ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E17-5

Output and Welfare Implications of
Oligopolistic Third-Degree Price
Discrimination
byTakanori Adachi
Michal Fabinger

August 2017

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Output and Welfare Implications of Oligopolistic
Third-Degree Price Discrimination∗
Takanori Adachi†

Michal Fabinger‡

July 31, 2017

Abstract
Using estimable concepts, we provide sufficient conditions for price discrimination to lower or raise aggregate output and social welfare under symmetrically differentiated oligopoly with general demand functions and costdifferences across separated markets. Assuming that all markets are open under uniform pricing, we show that if the markup ratio in the strong market(where the discriminatory price is higher than the uniform price) relative tothe weak market (where it is lower) is sufficiently large under uniform pricing,then social welfare will be lower if price discrimination is allowed. It is also
shown that if either the conduct ratio, the pass-through ratio, or the markup
ratio is sufficiently small in the strong market under price discrimination, thenit raises social welfare.
Keywords: Third-Degree Price Discrimination; Differentiated Oligopoly; Social Welfare; Pass-through.
JEL classification: D21; D43; D60; L11; L13.

∗
We are grateful to Nicolas Schutz for helpful comments. Adachi acknowledges a Grant-inAid for Scientific Research (C) (15K03425) from the Japan Society of the Promotion of Science.
Fabinger acknowledges a Grant-in-Aid for Young Scientists (A) (26705003) from the Society. Allremaining errors are our own.
†
School of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan.
E-mail: adachi.t@soec.nagoya-u.ac.jp‡
Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 1130033, Japan. Fabinger is also a research associate at CERGE-EI, Prague, Czech Republic. E-mail:fabinger@e.u-tokyo.ac.jp.

11Introduction

In this paper, we extend Aguirre, Cowan, and Vickers’ (2010) arguments of thewelfare effects of monopolistic third-degree price discrimination to the case of (symmetric) oligopoly. In particular, we emphasize the role of pass-through as in Weyland Fabinger (2013) and Adachi and Fabinger (2017). Furthermore, we allow costdifferentials across discriminatory markets as in Chen and Schwartz (2015). Assuming that marginal costs are constant and that all markets are open under uniform
pricing, we show that if the markup ratio in the strong market (where the discriminatory price is higher than the uniform price) relative to the weak market (where it
is lower) is sufficiently large under uniform pricing, then social welfare will be lower
if price discrimination is allowed. It is also shown that if either the conduct ratio,
the pass-through ratio or the markup ratio is sufficiently small in the strong market
under price discrimination, then it raises social welfare.
In almost all of the theoretical studies on price discrimination, researchers (manually) assume that there are no cost differentials across discriminatory markets tofocus on the demand side. However, in many real-world cases of price discrimination, cost differentials are quite often observed, not to mention the typical example
of a first-class seat and an economy-class seat (see Phlips 1983, pp. 5-7). In the narrow definition of price discrimination, this is not price discrimination because they
are considered different products. However, airlines are arguably motivated to offerdifferent types of seats because they aim to exploit consumer surplus by making
use of heterogeneity among consumers. Thus, ideally, a theoretical analysis of pricediscrimination should also allow cost differentials across discriminatory markets.
Even if costs differ across markets, sellers, in reality, may have to be engaged
in uniform pricing due to the universal service requirement, fairness concerns fromconsumers, and so on. Following Robinson (1933), we call one market s (strong),
where the equilibrium discriminatory price, ps , will be higher, if price discriminationis allowed, than the equilibrium uniform price, p, and the other w (weak), where theopposite is true, and the equilibrium discriminatory price is denoted by pw .1 In the1

In this paper, price discrimination is present when ps > pw . As Clerides (2004, p. 402) argues,

