ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E14-14

The Welfare Effects of Oligopolistic Third-Degree Price
Discrimination When Own and Cross Price Elasticities
Are ConstantsbyTakanori Adachi
Takeshi Ebina

December 2014

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITYThe Welfare E¤ects of Oligopolistic Third-Degree
Price Discrimination When Own and Cross Price
Elasticities Are Constants
Takanori Adachiy

Takeshi Ebinaz

November 28, 2014
Abstract
This paper studies the welfare e¤ects of oligopolistic third-degree pricediscrimination with constant own and cross price elasticities of demand under
product di¤erentiation. We verify the robustness of Adachi and Matsushima’s(2014) result on social welfare under linear demands: price discrimination ismore likely to improve social welfare for a higher value of the cross price elasticity in the “strong” market (where the discriminatory price is higher thanthe uniform price). In contrast to Aguirre and Cowan’s (2013) results in thecase of monopoly, social welfare can be higher with price discrimination even
if the relative share of the strong market under uniform pricing is su¢ ciently
small or the own elasticity di¤erence between the two markets is also su¢ ciently small. Consumer surplus can also be higher with price discrimination
if the cross price elasticities are su¢ ciently low. This suggests that Adachi andMatsushima’s (2014) result on consumer surplus (price discrimination neverimproves social welfare) hinges on the linearity assumption.

JEL classi…cation: D43; L11; L13.
We are grateful to Takeshi Ikeda, Tamotsu Kadoda, Stephen King, Toshihiro Tsuchihashi andconference and seminar participants at the 41th Annual EARIE Conference, Daito Bunka University and the 2014 Autumn Meeting of the Japan Association for Applied Economics for helpfulcomments discussions. Adachi acknowledges a Grant-in-Aid for Young Scientists (B) from theJapanese Ministry of Education, Science, Sports, and Culture (24730205) as well as the Kikawada
Foundation for their …nancial supports. Ebina acknowledges a Grant-in-Aid for Scienti…c Research(C) (24530264) and a Grant-in-Aid for Young Scientists (B) (24730224) from the Ministry. Anyremaining errors are our own.
ySchool of Economics, Nagoya University, Chikusa, Nagoya 464-8601, Japan. E-mail:adachi.t@soec.nagoya-u.ac.jpzFaculty of Economics, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan.
E-mail: ebina@shinshu-u.ac.jp

1

Introduction

This paper studies oligopolistic third-degree price discrimination when demand ineach discriminatory submarket has constant own and cross elasticities to examine animportant question since Pigou (1920) and Robinson (1933): under which conditions
third-degree price discrimination raises social welfare. In many applied studies of
welfare e¤ects of third-degree price discrimination, researchers typically assume thatdemand in each discriminatory submarket belongs to the same functional family. In
fact, linear demand is often assumed.1 In other theoretical studies, researchers consider nonlinear demands in a nonrestrictive way. For example, Aguirre, Cowan,
and Vickers (2010) consider the curvatures of submarket demands to synthesize theexisting studies2 of output and welfare e¤ects of monopolistic third-degree pricediscrimination. Notably, Aguirre, Cowan, and Vickers’(2010) Proposition 2 statesthat if the inverse demand in the weak market is more convex than the inverse demand in the strong market, then price discrimination raises social welfare. However,no comparable characterization has yet come to in the case of oligopoly with nonlinear demands.3 Thus, this paper aims to take one step forward to the study ofoligopolistic third-degree price discrimination by considering one of the most “popular” classes of nonlinear demand, namely, “log-linear” (Davis and Garcés (2010,
p.447)) demands that have constant own and cross elasticities. It also studies therobustness of Adachi and Matsushima’s (2014) study of oligopolistic third-degree
price discrimination with linear demands.
To understand Adachi and Matsushima’s (2014) necessary and su¢ cient condition for oligopolistic third-degree price discrimination to improve social welfare,consider the situation where there are two geographical markets: one is a hot resortarea, and the other is a city area. There are two beverage companies, and they
compete by selling their own (one) product (such as cola) in each market. Resale
1See, e.g., Layson (1988), Adachi (2002, 2004, 2005), and Bertoletti (2004).
See, e.g., Schmalensee (1981), Varian (1985), Nahata, Ostaszewski, and Sahoo (1990), Schwartz(1990), Layson (1998), Cowan (2007), Cowan (2012), and Cowan (2013). Armstrong (2006), Stole(2007), and Liu and Serfes (2010) are excellent surveys on third-degree price discrimination.
3
See, e.g., Holmes (1989), Corts (1998), Dastidar (2006), and Galera and Zaratiegui (2006) for
studies of oligopolistic third-degree price discrimination.
2

