Monopoly Price Discrimination and Demand
Curvature
Iñaki Aguirre, Simon Cowan and John Vickers�
Abstract
This paper presents a general analysis of the e¤ects of monopolis-
tic third-degree price discrimination on welfare and output when all
markets are served. Su¢ cient conditions involving straightforward
comparisons of the curvatures of the direct and inverse demand func-
tions in the di¤erent markets are presented for discrimination to
have negative or positive e¤ects on social welfare and output. (JEL
D42, L12, L13)
This paper develops general conditions that determine whether third-
degree price discrimination by a monopolist serving all markets reduces or
raises output and social welfare, de
ned as the sum of consumer surplus and
pro
t. A
rm practising third-degree price discrimination uses an exogenous
characteristic, such as the age or location of the consumer or the time of
�Aguirre: Departamento de Fundamentos del Análisis Económico I, University of
the Basque Country, Avda. Lehendakari Aguirre 83, 48015-Bilbao, Spain (e-mail: in-
aki.aguirre@ehu.es); Cowan: Department of Economics, University of Oxford, Manor Road
Building, Manor Road, Oxford OX1 3UQ, UK (e-mail: simon.cowan@economics.ox.ac.uk);
Vickers: All Souls College, Oxford OX1 4AL, UK (e-mail: john.vickers@all-souls.ox.ac.uk).
Financial support from the Ministerio de Ciencia y Tecnología and FEDER (SEJ2006-
05596), and from the Departamento de Educación, Universidades e Investigación del Go-
bierno Vasco (IT-223-07) is gratefully acknowledged. We are grateful to a referee who,
along with very helpful comments, recommended that we combine three working papers
(available on request) into one, as well as to three referees for comments on one of the
papers. We would also like to thank Mark Armstrong, Christopher Bliss, Ian Jewitt,
Meg Meyer, Volker Nocke, Ignacio Palacios-Huerta, Joe Perkins, John Thanassoulis and
Michael Whinston for their comments.
1
purchase, to divide customers into separate markets. The monopoly price can
then be set in each market if discrimination is allowed. Moving from non-
discrimination to discrimination raises the
rms pro
ts, harms consumers
in markets where prices increase and bene
ts the consumers who face lower
prices. The overall e¤ect on welfare can be positive or negative. The main
aim of this paper is to provide conditions based on the shapes of the demand
functions to determine the sign of the welfare e¤ect. We also address the
classic question of the e¤ect of discrimination on total output, and the paper
combines new
ndings with existing results in a uni
ed framework.
The e¤ect of discrimination on welfare can be divided into a misalloca-
tion e¤ect and an output e¤ect. With discrimination output is ine¢ ciently
distributed because consumers face di¤erent prices in di¤erent markets. This
negative feature of discrimination may, however, be o¤set if there is an in-
crease in total output, which is socially valuable since prices exceed marginal
costs. Arthur Pigou (1920) proved that if all demand functions are linear
and all markets are served at the non-discriminatory price then total output
remains at the no-discrimination level, in which case discrimination is bad
for welfare. Joan Robinsons (1933) pioneering analysis, taken forward by
Richard Schmalensee (1981), showed how the curvature of demands deter-
mines the sign of output e¤ect. Hal Varian (1985) proved very generally
that a necessary condition for welfare to rise with discrimination is that total
output increases (see also Marius Schwartz, 1990).
In this paper we explore the welfare e¤ect directly using the technique
developed by Schmalensee (1981) and Thomas Holmes (1989) to analyze the
output e¤ect.1 Throughout it is assumed that at the non-discriminatory
price all markets are served with positive quantities, so price discrimination
does not open up new markets.2 To simplify the exposition, but without loss
of generality, we explore the case with two markets. The
rm is supposed
initially to be required to set the same price in both markets i.e. the price
di¤erence is constrained to be zero. As this constraint is relaxed, the
rm
moves towards the laissez-faire outcome with price discrimination. As this
happens, output and welfare will increase in what Joan Robinson (1933)
termed the weakmarket, where the discriminatory price is below the non-
discriminatory price, and decrease in the strongmarket. The question is
1Earlier applications of the method are by Wassily Leontief (1940) and Eugene Silber-
berg (1970).
2See Jerry Hausman and Je¤rey Mackie-Mason, 1988, Stephen Layson, 1994, and Victor
Kaftal and Debashi Pal, 2008, for analyses of price discrimination that opens new markets.
2
how overall welfare and total output vary as the price-di¤erence constraint
is relaxed.
Central to our analysis is a (commonly met) condition on demand func-
tions the increasing ratio condition which ensures that welfare varies
monotonically with the price-di¤erence constraint, or else has a single in-
terior peak. Given the increasing ratio condition, discrimination is shown
to reduce welfare if the direct demand function in the strong market is at
least as convex as that in the weak market at the non-discriminatory price.