2

case of monopoly with constant marginal costs, Chen and Schwartz (2015) derive
sufficient conditions for consumer surplus to be higher under differential pricing. To
ensure that the strong market is indeed strong when cost differentials are allowed,
it is sufficient to assume that the marginal cost in the strong market is higher thanin the weak market: cs > cw (though cs should not be too much higher than cw ).
Then, under uniform pricing, the markup in the strong market p − cs is smaller thanthe markup in the weak market p − cw . Differential pricing allows the monopolist
to sell more products in the weak market which is more efficient than the othermarket. Chen and Schwartz (2015) find that while differential pricing with no costdifferentials (third-degree price discrimination in a traditional manner) tends toincrease the average price after differential pricing is allowed, differential pricing
with cost differentials does not. As in Chen and Schwartz (2015), this paper does
not have to make an explicit assumption on cs and cw as long as the second-orderconditions are satisfied and a large discrepancy between cw and cw does not change
the order of the discriminatory prices from the one with no cost differentials.2
We also emphasize that our sufficient conditions are related to estimable concepts. Interestingly, own- and cross-price elasticities per se do not play an importantrole in welfare evaluation. Our theoretical predictions, equipped with an estimable
framework, would be utilized to understand the mechanism behind an empiricalresult. For example, in their empirical analysis of within-store brand competition,
Hendel and Nevo (2013) show that social welfare is higher under third-degree pricediscrimination than with the case with no discrimination. However, it is not clear
what mechanism makes this empirical result. Although welfare evaluation is ultimately an empirical matter, one still wishes to know more about which force derives
once cost differentials are allowed, “there is no single, widely accepted definition of price discrimination.” To understand this issue, let mcs and mcw be equilibrium marginal costs in markets s
and w, respectively. Then, there are two alternative definitions. One is the margin definition:
price discrimination occurs when ps − mcs > pw − mcw . The other one is the markup definition:
price discrimination occurs when ps /mcs > pw /mcw . In this paper, we employ the simplest definition. As long as cost differentials are sufficiently small, these differences will not alter the results
significantly because these three definitions are equivalent if mcs = mcw .
2
In the context of reduced-fare parking as a form of third-degree price discrimination, Flores
and Kalashnikov (2017) characterize a sufficient condition for free parking (drivers receive a pricediscount in the form of complentary parking while pedestrians do not) to be welfare improving.

3the result. Thus, this paper also aims to fill the gap between theoretical predictions
and the empirical literature of price discrimination, where researchers often have toremain agnostic about the mechanism behind the result.
The rest of the paper is organized as follows. Section 2 presents our basic modelwith symmetric firms and constant marginal costs. Then, we derive output andwelfare implications in Section 3. Section 4 concludes the paper.

2

Symmetric Firms with Constant Marginal Costs

For ease of exposition, we, following Holmes (1989) and Aguirre, Cowan, and Vickers(2010), consider the case of two symmetric firms and two separate markets or consumer groups (simply called markets hereafter). Extending the following analysis
to the case of J (≥ 3) symmetric firms and M (≥ 3) separate markets is straightforward. As explained above, we call one market s (strong), where the equilibrium
discriminatory price will be higher than the equilibrium uniform price, and the otherw (weak), where the opposite is true. Two firms, A and B, have an identical cost
structure in each market. We assume that they have a constant marginal cost ineach market m, cm ≥ 0. In the spirit of Chen and Schwartz (2015), cs and cw canbe different.
In market m = s, w, given firms A and B’s prices pA,m and pB,m , the representative consumer consumes qA,m > 0 and qB,m > 0, and her (net) utility is written
as Um (qA,m , qB,m ) − pA,m qA,m − pB,m qB,m , where Um is twice continuously differen2
tiable, ∂Um /∂qjm > 0, ∂ 2 Um /∂qjm> 0, j = A, B, and ∂ 2 Um /(∂qA,m ∂qB,m ) < 0.

The direct demands in market m are derived from the representative consumer’sutility maximization: ∂Um (qjm , q−j,m )/∂qjm − pjm = 0, which leads to firm j’s
demand in market m, qjm = xjm (pjm , p−j,m ). We assume that xjm is twice continuously differentiable.

The corresponding inverse demand can be written as

pjm = pjm (qjm , q−j,m ). Because of the assumptions on the utility, firm j’s demand in
market m falls as its own price increases (∂xjm /∂qjm < 0), and it rises as the rival’s
price increases (∂xjm /∂q−j,m > 0; the firms’ products are substitutes). We assume4

that for a consumer’s perspective firms are symmetric: Um (q 0 , q 00 ) = Um (q 00 , q 0 ) for
any q 0 > 0 and q 00 > 0. Then, the firms’ demands in market m are also symmetric:
xA,m (p0 , p00 ) = xB,m (p00 , p0 ) for any p0 > 0 and p00 > 0. Because the firms’ technologies
are also identical, we, throughout this paper, focus on symmetric Nash equilibrium.
Under the regime of uniform pricing, the equilibrium uniform price for both markets
is p. If price discrimination is allowed, the equilibrium discriminatory prices are p∗s
in the strong market and p∗w in the weak market, and functional and parametricrestrictions are imposed to assure that p∗s > p > p∗w .3
We define the demand in symmetric pricing by qm (p) ≡ xA,m (p, p). Another interpretation of qm (p) is: both firms take 2qm (p) as the joint demand, ‘cooperatively’
choose the same price (behaving as an ‘industry’), and divide the joint demand
equally to obtain qm (p). Note that:
0
qm
(p) =