1

by non…rm agents for arbitrage is assumed impossible. The cost of production andsales is assumed common for both markets (for example, there are no di¤erencesin transportation costs between the markets). Although the …rms are symmetric(i.e., homogeneous), consumers are not indi¤erent (except the marginal consumers)
between the two products (given that the …rms’prices are the same). In fact, horizontal product di¤erentiation makes some consumers prefer one …rm’s product tothe other’s, and vice versa for the other consumers.4
Adachi and Matsushima (2014) …nd that price discrimination raises social welfare if and only if the degree of substitution between the two products in the “strong”
market (where the discriminatory price is higher than the uniform price) is su¢ ciently higher than in the “weak”market (where it is lower).5 If consumers care lessabout the …rms’brands in the hot resort area (because when people are thirstier it
is natural that they should be less concerned about which brand), then price discrimination may improve social welfare. One might think that substitution in theweak, not the strong, market should be su¢ ciently higher because a …ercer levelof competition (due to a higher level of substitution) in the weak market increases
the aggregate output more. However, weakening the misallocation e¤ect caused byprice discrimination is more important than this output e¤ect: if substitution is suf…ciently high in the strong market, then the price increase by price discriminationis smaller, which leads to a less number of consumers in the strong market give up
consumption after price discrimination is introduced. This is bene…cial for social
welfare because, on average, those who consume a product in the strong market
have higher willingness to pay than those in the weak market do both under uniform
4

The source of horizontal product di¤erentiation may vary. At the extreme, two similar products
may be recognized as very di¤erent by consumers due to advertising by manufacturers. To quote
Tremblay and Tremblay (2005, p.173), “[b]ecause their colas taste very much alike, Coke and Pepsi
use advertising to segment the market by creating images that appeal to di¤erent consumers.
Coke pursues an image of traditional family values, while Pepsi presents a more youthful andrebellious image. This strategy bene…ts both …rms by strengthening brand loyalty and reducing
price competition.”
5
The usage of “strong” and “weak” markets is in the tradition since Robinson (1933). A more
precise statement for the necessity and su¢ ciency is that given the other parameter values, thesubstitution parameter in the strong market exceeds a threshold (see Adachi and Matsushima’s(2014) Proposition 1).

2

pricing and under price discrimination.
Notice that the argument so far does not rely on the linearity of demands. We
thus conjecture that this result and the intuition by Adachi and Matsushima (2014)
on social welfare (their Proposition 1) are robust for nonlinear demands. In Adachi
and Matsushima’s (2014) formulation, the degree of substitution in each market ischaracterized by one constant parameter (because of linearity), which is separablefrom other variables and parameters. In equilibrium, this parameter, with normalization, coincides with the cross price elasticity, though it is not the elasticity per se.
In particular, the normalization needs to take into account the slope of the linear
demand, which itself is less relevant to the marginal conditions. Furthermore, thelinearity restriction imposes the own price elasticity to be always one in equilibrium.
Does the intuition above survive under nonlinear demands? By investigating constant own and cross price elasticities of demand, we provide a positive answer for thisconjecture. In contrast, Adachi and Matsushima’s (2014) result on aggregate consumer surplus (their Proposition 2) may depend on the linearity of demands. Adachi
and Matsushima’s (2014) Proposition 2 shows that price discrimination always lowers aggregate consumer surplus. This is true even if it raises social welfare. This
result implies that …rms ‘squeeze’all extra surplus generated by welfare-improving
price discrimination described as above. Is this still true if market demands are
nonlinear?
In this paper, we extend Aguirre and Cowan’s (2013) analysis on monopolistic third-degree price discrimination with constant elasticity demands6 to the caseof di¤erentiated oligopoly.7 As in Adachi and Matsushima (2014), we express thedegree of product di¤erentiation in a submarket by one parameter: in the present
6

Aguirre and Cowan (2013) explain the reasons why the well-known results on third-degree pricediscrimination and welfare (such as Schmalensee (1981), Varian (1985), Aguirre, Cowan and Vickers(2010), and Cowan (2012)) are less applicable to the case of constant elasticity demand. Aguirre
and Cowan (2013) also emphasize that Aguirre, Cowan and Vickers’(2010) su¢ cient condition for
their Proposition 2 does not hold if submarket demands belong to the class of constant elasticity
demands because they are all convex with respect to the own price.
7
One thing to keep in mind in an analysis of third-degree price discrimination with linear
demands (as in Adachi and Matsushima (2014)) is to guarantee conditions for all submarkets tobe open under uniform pricing (the issue of ‘market opening’). However, under constant elasticity
demands, all submarkets are necessarily open under uniform pricing because there are no intercepts.

