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., [email protected]. z Department of Economics, Princeton University, Princeton, U.S.A., [email protected]. x Department of Economics, Princeton University, Princeton, U.S.A., [email protected].

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

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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)

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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

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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

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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 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
> < 0 > > :

p

,

v) if v > p

if

p v2

= 4p,

0

vp

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 12 , we have E [p jp < v ] = We can then plot the interim utility and get u (v) = and w (v) =

8
, 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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