∂xA,m
∂xA,m
+
(pA , p)(p, pB )
.
∂pA
∂pB
pA =p
pB =p
{z
} |
{z
}
|
<0

>0 if substitutes

0(p) to be negative, we assume that |∂xA,m (p, p)/∂pA | > ∂xA,m (p, p)/∂pB .
Thus, for qm

Note also that by symmetry the following relationship also holds:
∂xA,m
∂xB,m
0
(p, p) = qm(p) −(p, p),| {z }
∂pA
∂pA
|
{z
}
|
{z
}
industryownstrategic effects

which corresponds to Holmes’ (1989) equation (4). This exchangeability is to key
in Holmes’ (1989) derivation below. Intuitively, under symmetry, each firm treats
the industry demand qm (p) as if it is its own demand. Thus, how a firm’s pricing
behavior affects its own demand as an industry demand has the following two effects:
a small decrease in pA by firm A by deviating from the ‘coordinated’ price p (i) not
only raises its own demand by ∂xA,m /∂pA as the residual monopolist (taking therival’s pricing as fixed; intrinsic effects), (ii) firm A can now also obtain some of the3

However, see Nahata, Ostaszewski, and Sahoo (1990) for an example of all discriminatory
prices being lower than the uniform price with a plausible demand structure under monopoly. In
the case of oligopoly, Corts (1998) show that best-response asymmetry, by which firms differ inranking strong and weak markets, is necessary to all discriminatory prices to be lower than theuniform price (“all-out price competition”).

5consumers originally attached to firm B, and this amount is ∂xB,m /∂pA (strategic
effects).
Under symmetric pricing, we define, following Holmes (1989, p.245), the price0
elasticity of the industry’s demand by εIm (p) ≡ −pqm(p)/qm (p). This corresponds

to D in Weyl and Fabinger (2013, p.542), and it should not “be confused with theelasticity of the residual demand that any of the firms faces.” Similarly, the ownprice and the cross-price elasticities of the firm’s demand are defined by εFm (p) ≡−(p/qm (p))(∂xA,m (p, p)/∂pA ) and by εCm (p) ≡ (p/qm (p))(∂xB,m (p, p)/∂pA ), respectively. Then, Holmes (1989) shows that under symmetric pricing, εFm (p) = εIm (p) +εCm (p) holds. This implies that the own-price elasticity must be greater than the22cross-price elasticity (εFm (p) > εCm (p)). Here, ∂ xA,m (p, p)/∂(pA ) can be positive,

zero or negative. Following Dastidar’s (2006, p.234) Assumption 2 (iv), we assume
that ∂ 2 xjm (p, p)/∂p2j + ∂ 2 xjm (p, p)/∂pA ∂pB ≤ 0.
Firm j’s profit in market m is written as π jm (pjm , p−j,m ; cm ) = (pjm −cm )xjm (pjm , p−j,m ).
As in Dastidar’s (2006, pp.235-6) Assumptions 3 and 4, for the existence and theglobal uniqueness of pricing equilibrium under either uniform pricing or price discrimination, we assume that for each firm j = A, B, ∂ 2 π jm /∂p2jm < 0, ∂ 2 π jm /(∂pjm ∂p−j,m ) >0, and −[∂ 2 π jm /(∂pjm ∂p−j,m )]/[∂ 2 π jm /∂p2jm ] < 1 (see Dastidar’s (2006) Lemmas 1
and 2 for the existence and the uniqueness). We then define the first-order partial
derivative of the profit in market m, evaluated at a symmetric price p, by∂p π m (p; cm ) ≡∂π jm (pjm , p−j,m ; cm )∂pjm
pjm =p−j,m =p= qm (p) + (p − cm )∂xA,m(p, p).
∂pA

Then, under symmetric discriminatory pricing, p∗m = p∗m (cm ) satisfies ∂p π m (p∗m ; cm ) =0 for m = s, w. Under symmetric uniform pricing, p = p(cs , cw ) is a (unique) solution of ∂p π s (p; cs ) + ∂p π w (p; cw ) = 0.4 Throughout this paper, we consider the4