3paper, in contrast to Adachi and Matsushima (2014), it is the cross price elasticity
itself. Aguirre and Cowan (2013) show that price discrimination can raise social
welfare if the output share of the strong market under uniform pricing ( in theirand our notation) is su¢ ciently large and the (own) elasticity di¤erence between thetwo markets is su¢ ciently large ( in their and our notation, which is the elasticity
di¤erence between the two markets). Aguirre and Cowan (2013) also show thatprice discrimination can raise consumer surplus under stricter conditions. Noticethe similarity between Aguirre and Cowan’s (2013) su¢ cient condition and Adachi
and Matsushima’s (2014) necessary and su¢ cient condition. First, a …ercer levelof competition in the strong market would push the amount of production in thatmarket. Second, Adachi and Matsushima’s (2014) Proposition 1 implies that theelasticity di¤erence must be su¢ ciently high. In essence, the substitution parameter
plays an important role to a¤ect the equilibrium elasticity in each submarket. Thus,in equilibrium, oligopolistic …rms can been seen as monopoly where strategic interactions are already incorporated: the intuition for welfare improvement in the caseof oligopoly is similar to that in the case of monopoly once strategic interactions
are taken into account. We also verify that consumer surplus can also be higher
with price discrimination if he cross price elasticities are su¢ ciently low. This suggests that Adachi and Matsushima’s (2014) result on consumer surplus hinges on
the linearity assumption.8
The rest of the paper is organized as follows. The next section presents themodel. We then provide welfare analysis in Section 3. Most of arguments are basedon numerical analysis. Section 4 concludes the paper.

2

The Model

Consider J (2) oligopolistic …rms producing (horizontally) di¤erentiated products

to compete in price to sell their products (directly) to consumers. Each …rm produces
8

Aguirre (2011) studies the situation where a multimarket …rm and a local …rm compete in one
market while the multimarket is a monopolist in another market. As in Adachi and Matsushima(2014), Aguirre (2011) assumes the linearity of demands and shows that price discrimination can
improve social welfare in this setting as well.

4and sells only one type of product, which is interpreted as the …rm’s brand. The
whole market can be segmented into independent M (2) separate sub-markets(hereafter, just called markets if there arises no ambiguity) according to identi…able
signals (e.g., geography, age, and gender). If a …rm implements uniform pricing, the…rm’s price that consumers face is common across all markets. On the other hand,
if a …rm price discriminates across markets, consumers may face di¤erent unit pricesof the …rm’s product, depending on which market they belong to. We assume that
resale of a product of the price discriminating …rm across markets is not possible.
In this paper, we consider symmetric …rms as in Holmes (1989), and thus assume
the marginal cost is common for all …rms. In addition, we assume that the marginal
cost is constant, c > 0. For further simplicity, we mainly work on the case of two…rms (A and B) and two markets in the following analysis. Speci…cally, indicesi; j 2 fA; Bg are used for …rms, and index m 2 fs; wg is used for markets (s denotes(the set of) the strong markets and w the weak markets; these meanings will beclear below). Notice that, however, our main results can be extended to the caseof J (

3) …rms and M (3) markets as long as the …rms are symmetric, and thefollowing s and w are considered as (arbitrary) two representatives of all markets.
The demand function of …rm i in market m is given byiqm(pim ; pjm ) = am (pim ) "m (pjm ) m ,where ai > 0 is a measure of market size, "m > 1 is the constant own price elasticity
ii(notice that (@qm=@pim )(pim =qm)="m ) andm , which is assumed to be less than"m (the next paragraph explains the reason for this restriction) is the constant crossiiprice elasticity (notice that (@qm=@pjm )(pjm =qm)=m ), which captures the degreeof product di¤erentiation (note that our demand form is equivalent to the followingifamiliar form of log-linear demand: ln qm= ln amassumption of identical …rms requires that "m and"m ln pim +jm ln pm ).

Them are common for all …rms.

