniform monopoly pro…ts
and consumer surplus
j=i
0
=@
K
X
j=i
0
u =@
0
@
1
qj A vi
K
X
j=i
qj (vj
K
X
j=k
1
0
qj A vk
0
C@
K
X
j=i
1
C@
1
qj A
vi )A .
K
X
j=k
1
qj A
The Limits of Price Discrimination March 9, 2013
39
And there is an e¢ cient production decision, i.e., a unique i and 2 (0; 1] maximizing
0
1
K
K
X
X
qi vi +
qj vj C @ qi +
qj A
j=i+1
j=i+1
and, under perfect price discrimination, the monopolist would extract this surplus.
If marginal cost is strictly increasing, we can say a bit more about what is not possible. In particular, it
is impossible to achieve any improvement in total welfare while producer surplus equals
consumer surplus to 0 while holding total surplus down to
the monopolist to be held down to
, nor can we reduce
. To see why, …rst observe that it is impossible for
with a total output that di¤ers from the uniform price monopoly output. For
with strictly increasing marginal costs, pro…ts will be strictly concave between segmentations with di¤ erent total
output. Hence, if the monopolist attained surplus
with a di¤erent output level, he could deviate by randomly
o¤ering vi to a share of consumers, and pro…ts would rise.
Second, since the allocation is conditionally e¢ cient at the uniform monopoly price allocation, for total welfare
to increase it must be that output goes up. Hence, there cannot be any welfare improvement while the monopolist
is held down to
. It is possible for welfare to fall without a change in output because of misallocation. However,
the combination of zero consumer surplus and producer surplus of
, and output remaining at the uniform
monopoly price level cannot happen: if welfare falls and consumer surplus is zero, then some consumer with a
valuation less than vi must buy the good at his own valuation. If the monopolist charged vi on this segment,
revenue and output would be zero. Hence, the monopolist cannot be indi¤erent to charging vi on every segment,
since he could do strictly better by still selling to some of those with valuations below the uniform monopoly price.
Information and Segmentation It was the objective of this paper to study the impact of information and in
particular of additional information over and above the prior distribution on the e¢ ciency and the distribution of
the surplus in a canonical setting of optimal pricing. We showed that the impact of additional information can be
substantial, both in terms of the changes on the revenues and the consumer surplus. Importantly, the direction
of the change is in general di¢ cult to determine. By construction, any additional information can never hurt the
seller, but it can lead to increases in both the social and the consumer surplus, or it could lead to decreases in
social as well as consumer surplus, but the direction of change for social and consumer surplus could also point in
opposite directions. The range of these predictions is established without further restrictions on the nature of the
distribution in the aggregate market, and in particular does not rely on any regularity or concavity assumption with
respect to the aggregate distribution. This suggests that policy recommendations tailored to speci…c conditions of
the aggregate market are di¢ cult to justify as a wide range of informational structure are associated with a wide
range of changes in the distribution of surplus across buyers and sellers.
The Limits of Price Discrimination March 9, 2013
6
40
Appendix
Proof of Lemma 2.. Let i 2 f1; ::; Kg and
2 (0; 1] to solve
qbi vbi +
K
X
qj vj =
= vi
j=bi+1
K
X
qj .
(53)
j=i
Write bi and b for the unique solutions to this equation. In any conditionally e¢ cient, zero consumer surplus and
uniform monopoly pro…t segmentation, consumers with valuations strictly above bi must always purchase the good
at a price equal to their valuation, and consumers with valuation bi must purchase the good with probability b ,
paying their valuation if they purchase the good, and consumers with valuations below bi will not purchase the
good.
We now de…ne a particular conditionally e¢ cient, zero consumer surplus and uniform monopoly pro…t segmentation. For j > bi, let
let
and iteratively de…ne for j = bi
xij =
8
>
>
< 0, if i < j
>
>
:
qi , if i = j ;
0, if i > j
8
>
0, if i < bi
>
>
>
>
>
< b qbi , if i = bi
xibi =
xii (vi vbi )
>
(1
>
K
X
>
>
>
>
xll (vl vbi )
:
1; bi
l=b
i+1
b )qb, if i > bi ;
i
2; :::; 1
8
>
0, if i < bi
>
>
>
>
K
>
X
>
<
xik
vi xii vj
xij =
k=j+1
1 qj , if i
0
>
>
K
K
>
X
X
>
>
B
C
>
xlk A
@vl xll vj
>
:
l=b
i
k=j+1
bi
.