If one considers quantity-setting, rather than price-setting, firms, as in Agurre (2017), thenfirm j’s profit in market m is defined by π jm = pjm (qjm , q−j,m )qjm − cm (qjm ), and thus
the first-order partial derivative in symmetric equilibrium is ∂π jm /∂qjm |qjm =q−j,m =q , which isequivalent to pm (q) − mcm (q) + q(∂pA,m (q, q))/∂qPA ) = 0. Under uniform pricing, firm j’s
quantity-setting problem is formulated as maxqs ,qw m=s,w [pjm (qjm , q−j,m )qjm − cm (qjm ) subject
to pj,s (qj,s , q−j,s ) = pj,w (qj,w , q−j,w ).

6

situation where the weak market is open under uniform pricing: qw (p) > 0.5
Let the equilibrium profit in market m in symmetric equilibrium under uniform
pricing and under price discrimination be denoted by π ∗m and π m , respectively. Accordingly, the aggregate profits are defined by Π∗ ≡ π ∗s + π ∗w and Π ≡ π s + π w ,respectively. If price discrimination lowers firms’ profit in symmetric pricing equilibrium (i.e., Π∗ < Π), the firms may want to agree not to price discriminate even if
they are allowed to do so. This is a situation of Prisoners’ Dilemma: one firm’s deviation is profitable.6 We assume that firms cannot commit to not engaging in pricediscrimination even if Π > Π∗ : we consider Nash equilibrium under each regime ofpricing (uniform pricing or price discrimination).
The equilibrium discriminatory price in market m = s, w, p∗m ≡ p∗m (cm ), satisfies the following Lerner formula: εFm (p∗m )(p∗m − cm )/p∗m = 1. This shows that thediscriminatory price in market m approaches to the marginal cost as the own-priceelasticity for the firm, εFm (p∗m ), becomes large. Because of Holmes’ (1989) elasticity
formula explained above, εFm (p∗m ) can be large (i) when εIm (p∗m ) is very large even
∗
I
∗
C
∗
if εC
m (pm ) is close to zero, or (ii) when εm (pm ) is very large even if εm (pm ) is close

to zero. These are two polar cases of a large εFm (p∗m ): of course if both εIm (p∗m ) and∗F∗εCm (pm ) are very large, then εm (pm ) is also very large. Case (i) is where the industry

is under strong pressure from other substitutable industries7 so that a small priceincrease in symmetric pricing causes a large number of consumers switching to purchasing a product in other industries instead, although any consumers are very loyal
to either firm so that a small price increase by one firm causes a very small number
Note that qw (p) > qw (p∗s ) because qw (·) is strictly decreasing and p∗s > p. By Assumption,qw (p∗s ) > 0. Thus, the weak market is open under uniform pricing, i.e., qw (p) > 0. Alternatively,
we would be able to show that there exist cs and cs , cs < cs , such that p∗s > p∗w and qw (p) > 0 forcs ∈ (cs , cs ) in a similar spirit of Adachi and Matsushima (2014).
6
Dastidar (2006) provides a sufficient condition for firms’ equilibrium profit to be higher underprice discrimination. More specifically, define the price difference between the discriminatory priceand the uniform price in market m by ∆p∗m ≡ p∗m − p. Then, Dastidar’s Proposition 2 (2006,
p. 241) shows that if |∆p∗s | ≥ |∆p∗w | and ∂xB,s (p, p)/∂pA ≥ ∂xB,w (p, p)/∂pA for all p ∈ (p∗w , p∗s ),then the per-firm profit difference, ∆Π ≡ Π∗ − Π, is positive.
7
In the demand construction based on random utility and discrete choice, this would be interpreted as an outside option. Note also that if the industry is defined by the SSNIP (“Small but
Significant and Non-transitory Increase in Price”) test, then by definition, εIm (p∗m ) is close to zero,∗C∗and thus, εFm (pm ) is closely approximated by εm (pm ).
57

of consumers switching to other rivals’ product in the same industry (most of themleave the industry to purchase something outside the industry). For example, in a
residential area, consumers (especially young consumers) would have a strong taste∗
for their favorite soda (thus, εCm (pm ) is close to zero), although soda is highly sub-stitutable by mineral water (thus, εIm (p∗m ) is very large) if consumers just want toquench their thirst (whether soda or water does not matter).
On the other hand, case (ii) is where the competitive pressure from other industries is weak, although inside the industry, firms are fiercely competing for consumers. For example, in a resort, consumers may not care much about the difference∗
between Coke and Pepsi (thus, εCm (pm ) is very large), though soda would not be eas-