Second, welfare is higher with discrimination if the discriminatory prices are
not far apart and the inverse demand function in the weak market is locally
more convex than that in the strong market: total output then rises while
the misallocation e¤ect is relatively small. Outside these cases, welfare
rst
rises but then falls as the price-di¤erence constraint is relaxed, so an inter-
mediate degree of discrimination would be optimal, and the overall e¤ect on
welfare of unfettered discrimination can be positive or negative. Its sign can
however be determined in important special cases: (i) when inverse demand
curvature is constant, welfare falls with discrimination if curvature is su¢ -
ciently below unity and rises if curvature is su¢ ciently above unity, and (ii)
when demands have constant elasticities, although total output rises with
discrimination (Iñaki Aguirre, 2006), welfare falls if the di¤erence between
the elasticities is no more than one. In parallel to the welfare analysis, we
also obtain new results on how discrimination a¤ects total output, which
rises if both inverse and direct demand in the weak market are more convex
than those functions in the strong market, but not if both inverse and direct
demand in the strong market are at least as convex as those in the weak
market.3
The broad economic intuition for why the di¤erence between the curva-
tures of demand in weak and strong markets is important for welfare and
output is as follows. A price increase when demand is concave has relatively
little e¤ect on welfare (the extreme form of concavity is when the demand
function is rectangular and there is no deadweight loss from monopoly pric-
ing). If at the same time price falls in a market with relatively convex de-
mand, there is a large increase in output and thus in welfare in that market.
This is the insight of Robinson (1933), who showed that total output rises
3These output results build on, and encompass, those of Robinson (1933), Schmalensee
(1981), Jun-ji Shih, Chao-cheng Mai and Jung-chao Liu (1988) and Francis Cheung and
Xinghe Wang (1994).
3
when discrimination causes prices to rise in markets with concave demands
and prices to fall in markets with convex demands, and David Malueg (1994)
explored further the relationship between the curvature of the demand func-
tion and the deadweight loss from monopoly pricing.4
The paper is organized as follows. Section I presents the model of monopoly
pricing with and without third-degree price discrimination. Section II con-
tains the welfare analysis using the price-di¤erence technique. The e¤ect of
discrimination on total output is considered in Section III. Section IV presents
the results of the welfare analysis using a restriction on how far quantities
can vary from their non-discriminatory levels, and considers the important
special case where demands have constant elasticities. Conclusions are in
Section V.
I. The Model of Monopoly Pricing
Amonopolist sells its product in two markets and has a constant marginal
cost, c � 0. The assumption of two markets is made for simplicity all
the results can be generalized to the case of more than two markets and
the method for doing this is discussed later. Utility functions are quasi-
linear. Demand in a representative market with price p is q(p), which is
twice-di¤erentiable, decreasing and independent of the price in the other
market. (To avoid notational clutter we omit subscripts where it is not
necessary to indicate which market is which.) The price elasticity of demand
is � � �pq0=q. The pro
t function in a market is � = (p � c)q(p). Assume
that
�00(p) = 2q0 + (p� c)q00 =
�
2 + (p� c)q
00
q0
�
q0 < 0;
so the expression in square brackets is positive and the pro
t function is
strictly concave.5 With strict concavity the second-order conditions hold
for the maximization problems that follow. De
ne �(p) � �pq00=q0 as the
convexity (or curvature) of direct demand, which is analogous to relative risk
aversion for a utility function and is the elasticity of the slope of demand. The
4See Glen Weyl and Michal Fabinger (2009) for a general analysis of demand curvature
and social welfare with imperfect competition.
5Part A of the Appendix discusses conditions that ensure strict concavity. See Babu
Nahata, Krzysztof Ostaszewski and Prasanna K. Sahoo (1990) for an analysis of price
discrimination when pro
t functions are not concave in prices.
4
Lerner index, the mark-up of price over marginal cost, is L(p) � (p�c)=p and
2 + (p� c)q00=q0 = 2�L� > 0 by strict concavity. Similarly the curvature or
convexity of the inverse demand function p(q) is �(q) � �qp00=p0 = qq00=[q0]2.
The two curvature measures are related to the price elasticity by � = �=�.
The values of � and of � play key roles in the analysis.