Note here that "m per se indicates that how many percent of the customers
leave the …rm if it raises its price by one percent, but not that how many of themswitch to the other …rm. However, it is seen that the other …rm gains

m percentmore of the existing customers as new customers. Thus, as a response to the one
5percent increase of the …rm’s price, ("mm ) percent of the customers leave themarket (i.e., purchase no products), andm percent of them switch to the other…rm (i.e., now purchase the other …rm’s product). As a natural restriction on the
demand, ("mIfm ) should be positive (thus, "m >9m ).

m = 0, the products produced by two …rms are independent in each market,

in the sense that one …rm’s demand is not a¤ected by the other …rm’s price. In otherwords, no marginal consumers are better o¤ by switching to the other …rm, and thus
they leave the market if the price goes up. In this case, the two …rms behave as an
identical monopolist in each market: the demand function is identical (with rescaling) as the one in Aguirre and Cowan’s (2013) analysis of monopoly. If
the products are substitutes. As

m > 0,

m approaches to "m , competition in market m

becomes …ercer. In the extreme case ofm"m , the marginal consumers all switch

to the rival: the two products are homogeneous and (almost) perfect substitutes.
In the present paper, we consider the case of substitutes only (i.e.,

m is positive):

complementarity is assumed away as opposed to Adachi and Matsushima’s (2014)
analysis with linear demands. This is because consumer surplus de…ned in the next
section assumes that consumers are segment into three groups: (i) those who purchase product A, (ii) those who purchase product B, and (iii) those who purchasenothing. If complementarity between two products is allowed, it is necessary to consider another type of consumers: those who purchase both products. As consumer
surplus used in this paper is a ‘naive’one, namely, the integral of the positive difference between the inverse demand and the price, it is less obvious to de…ne proper
consumer surplus in the case of complementarity. Thus, we simply assume

m > 0

10

for m 2 fs; wg throughout the paper.

Following Aguirre and Cowan (2013), we assume that the elasticity in market w
9

Holmes (1989) shows that with any symmetric price under oligopoly p, a …rm’s price elasticity is equal to the sum of the industry-demand elasticity and the cross-price elasticity. Thei
industry demand in market m is given by 2qm(pm ; pm ), and the industry-demand elasticity isii(pm =2qm )(2dqm =dpm ) = "mm.
10Adachi and Ebina (2014c) argue that there is no repsentative consumer’s utility that can beconsistent with the log-linear demand system. Thus, in contrast to the case of linear demands, wecannot use the representative consumer’s utility to derive consumer surplues under the log-linear
demands,6

is greater than that in market s: "w = "s + where> 0 is the own price elasticity

di¤erence (more precisely, this condition that "w > "s makes market w the weakmarket).
We consider two regimes, uniform pricing (r = U ) and price discrimination(r = D): under uniform pricing, …rms set a common unit price for all separate
markets. Under price discrimination, they can set a di¤erent price in each market.11j(p0 ; p00 ) = qmj (p00 ; p0 ) to focus
Furthermore, we impose the symmetry on demands qm

on a symmetric equilibrium where all …rms set the same price in one market. WithA
(p; p).
little abuse of notation, let qm (p) = qm

2.1

Price Discrimination

First, suppose that …rm i can discriminate a price pim in each market m. The pro…tfunction of …rm i from market m is given byiijim (pm ; pm ) = (pmc)am (pim ) "m (pjm ) m .

From the …rst-order condition of pim , the equilibrium discriminatory price is obtainedby@ im= am (pim ) "m (pjm ) m + (pim@pim"m
) (pim ) =c,"m 1
where superscriptc)am ( "m )(pim ) "m 1 (pjm ) m = 0

denotes the equilibrium under price discrimination.12 Here, thediscriminatory price in the symmetric equilibrium is (a bit surprisingly) independentof the cross price elasticity,

m , and coincides with the monopolistic discriminatory

11

Note that all markets are served under both regimes: market opening is not an issue withconstant elasticity demands because a positive number of consumers demands the product even if
the price is tremendously high.
12Note that the second-order condition is satis…ed at the equilibrium price because@ 2 im<0@(pim )2,

2pim + ("m + 1)(pimc) < 0 , pim <"m + 1c,"m 1which implies that the …rst-order conditions attains the unique solution.