ily substitutable by mineral water (thus, εIm (p∗m ) is close to zero) because consumers
want to get perfectly refreshed: having water instead of soda does not relieve theirthroat. From a firm’s perspective, these two polar cases are equivalent with respect
to pricing in the sense that if it raises its price by even a small amount, it loses
a large number of consumers: whether they leave the industry or switch to rivals’
products does not matter to that firm. Thus, in the examples above, the firms’(discriminatory) prices are close to marginal cost both in a residential area and ina resort due to different reasons. Recall again that these are two polar cases: in∗reality, εFm (p∗m ) may be large because both εIm (p∗m ) and εCm (pm ) are large. We canalso think of the following alternative possibility: in a resort filled by young visitors,
Coke’s and Pepsi’s prices are close to the marginal cost because consumers do notcare about water or soda as long as they can relieve their throat (i.e., εIm (p∗m ) is very∗large), although they are very loyal to either brand once they choose soda (εCm (pm )

is close to zero).
Next, let ym be per-firm (symmetric) share output in market m,8 that is,ym (ps , pw ) ≡ qm (pm )/[qs (ps ) + qw (pw )]. Then, the equilibrium uniform price, p ≡Xp(cs , cw ), satisfies:y m εFm (p)(p−cm )/p = 1, where y m ≡ ym (p(cs , cw ), p(cs , cw ))m=s,w

for m = s, w.98Of course, the total output in market m is 2qm (pm ), aggregated across symmetric firms.
If
there are no cost differentials, i.e., cs = cw (≡ c), then the formula is simpler:PF
m=s,w xm εm (p)(p − c)/p = 1 as shown by Holmes (1989, p. 247): the markup rate (common
9

8

3

Output and Welfare

In the analysis below, we, following Schmalensee (1981), Holmes (1989), and Aguirre,
Cowan, and Vickers (2010), add the constraint ps −pw ≤ r, where r > 0, to the firms’
profit maximization problem (under symmetric pricing). Then, r = 0 corresponds
to uniform pricing, and r = r∗ ≡ p∗s − p∗w to price discrimination. We express social
welfare (and aggregate output) as a function of r in [0, r∗ ]. Note that under thisconstrained problem of profit maximization, pw satisfies ∂p π s (pw +r)+∂p π w (pw ) = 0.
Thus, we write the solution by pw (r). Then, we define ps (r) ≡ pw (r) + r. Applying
00

00

00

the implicit function theorem to this equation yields to p0w (r) = −π s /(π s + π w ) < 0
00

00

00

and p0s (r) = π w /(π s + π w ) > 0.
00

Here, note that π m (p, cm ) and ∂p2 π m (p, cm ) are different:


∂xA,m
d ∂xA,m00
0π m (p, cm ) = qm (p) +(p, p) + (p − cm )(p, p)
∂pA
dp
∂pA
∂xA,m
∂ 2 xA,m= ∂p2 π m (p) +(p, p) + (p − cm )(p, p),∂pB∂pB ∂pA
where ∂p2 π m (p, cm ) is defined by∂ 2 xA,m (p, p)/∂p2A ∂xA,m2(p, p),∂p π m (p) ≡ 2 + (p − cm )∂xA,m (p, p)/∂pA∂pA
00

which corresponds to Aguirre, Cowan, and Vickers’ (2010) π m (p). We assume that00

π m (p, cm ) < 0 for all p ≥ 0.10
em (q) =We define the representative consumer’s utility in symmetric pricing by U
Um (q, q). Aggregate output under symmetric pricing is given by Q(r) = Qs (r) +Qw (r) = 2 (qs (ps (r)) + qw (pw (r)). Social welfare under symmetric pricing as a funces (qs (ps (r))) + U
ew (qw (pw (r))) − 2cs · qs (ps (r)) −tion of r is written as W (r; cs , cw ) ≡ U
e 0 − 2cs ) · q 0 · p0 (r) + (Ue 0 − 2cw ) · q 0 · p0 (r).
2cw · qw (pw (r)), which implies W 0 (r) ≡ (U
s
s
s
w
w
w
0
e
Now, note that Um = ∂Um /∂qA + ∂Um /∂qB = 2∂Um /∂qA (by symmetry). Thus,
W 0 (r) = 2 (ps (r) − cs ) · qs0 · p0s (r) + 2 (pw (r) − cw ) · qw0 · p0w (r).
to all markets) is equal to the inverse of the average of own-price elasticities weighted by theoutput shares.
10
Appendix A of Aguirre, Cowan, and Vickers (2010) discusses the concavity of the profit function.