Since this segmentation satis…es feasibility (4) by construction, it remains to verify incentive compatibility. The
non-trivial conditions we need to check are that pro…ts cannot be increasing by deviating from price vi for i
bi
The Limits of Price Discrimination March 9, 2013
41
to a lower price. Consider the case where i > bi. First, observe that for each j = 1; ::; bi
1
0
K
K
K
K
X
X
X
X
@vl xll vj
xlk A =
vl xll vj
qk , by feasibility (4)
l=bi
l=bi
k=j+1
= vi
k=j+1
K
X
qk
vj
k=i
vj
K
X
qk
k=j
It follows that
for all i > bi
K
X
k=j+1
vj
K
X
qk , by construction of bi
qk , by incentive compatibility
k=j+1
= qj vj .
vi xii
vj
xij
1,
K
X
xik
k=j+1
j. Re-arranging this expression, we have
vj
K
X
xik
vi xii ,
k=j
verifying incentive compatibility for i > bi and j < bi. The same argument goes through with i = bi or j = bi, with
suitable allowance for the fact that xbb = b qb.
ii
i
Proof of Theorem 3.. We verify that the solution (25) of the density:
h (p) =
(1 F (p )) f (p) p
e
F (p )) p
(1 F (p)) p
(1
solves (55) the balancing condition:
Z v
p
f (v) h (p) dp + 1
p
(1
F (p ))
v
Zp
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
v (1
p (1
.
F (v))
F (p ))
(54)
h (v) = f (v) .
(55)
Thus inserting (54) into (55) we get:
Z
v
v
+
p (1
p
f (v)
p (1 F (p ))
(1
F (p )) v (1 F (v))
p (1 F (p ))
(1 F (p )) f (p) p
e
F (p )) p
(1 F (p)) p
(1
or
Z
v
v
(1
pf (p)
F (p )) p
(1
F (p)) p
e
Zp
s=0
(1 F (p )) f (p) p
e
F (p )) p
(1 F (p)) p
Zp
s=0
sf (s)
ds
(1 F (p ))p
(1 F (s))s
dp
Zp
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
= f (v) ,
sf (s)
ds
(1 F (p ))p
(1 F (s))s
dp = 1
+
e
Zv
s=0
sf (s)
ds
(1 F (p ))p
(1 F (s))s
.
The Limits of Price Discrimination March 9, 2013
42
So, after integration by parts, we get:
Z
v
v
(1
pf (p)
F (p )) p
(1
F (p)) p
e
Zp
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
dp = 1
So, if we de…ne
H (p) = 1
e
Zp
and so
(1
Z
v
0
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
.
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
then
h (p) = H 0 (p) =
e
Zv
pf (p)
F (p )) p
(1
F (p)) p
e
H 0 (p) dp = [H (p)]v0 = H (p)
Zp
sf (s)
ds
(1 F (p ))p
(1 F (s))s
s=0
H (0) .
The distribution function H (p) is everywhere continuous, and in particular does not have a mass point at p = p
as the integral in the exponential diverges, that is
Z p
sf (s)
lim
p!p
F (p )) p
(1
0 (1
F (s)) s
ds = 1.
For the divergence of the integral, it is su¢ cient to establish that the term inside the integral grows su¢ ciently
fast as p ! p :
(1
sf (s)
F (p )) p
(1
By the p test for divergence:
Z
1
0
F (s)) s
:
1
dx
xp
is convergent if and only if p < 1. It thus follows that the integral always diverges, (as it relies on the square
rather than the linear term, due to the …rst condition), and hence there is no mass point at the optimal price v .
Namely, we can approximate the above ratio, using the quadratic polynomial:
f (x) = f (x0 ) + f 0 (x0 ) (x
1
x0 ) + f 00 (x0 ) (x
2
x0 )2
and applying it to the function X (s) as de…ned below:
X (s) , (1
F (s)) s;
we get
X 0 (s) = (1
F (s))
f (s) s; X 00 (s) =
2f (s)
f 0 (s) s;
The Limits of Price Discrimination March 9, 2013
43
and thus we have the following approximation, using the fact the 0 th term and the 1 st term vanish, (the later
due to the …rst order condition):
pf (p)
(2f (p ) + f 0 (p ) p ) (p
p )2
;
and, the rate is quadratic rather, p = 2, rather than sublinear, p < 1, and hence the integral diverges as p ! p .
The Limits of Price Discrimination March 9, 2013
44
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