7

price in Aguirre and Cowan (2013). To understand why, the …rst-order condition in
general is written byi@qm(pi ; pj ) = 0@pim m m
i
pi
c
qm
=pim
, mi
=
,
i =@pi
pm
@qm
m

i
qm
(pim ; pjm ) + (pimc)

which, known as the Lerner condition, essentially implies that the competing …rms’
problem can be seen as the monopolist’s problem under the residual demand given
the other …rms’ prices. The right hand side is the inverse of the …rm’s own priceelasticity. It is in general a function of pim and pjm . In our demand speci…cation,
however, …rm i’s own elasticity is independent of pjm (moreover, it is a constant, "m ,which is also independent of pim ). This also makes …rm i’s optimal price independentof its belief about pjm .
Accordingly, …rm i’s output in market m, the aggregate output in market m,
and the aggregate output in the industry are"m(Qm ) = 2am(Q) = 2

("m

"m

i
(qm
) = am

X

1

,
("m

"m
"m
am

m=s;w

m)

c

1

m)

,

c
("m

"m
"m

1

m)

,

c

respectively. The equilibrium pro…t of …rm i from market m is
( im )(pim )ic (qm) = am c1 ("mand thus, the total pro…t of …rm i is written byXX( i)( im ) =am c1 ("m
m=s;w

2.2

m=s;w

m)

m)

("m ) ("m m ),("m 1)1 ("m m )("m ) ("m m ).
("m 1)1 ("m m )Uniform Pricing

Next, consider the case of uniform pricing. The pro…t function of …rm i is given by
X
X
i
i j
i
i j
i
i
i j
i
(p
;
p
)
+
(p
;
p
)
=
(p
c)
q
(p
;
p
)
=
(p
c)
am (pi ) "m (pj ) m .
s
w
m
m=s;w

m=s;w

8

By the …rst-order condition of pi , we have13@( is +
@pi

i
w)

=

X

am (pi ) "m (pj ) m + (pic)am ( "m )(pi ) "m 1 (pj ) m = 0.

m=s;wNotice that the …rst-order condition is reformulated as the Lerner condition underuniform pricing:
pi

c
pi

1
= P
i
i j ,
m=s;w "m qm (p ; p )Pii jm=s;w qm (p ; p )where the right hand side is no longer independent of pj . Here, the elasticity that…rm i takes into account is the average sum of the own elasticities over markets
weighted by the outputs. The symmetric equilibrium, where pi = pj , satis…es
p0 c
1
=
,
p0
"(p0 )
where

(1)

P
0 ("m
m)
m=s;w "m am (p ),"(p ) = P
0 ("mm)
m=s;w am (p )0and superscript 0 denotes the equilibrium outcome under uniform pricing. Therefore,…rm i’s output in market m, the aggregate output in market m, and the aggregate
output in the industry are
i 0(qm) = am qm (p0 ; p0 ) = am (p0 ) ("m

m)

,

i 0
(Qm )0 = 2(qm) = 2am (p0 ) ("m m ) ,X
X(Q0 ) =(Qm )0 = 2
am (p0 ) ("m
m=s;w

m)

,

m=s;w

13

This pro…t function is not necessarily quasi-concave, so that it may have several peaks. Similararguments for the optimal price with uniform pricing (pi )0 also hold as in Aguirre and Cowan(2013). Thus, we make almost the same assumptions and (pi )0 satisfying the …rst-order condition
has maxima:
AA@qm@ 2 ( is + iw )@ 2 qm
i
=
2
+
(p
c)
@(pi )2@pA@(pA )2
X="m am (pi ) "m 1 (pj ) m
m=s;w

9

(1

"m )

"m + 1
c < 0.
pi

respectively. The di¤erences between the quantities under price discrimination andunder uniform pricing are given by"i 0) = am
(qm

i
i
)
= (qm
qm

(Qm )0 = 2am

Qm = (Qm )
Q=2

X

am

m=s;w

"

"m

"m
"

1

1

m)

(p0 ) ("m

c
("m

"m
"m

("m

"m

("m

"m

1

;

m)

c

(p0 ) ("m(p0 ) ("m

#

m)

c

m)

#

m)

m)

#

;

;

respectively.
Now, following Formby, Layson and Smith (1983) and Aguirre and Cowani 0(2013), we normalize c so that p0 = 1. In this case, (qm) = am , (Qm )0 = 2am ,and Q0 = 2(as + aw ). The averaged elasticity is also simpli…ed as"(1) =" s as + " w aw.
as + aw

Given Equation (1) and p0 = 1, the marginal cost satis…esc=

as ("s1) + aw ("was " s + aw " w

1).