9

3.1

OutputNow, we can further proceed:
W 0 (r)= (ps (r) − p + p − cs ) qs0 (ps (r))p0s (r)2+ (pw (r) − p + p − cw ) qw0 (pw (r))p0w (r)= (ps (r) − p) qs0 (ps (r))p0s (r) + (pw (r) − p) qw0 (pw (r))p0w (r)
|
{z
} |
{z
}
<0
<0
X
0
+
(p − cm ) qm
(pm (r))p0m (r).
m=s,wThis derivation coincides with the case of monopoly as shown in Aguirre, Cowan, andVickers’ (2010, p. 1604) equality (3) if there are no cost differentials (i.e., cs = cw ≡c), with two minor modifications: (i) the left hand side is W 0 (r)/2 rather than W 0 (r)itself, and (ii) the last term of Aguirre, Cowan, and Vickers’ (2010) equality (3) is reP
0placed by Q0 (r)/2 rather than Q0 (r) because (1/2) m=s,w (p − cm ) qm
(pm (r))p0m (r) =(p − c) (Q0 (r)/2). If cost differentials are allowed, it is observed that an increase inP00
the weighted aggregate output,
m=s,w (p − cm ) qm (pm (r))pm (r), is necessary forprice discrimination to raise social welfare, as in the case of monopoly.11
To proceed further, we define the curvature of the firm’s (direct) demand in
market m by
αFm (p) ≡ −p∂ 2 xA,m(p, p)∂xA,m (p, p)/∂pA ∂p2A

(which measures the concavity/convexity of the firm’s direct demand, and corresponds to αm (p) in Aguirre, Cowan, and Vickers, 2010, p. 1603), and the elasticityof the cross-price effect of the firm’s direct demand in market m by
αC
m (p)

p
∂ 2 xA,m≡ −(p, p)∂xA,m (p, p)/∂pB ∂pB ∂pAp∂ 2 xB,m(p, p),= −∂xB,m (p, p)/∂pA ∂p2A
{z
}
|
{z
}|
>0

11

<0

However, if externalities across consumers, such as network externalities and congestion, exist,
then an increase in aggregate output would be no longer a necessary condition, as implied byAdachi (2002, 2005), who studies monopoly with linear demands. See also Czerny and Zhang(2015) as a recent study of price discrimination and congestion.

10

which is new to oligopoly.12 Here, αFm (p) is positive (resp. negative) if and only if∂ 2 xA,m (p, p)/∂p2A is negative (resp. positive), while αCm (p) is always positive (becauseof our assumption, ∂ 2 xjm (p, p)/(∂pA ∂pB ) < 0). Note that the sign of αFm (p) indicates
whether the firm’s own part of the demand slope under symmetric pricing given therival’s price being p, ∂xA,m (·, p)/∂pA , is convex (αFm (p) is positive) or concave (αFm (p)is negative). On the other hand, αCm (p) measures how the rival’s price level matters
to how many of the firm’s customers switch to the rival’s product when the firmraises its own price (∂xB,m /∂pA ). Thus, a large αCm (p) implies that ∂xB,m /∂pA isvery responsive to a change in pB , and vice versa.
Next, we define the conduct index (see, e.g., Bresnahan 1989; Genesove andMullin 1998; and Corts 1999) in market m by θm (p) ≡ 1 − Am (p),13 where Am (p)is the aggregate diversion ratio (Shapiro 1996) in market m, which is is definedF
by Am (p) ≡ −(∂xB,m (p, p)/∂pA )/(∂xA,m (p, p)/∂pA ) = εCm (p)/εm (p). Here, Am (p)measures the degree of rivalness: if Am (p) is close to one, consumers who leave afirm as a response to an increase in its price are nearly all switching to its rival’s
product. In this way, Aguirre, Cowan, and Vickers’ (2010) derivation in the caseof monopoly (where the method by Schmalensee (1981) is utilized) is connected
to Weyl and Fabinger’s (2013) condition in the case of symmetric oligopoly. In
particular, Aguirre, Cowan, and Vickers’ (2010, p.1606) Proposition 2 (a sufficientcondition for price discrimination to raise social welfare) is extended to the case of
12This is because ∂ (∂xA,m (p, p)/∂pB ) /∂pA = ∂ (∂xB,m (p, p)/∂pA ) /∂pA .
Alternatively, Weyl and Fabinger (2013, p. 531) define the conduct index in a market (which,in our interest in price discrimination, can be indexed by m) by θm ≡ [(p − cm )/p]εIm (their mc
and εD are replaced by our cm and εIm , respectively) as the Lerner index adjusted by the elasticityof the industry’s demand. If the first-order condition is given for each market (that is, if full
price discrimination is allowed), then θm (p) defined as in Weyl and Fabinger (2013) coincides with
1 − Am (p) because εFm [(pm − cm )/pm ] = 1 and thusp − cm1p
εIm (p)=−q 0 (p)p
εF
qm (p) m
m (p)