When the
rm discriminates, the
rst-order condition for its problem in
each market is
�0(p�) = q(p�) + (p� � c)q0(p�) = 0;
where p� > c is the pro
t-maximizing price and the star denotes the value
that applies with full discrimination. From the
rst-order condition comes the
Lerner condition for monopoly pricing L� = 1=��. Thus L��� = ��=�� = ��
and, with strict concavity, 2�L��� = 2��� > 0. The subscript w denotes the
weak market, where the discriminatory price is below the non-discriminatory
one (see below), and subscript s denotes the strong market, where the price
is higher with discrimination. The classi
cation of a market as strong or
weak is endogenous. It is assumed that both markets are served at the non-
discriminatory price a su¢ cient condition for this is that qw(p�s) > 0.
When the
rm cannot discriminate it chooses the single price p that
maximizes aggregate pro
t, which is de
ned by the
rst-order condition
�0w(p) + �
0
s(p) = 0. The
rst-order condition and the assumption that
both markets are served at the non-discriminatory price imply that �0w(p) =
qw(p)[1 � L(p)�w(p)] < 0 and �0s(p) = qs(p)[1 � L(p)�s(p)] > 0, so �w(p) >
�s(p). The weak market has the higher elasticity at the non-discriminatory
(or uniform) price. With strict concavity of each pro
t function it follows
that p�s > p > p
�
w: Social welfare W is the sum of consumer surplus and pro-
ducer surplus (or gross utility minus cost) so the marginal e¤ect of price on
social welfare in a market is dW
dp
= (p� c)q0(p), i.e. the e¤ect on the quantity
multiplied by the price-cost margin.
II. The E¤ect of Discrimination on Welfare
The method used by Schmalensee (1981), Holmes (1989) and Lars Stole
(2007) to consider the output e¤ect is adopted here to analyze the welfare
e¤ect. In the following section we use it to re-examine the output e¤ect.
Initially the
rm is not allowed (or is unable) to discriminate and thus sets
the uniform price p. Then the constraint on the
rms freedom to discrimi-
nate is gradually relaxed until the
rm can discriminate as much as it likes.
5
Our approach is to calculate the marginal e¤ect on welfare of relaxing the
constraint; if this keeps the same sign as more discrimination is allowed, then
the overall e¤ect of discrimination can be found.
In particular, we assume that the
rm chooses its prices to maximize
pro
t subject to the constraint that ps� pw � r where r � 0 is the degree of
discrimination allowed. The objective function is �w(pw)+�s(pw+r) and the
rst-order condition is �0w(pw) + �
0
s(pw + r) = 0 when the constraint binds.
When r = 0 the
rm sets the non-discriminatory price. As r rises more
discrimination is allowed, the price in the weak market falls and that in the
strong market rises:
(1) p0w(r) =
��00s
�00w + �00s
< 0; p0s(r) =
�00w
�00w + �00s
> 0:
When the constraint does not bind the
rm sets the discriminatory prices.
The marginal change in social welfare W as more price discrimination is
allowed is
(2) W 0(r) = (pw � c)q0w(pw)p0w(r) + (ps � c)q0s(ps)p0s(r):
A relaxation of the constraint alters prices and thus the quantities demanded,
and each additional unit of output has social value equal to the price-cost
margin in that market. For r > r� = p�s � p�w the marginal welfare e¤ect
is zero because the prices remain at the discriminatory levels. De
ne W 0(0)
and W 0(r�) as right- and left-derivatives respectively. The marginal e¤ect
on total output is Q0(r) � q0wp0w + q0sp0s so, following Schmalensee (1981), (2)
may be written as:
(3) W 0(r) = (pw � p)q0w(pw)p0w(r) + (ps � p)q0s(ps)p0s(r)| {z }
Misallocation e¤ect
+ (p� c)Q0(r)| {z } :
(Value of) output e¤ect
The
rst two terms equal zero at r = 0 and are negative for r > 0. Together
they represent the marginal misallocation e¤ect. The
nal term is the value
of the change in output. At the non-discriminatory price, because there
is no misallocation e¤ect, the marginal welfare e¤ect is proportional to the
marginal change in aggregate output. Integrating (3) over [0; r�] gives the
total welfare e¤ect as two negative terms (the total misallocation e¤ect) plus
6
(p� c) times the total output change. This con
rms that an output increase
is necessary for social welfare to rise.6
In our analysis a crucial role is played by
(4) z(p) � (p� c)q
0(p)
2q0 + (p� c)q00 =
p� c
2� L�;
the ratio of the marginal e¤ect of a price increase on social welfare to the
second derivative of the pro
t function. Substituting the comparative statics
results for prices, (1), into (2) and using (4) gives the marginal welfare e¤ect:
(5) W 0(r) =
� ��00w�00s
�00w + �00s
�
| {z }
>0
[zw(pw(r))� zs(ps(r))] :
The marginal welfare e¤ect thus has the same sign as [zw(pw(r))� zs(ps(r))].
The following assumption is made for the three propositions in this section.