Each …rm’s equilibrium aggregate output under uniform pricing is given by
as + aw . Let the share of the strong market under uniform pricing be de…ned byas =(as + aw ). Analogously, the share of weak market is de…ned by 1=

aw =(as + aw ). As a further normalization, we also assume that as + aw = 1. Thus,
under this normalization am denotes the relative share of market m under uniform
pricing. The above equation can be written as:c="s + (1)
1,"s + (1)

which is less than p0 = 1.
In the literature of third-degree market price discrimination, a market is calledstrong if the discriminatory price is higher than the uniform price, and it is called
weak if the opposite is true. Now,
(ps )

=

"s
"s

"s + (1)
11"s + (1)
10"2s + (1) "s "s2"s + (1) "s "s (1)
0> 1 = p (because > 0)"s "w + (1) " w "s>"s "w + (1) " w "s+== (pw ) > c.

Thus, market s and market w are veri…ed to satisfy the de…nition. Note here thatas

! 0, (ps ) ! 1 and (pw ) ! 1.
Finally, the equilibrium pro…t of …rm i from market m is(pim )0( im )0i 0) = (1c (qmc)am ,and thus, the total pro…t of …rm i is written by
(

i 0

)

X

( im )0 = (1c)(as + aw ) = 1

c.

m=s;w

Thus, the pro…t changes between the two regimes is
= (

i

)

(

= c1 ("m

3

m)

i 0

)
X

am ("m ) ("m m )("m 1)1 ("m m )m=s;w

1 + c.

Welfare Analysis

In the case of monopoly with constant elasticity submarket demands, Aguirre andCowan (2013) show that price discrimination raises social welfare if both the share of
the strong market under uniform pricing ( )14 and the elasticity di¤erence between
markets ( ) are su¢ ciently high. If either parameter is further high, price discrimination raises consumer surplus as well. More precisely, Aguirre and Cowan (2013)
show that ifW

0 then> 1. Intuitively, it is necessary forlarge for a nonnegative welfare change. If

to be su¢ ciently

is large, it means that the price elas-ticity in the weak market is su¢ ciently large relative to that in the strong market.
In other words, the strong market is su¢ ciently price inelastic relative to the weak14
Due to the normalization, the share of the strong market under uniform pricing, which isclearly an endogenous variable, can be expressed by one parameter, .

11market. Similar to the argument in Adachi and Matsushima’s (2014) analysis ofoligopoly, this is bene…cial to weaken the distortion in the strong market (i.e., theoutput decrease in the strong market is kept small relative to the output increase in
the weak market). In this section, we argue that

> 1 is not necessary for pricediscrimination to raise social welfare in the case of oligopoly. As expected, the twocross-elasticity parameters,s andw , play an important role.

First, note that under symmetric equilibrium, the equilibrium demand for …rm
i in market m is given byiqm(pm ; pm ) = am (pm ) "m (pm ) m= am (pm ) ("mm),

where pm is the discriminatory or the uniform price. This provides the inverse
demand function in symmetric equilibrium for …rm i:
1"mpm (qm ) = am

1
m

qm "m

m

.

Following Aguirre and Cowan (2013), let SWmr be social welfare in market m =s, w under regime r = U , D. Then, because of the symmetry,15
Z qm
r
1
1
"m
r
m
q "m m c dq.
SWm = 2am
0Therefore, a per-…rm change in social welfare is written as
SW
2

"s

1

= as"s

s

"s

=

"s

s

s

"s

qs0

1
"w

w

"w

w

1

cqs + aw"w

s

"s
"s

"w

1

1

s

11
"s c

s

)

1
"s

s

(

[qs ] "s

s

"s

"s

+(1

1

s

1

as"s
(

"s

s

1

s
s

"s

w

"w
1

+ cqs0

aw"w

1

"s

s

"s
"w 1 1
"w c

15

"w

w

w

w

1

"w

w

"w

1

[qw ] "w
1

w

s
s

"w

w

c

1

"s

"w
"w

11
"s c

w
w

qw0

1

c

w 1
w

cqw

"w
"w

w 1
w

"s

s

+ cqw0)+c"w 1 1"w c

Adachi and Ebina (2014c) argue that the log-linear demand system as employed in this paper
cannot be generated from the representative consumer’s utility. Thus, consumer surplus used inthis paer is not linked to consumers’utility.

12

"w

w

+c

)

=

"s
+

"

1

s

"s
"s
"

1
"s +

"s

"s + (1)"s + (1
)
1
"s +
"s +

1

w

1
"s + (1

1

1

1

s

2"s
"s

"s

1

s

"s +

"s + (1)"s + (1
)
1
1

)

1

s

"s +

1

w

w

1

#
2"s + 2
"s +

1

w

#

.