qm (p)
1
p
∂xA,m
∂xA,m
= −
−
(p, p) +(p, p)
p ∂xA,m (p, p)/∂pA
qm (p)∂pA∂pB∂xA,m (p, p)/∂pA + ∂xB,m (p, p)/∂pA=(by symmetry).
∂xA,m (p, p)/∂pA
13

See also Adachi and Fabinger (2017) for a generalized definition of the conduct index that allows
for the possibly of non-zero specific and ad valorem taxes.

11

oligopoly in a simpler manner, using the concept of pass-through introduced in thenext subsection.
As Weyl and Fabinger (2013, p. 544) argue, θm (p) captures the degree of industrylevel brand loyalty or stickiness 14 in market m: if θm (p) is zero (close to one), market
m is captured by perfect competition (almost monopoly): firms’ products are perfect substitutes (nearly non-substitutable products).15 The markup rate (the Lernerindex), Lm (pm , cm ) ≡ (pm − cm )/pm , alone is not appropriate to measure the rivalness within market m because it can be the case that pm is close to cm (the markup
rate is close to zero) simply because the price elasticity of the industry’s demandεIm (pm ) is very large while the brand rivalness is so weak that the cross-price elasF
ticity, εCm (pm ), remains very small (as a result, in total, εm (pm ) is very large, whichF
is actually reason for the low markup rate). However, if εCm (pm ) is close to εm (pm )(i.e., almost of all consumers who leave a firm as a response to its price increase areswitching to other rivals’ products), then θm becomes close to zero irrespective of
the value of the markup rate. Thus, θm (p), which ranges between 0 and 1, better
captures the brand stickiness than Lm (p, cm ) does.
Now, we consider the effects of price discrimination on aggregate output. First,
note that
Q0 (r)= qw0 · p0w + qs0 · p0s
2


00 00
πs πw
=
− 00
π s + π 00w
|
{z
}
>0

1 − Ls (ps (r))[αFs (ps (r)) − (1 − θs (ps (r))αCs (ps (r))]×θs (ps (r))
F
1 − Lw (pw (r))[αw (pw (r)) − (1 − θw (pw (r))αCw (pw (r))]−.
θw (pw (r))
I
Note also that εCm (pm )/εm (pm ) = [1 − θ m (pm )]/θ m (pm ) measures the substitutability
14

Even if the firms’ products have the same characteristics across different markets (with noproduct differentiation), the degree of brand loyalty may differ across markets, reflecting differences
in market characteristics (summarized in demand functions).
15
F
IC
Because (pm − cm )εF
m (pm )/pm = 1 and εm (pm ) = εm (pm ) + εm (pm ), it is verified thatCθm (pm , cm )+εm (pm )(pm −cm )/pm = 1. Thus, as long as the products are substitutes (εCm (pm ) > 0),θm (pm , cm ) is less than one.

12

between brands (adjusted by the elasticity of the industry’s demand): if the brandI
stickiness is very strong (i.e., θm (p) is close to one), εCm (pm )/εm (pm ) is close toone, while if the brand stickiness is very weak (i.e., θm (p) is close to zero), thenIεCm (pm )/εm (pm ) becomes infinitely large. Then, the following lemma holds with costdifferentials being allowed.
Lemma 1. Q0 (r) > 0 if and only if (suppressing the dependence on pm (r) and cm )Lw ·αFw − (1 − θw )αC
αF − (1 − θs )αC
11ws− Ls · s
+ −
> 0.
θw
θs
θs θw

(1)

Now, suppose that the brand stickiness in the weak market is so weak thatθw (pw ) is close to zero (θw (pw ) ≈ 0), while the brand stickiness in the strong market
is moderate or strong (θs (ps )  0). Then, the left hand side of the inequality above
is approximated by