The increasing ratio condition (IRC): z(p) is increasing in p in each
market.
This holds for a very large set of demand functions. These include:
functions that are linear; inverse demands with constant positive curvature,
including the exponential and constant-elasticity functions; direct demand
functions with constant curvature (whether positive or negative); probits
and logits (derived from the normal and logistic distributions respectively);
and demand functions derived from the lognormal distribution. Part B of the
Appendix presents su¢ cient conditions for the condition to hold and gives a
fuller list of the demand functions to which it applies. While the increasing
ratio condition holds very commonly and always it holds locally in the
region around marginal cost it is not universally applicable. For example
when inverse demand has constant negative curvature the condition does not
hold for high enough prices and di¤erent techniques are necessary to deal
with this case.
6It should be noted that the decomposition of the total welfare e¤ect into an output
e¤ect and a misallocation e¤ect is not unique. See Aguirre (2008) for a graphical analysis
based on a di¤erent decomposition.
7
Lemma Given the IRC, if there exists br such that W 0(r^) = 0 then
W 00(r^) < 0:
Proof. From (5)
W 00(r) =
� ��00w�00s
�00w + �00s
�
[z0wp
0
w � z0sp0s] + [zw � zs]
d
dr
� ��00w�00s
�00w + �00s
�
;
which is negative if W 0 = 0 because z0wp
0
w < 0 and z
0
sp
0
s > 0, and zw = zs
where W 0 = 0.
The IRC therefore implies thatW (r) is strictly quasi-concave, and thus is
monotonic in r or has a single interior peak. Only three outcomes are possi-
ble: either welfare, as a function of r, is everywhere decreasing, or everywhere
increasing, or it
rst rises then falls. Which holds depends on the signs of
W 0(0) and W 0(r�). First, if W 0(0) � 0, then W (r) is decreasing for r > 0
and discrimination therefore reduces welfare.
Proposition 1 Given the IRC, if the direct demand function in the
strong market is at least as convex as that in the weak market at the non-
discriminatory price then discrimination reduces welfare.
Proof. The Lemma implies that discrimination reduces welfare ifW 0(0) �
0. At the non-discriminatory price, where r = 0, pw�c = ps�c and Lw = Ls.
So from (5), [zw(p)� zs(p)] and henceW 0(0) have the sign of [�w(p)��s(p)],
the di¤erence in curvatures of direct demand, which is non-positive under
the condition stated in the proposition.
The condition on the di¤erence in the demand curvatures implies that
locally output does not increase, and since at the non-discriminatory price the
marginal misallocation e¤ect is zero a local output e¤ect that is negative or
zero implies that the welfare e¤ect has the same sign. The IRC then extends
this local result to all additional increases in the amount of discrimination,
and thus acts as a sign-preserver.
Proposition 1 encompasses the results of Simon Cowan (2007), who has
demand in the strong market being an a¢ ne transformation of demand in
the weak market, i.e. qs(p) =M +Nqw(p) whereM and N are positive (and
demand in both markets is zero at a su¢ ciently high price). At the same
8
price the direct demand functions, by construction, have the same curvature.
This is analogous to the result in expected utility theory that the coe¢ cients
of absolute and relative risk aversion, at a given income level, are invariant
to positive a¢ ne transformations of the utility function. An example is when
the direct demand functions have constant and common curvature, �, and a
special case is when both demand functions are linear (� = 0). Proposition
1 is more general because it allows the demand functions to have di¤erent
parameters or di¤erent functional forms, as in the following example.
Example 1: exponential and linear demands. Demand in market 1 is
q1(p) = Be
�p=b (with B and b positive), so �1 = 1; �1 = �1 = p=b > 0 and
p�1 = b + c. Demand in market 2 is q2(p) = a � p so �2(p) = p=(a � p);
�2 = �2 = 0 and p�2 = (a + c)=2. Proposition 1 applies if b > (a � c)=2,
which is the condition for market 1 to be the strong one. The weak market
is served with non-discriminatory pricing if (but not only if) a > b+ c.
If discrimination is to raise welfare, given the IRC, direct demand in
the weak market must be strictly more convex than demand in the strong
market at the non-discriminatory price. Only then does a small amount
of discrimination causes total output to rise. This is a local version of the
condition that for welfare to rise total output must increase.
Figure 1 shows, in a standard monopoly diagram, that as demand in the
weak market becomes more convex the welfare gain in this market from dis-
crimination rises. Initially inverse demand is the linear function p1(q) and
its associated marginal revenue curve is MR1(q) = p1(q) + qp01(q). The non-
discriminatory quantity is q and the discriminatory quantity is q1. Suppose
that demand becomes more convex, while retaining the same slope and po-
sition at the no