Similarly, per-…rm changes in consumer surplus, pro…t, and output are1CS"s= as"s s
2
"s
1

+aw"w

"s

s
s

[qs ] "s

1

"w

1

s

p s qs

s

"w

w

w 1
w

[qw ] "wp w qw
"w
1
w
1
"s
s 1
"s
s
as"s sqs0 "s s + p0 qs0
"s
1
s
1
"w
w 1
"
w
w
aw"w w
qw0 "w w + p0 qw0
"w
1
w
"s
"s 1 "s + (1)="s1"s "s + (1
)
1
s
+

1
"s +

+1

2

w

w

("s
"s

s

1
s)
1

1

"s +
1 "s + (1)"s +"s + (1)1(1)("s +
w)
,
"s +
1
w

CS
2
"s 1 "s + (1)="s"s "s + (1)
11,
"s + (1
)
=

s

"s +

s

1

W
2

"s

s

1

+

1
"s +

"s +
"s +

1

"s + (1)"s + (1
)
1

"s +

and
Q
=
2

"s

1
"s

"s + (1)"s + (1
)
1

"s

s

+(1

)

"s +
"s +

1

"s + (1)"s + (1)
1"s +respectively. With a little abuse of notation, we consider these per …rm measures
below, and denote them as

SW ,

CS,

and

Q.

We now conduct a number of numerical analyses of changes in social welfare,consumer surplus and output from uniform pricing to price discrimination More
13

w

1,

w

1

speci…cally, we investigate how the cross price elasticities,s andw , a¤ect welfarechanges from uniform pricing to price discrimination. In particular, our results below
are in accordance with Adachi and Matsushima (2014): the greaters and the lessw (i.e., competition is …ercer in the strong market than in the weak market) the

more likely a positive change in social welfare (i.e.,
Now, we consider the case of

SW > 0).

1, where price discrimination never improvessocial welfare in the case of monopoly (see Aguirre and Cowan (2013)). The following
examples show that in the case of oligopoly social welfare can be higher with pricediscrimination even if

1. Table 1 shows numerical values for Figures 1 and 2.
"sCase 1 2

0:9 f0:05; 0:95g

Table 1: Parameter Values (for1)

First, Figure 1 shows that social welfare under price discrimination is higher inthe lower right area below the boundary (for= 0:95 and 0:05). This example showsthat, in contrast to the case of monopoly analyzed by Aguirre and Cowan (2013), theelasticity di¤erence, , does not always have to be large for price discrimination tobe bene…cial for social welfare under oligopoly. In line with Adachi and Matsushima(2014), the cross price elasticity in the strong market,

s , must be su¢ ciently high.

Note also that the relative share of the strong market under uniform pricing, , canbe relatively small.
However, it is veri…ed that a change in consumer surplus is negative for all ( s ,w ) ifis equal to either 0:95 or 0:05. Figures 2 and 3 depict the area ofand that of

Q > 0 for

= 0:95, 0:05, respectively. It seems thatnecessary for

SW > 0 in general.

>0

Q > 0 isNow, we consider numerical values in Table 2. Figure 4 is the corresponding…gure.
"s
Case 2 2

f0:9:0:3g 0:95

Table 2: Parameter Values (for141)

Figure 1: Area for

SW > 0 (in the case of "s = 2 andFigure 2: Areas for SW > 0,
= 0:9 and = 0:95).

> 0 and15= 0:9).

Q > 0 (in the case of "s = 2 andFigure 3: Areas for SW > 0,
= 0:9 and = 0:05).

Figure 4: Area for> 0 andQ > 0 (in the case of "s = 2 andSW > 0 (in the case of "s = 2 and16= 0:95).

Based on these numerical examples above, we conjecture the following resultson the condition for

W > 0 hold in general: price discrimination is more likely to

improve social welfare ( W > 0)
1. as the equilibrium share of the strong market under uniform pricing ( ) becomes larger (with other parameters held constant),
2. as the own price elasticity di¤erence ( ) becomes larger,
3. as the cross price elasticity in the strong market ( s ) becomes larger, and4. as the cross price elasticity in the weak market ( w ) becomes smaller.
In particular, the third and the fourth points correspond to Adachi and Matsushima’s (2014) necessary and su¢ cient condition in their Proposition 1 whichroughly states (in the notations of the present paper) that there exists a thresholdfors , and for a larger

fare. Notice here thats than the threshold price discrimination raises social wel-

W is not monotonically increasing in

s in either case. Thisshould be further investigated to establish results similar to Adachi and Matsushima(2014).
Now, we analyze the case of> 1 to see whether price discrimination canraise consumer surplus. First, numerical values in Table 3 consider di¤erent valuesof the own price elasticity in the strong market, "s . In Fugure 5, the area where
price discrimination raises social welfare is right below the boundary for each "s . In
contrast to the case of

1 above, price discrimination can raise social welfare

as Figure 6 shows: a change in consumer surplus is positive in the area left below
the boundary for each "s . It appears that boths andw must be su¢ ciently small

for price discrimination to raise consumer. It is conjectured that if
SW > 0 (andsocial welfare

CS > 0 thenis necessarily positive). Notice that for a positive change inw can be kept relatively small as long as

would be probably the reason why the region of ( s ;w is su¢ ciently large. Thatw ) where price discrimination

raises social welfare is contained in the region where price discrimination raises social
welfare.
17

"sCase 3 f1:05; 1:2; 1:35g

3

0:95

Table 3: Parameter Values (forFigure 5: Area for

SW > 0 (in the case of> 1)= 3 and= 0:95).

Finally, …xing the value of "s , we consider the e¤ects of di¤erent values ofandseparately. First, Table 4 indicates di¤erent values for the di¤erences in theown price elasticities. Interestingly, in Figure 7, the boundaries (corresponding to
SW = 0) are quite homothetic to changes in . Figure 8 shows that the region of( s;w ) where consumer surplus is higher under price discrimination is larger with= 4 than with= 3 (price discrimination never raises consumer surplus when

= 2). As Figure 9 (wherecase most favorable for a positive> 0 for any ( s ;w )) shows in the case of the

SW ( = 4), consumer surplus is higher underprice discrimination for a wide range ofs as long asw

is kept small (i.e., lesscompetition in the weak market). This would be probably because the own priceelasticity in the weak market ("s + ) is already su¢ ciently large, suggesting thatcompetition may work against consumer surplus (due to the lack of coordination) if
18

Figure 6: Area for

CS > 0 (in the case of= 3 and= 0:95).

the industry as a whole faces a su¢ ciently elastic demand in the weak market.
"sCase 4 1:2 f2; 3; 4g 0:95

Table 4: Parameter Values (for> 1)Lastly, we consider the numerical values in Table 5 to see the e¤ects of di¤erent
shares of the strong market. Figure 10 depicts the areas where price discrimination
raises social welfare. As expected, a higher

is favorable for a positive change

in social welfare. However, Figure 11 shows that a change in consumer surplus isnegative for any ( s ;w ) if= 0:5 or 0:95. In particular, the case of= 0:5 satis…es> 1. Thus, we predict that a higher value of the own elasticity di¤erence ( )is important then a higher share of the strong market for price discrimination to
improve consumer surplus.
"s
Case 5 1:2

3 f0:05; 0:5; 0:95g

Table 5: Parameter Values (for
19

> 1)

Figure 7: Area for

SW > 0 (in the case of "s = 1:2 and= 0:95).

Figure 8: Area for

CS > 0 (in the case of "s = 1:2 and= 0:95).

20

Figure 9: Areas for
and = 0:95).

SW > 0,Figure 10: Area for> 0 andQ > 0 (in the case of "s = 1:2,

SW > 0 (in the case of "s = 1:2 and

21

= 3).

=4

Figure 11: Area for

4

CS > 0 (in the case of "s = 1:2 and= 3).

Concluding Remarks

This paper has investigated the welfare consequences of oligopolistic third-degree
price discrimination with constant own and cross price elasticities of demand. We…nd that the key parameters for price discrimination to improve social welfare andeven consumer surplus are the cross price elasticities. If this parameter in the strong
market is su¢ ciently large (i.e., competition in the strong market is …erce) withthe corresponding parameter in the weak market kept su¢ ciently small, then pricediscrimination is more likely to be preferable. In comparison to the case of monopoly
analyzed by Aguirre and Cowan (2014), price discrimination can improve social
welfare even with the parameter values with which it does not under monopoly.
This result is consistent with Adachi and Matsushima’s (2014) on social welfare
with linear demands. In addition, our result that consumer surplus can be higher
with price discrimination shows that Adachi and Matsushima’s (2014) result on
consumer surplus (price discrimination never improves social welfare) hinges on the22

linearity assumption.
For future research, it is important to explore the conditions for price discrimination to improve social welfare and consumer surplus with general nonlinear demands.
As in Adachi and Ebina (2014a,b) in the context of cost pass-through (see, e.g., Bulow and Klemperer (2012) and Weyl and Fabinger (2013)) in vertical relationships,
one would be able to conduct welfare analysis with exponential demands, logistic
demands, and type I extreme demands.16

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