 1
1 − Ls αFs − (1 − θs )αCs− 1 − Lw αFw − αC
.
w
θs
θw
Thus, as long as 1 > Lw [αFw − αCw ], the left hand side becomes infinitely negative asθw (pw ) approaches to zero (assuming the first term is bounded). Counterintuitively,in the weak market, where price discrimination lowers the price, the brand rivalness
has a negative effect on an increase in aggregate output by price discrimination. Thisis because the uniform price is already very low due to the fierce level of competition
and thus there is little room for a price reduction by price discrimination to increase
the output in the weak market. The opposite argument holds if the strong market ischaracterized by a low brand stickiness (i.e., θs (ps , cs ) ≈ 0). This implies that, as is
expected, a fierce level of competition in the strong market has a positive effect on
an increase in aggregate output by price discrimination. The rivalness in the strong
market keeps the price increase by price discrimination small, and thus a reductionin output in the strong market is also kept small.
Following Holmes (1989), we call the first and the second terms in the left
hand side of (1) the adjusted-concavity part, and the third and the fourth terms the13elasticity-ratio part:Lw ·|αFw − (1 − θw )αC
αF − (1 − θs )αC
11ws− Ls · s
+
−
.
θw
θs
θ
θ
{z
} | s {z w}adjusted-concavity(1’)

elasticity-ratio

First, look at the elasticity-ratio part. If the difference θw − θs is greater, it is more
likely that Q0 (r) > 0. Thus, competitiveness in the strong market, rather than in the
weak market, is important. Next, look at the adjusted-concavity part. A larger αFw
0F
and/or a smaller αC
w make a positive Q (r) more likely. A larger αw means that thefirm’s own part of the demand in the weak market (∂xA,w /∂pA ) is more convex. On
the other hand, a smaller αC
w means that how many of the firm’s customers switch
to the rival’s product as response to the firm’s price increase is not so affected by
the current price level. In this sense, the strategic concerns in the firm’s pricing aresmall. Thus, both a larger αFw and a smaller αC
w indicate that the weak market iscompetitive. Even if ∂xA,w /∂pA is not so convex, a smaller αC
w (i.e., ∂xB,m /∂pA is
not responsive to the level of pB ) can substitute it. A similar argument also holds
0
for αFs and αC
s . In the Appendix, we show that Holmes’ (1989) expression for Q (r)(expression (9) in Holmes (1989, p. 247)) is equivalent to (1’).
000(p)/π m (p, cm )) > 0 so that
Now, define hm (p, cm ) ≡ 1/(qm
0 0
Q0 (r)
qs qw= − 00[hs (ps (r), cs ) − hw (pw (r), cw )] .
2
π s + π 00w
|
{z
}
>0

We assume that hm (·, cm ) is decreasing (and call it the Decreasing Inverse Ratio
Condition: DIRC).16 It is also shown that




00
0 0
0 0
Q (r)
qs q w
d
qs qw
0 0
0 0
[hs ps − hw pw ] + [hs − hw ].
= − 00− 00
2
π s + π 00w
dr
π s + π 00w
1600
0
It is verified that h0m < 0 is equivalent to qm> (qm/π 00m )π 000
m because
!
0
00
0
00 00
π m (p, cm )π 000
m qm − π m qm
0hm (p, cm ) =
=
.
0 (p)
0 ]2
qm
[qm

Thus, DIRC states that the profit function decreases quickly enough as p increases. To see this,0000> 0, then it is sufficient to assume π 000
if qm
m < 0. This means that π m , which is negative, should00
00
be smaller, that is, the the negative slope of π m should be steeper, as p increases. If qm≤ 0, then
00
000
00
0
00
π m be should be not only negative but sufficiently small that π m < qm /(qm /π m ). In both cases,
π m should decrease quickly as p increases.

14

00

0

0

00

00

Then, there exists rb such that Q0 (br) = 0 and Q (br)/2 = [−qs qw /(π s +π w )] [h0s p0s − h0w p0w ] <0 because h0s p0s < 0 and h0w p0w > 0. Then, (1/2)Q(r) behaves on [0, r∗ ] in either manner:17
1. If Q0 (0) ≤ 0, then (1/2)Q(r) is monotonically decreasing in r, and a result∆Q/2 = [Q(r∗ ; cs , cw ) − Q(0; cs , cw )]/2 < 0; price discrimination lowers aggregate output.
2. If Q0 (0) > 0, then (1/2)Q(r) either(a) is monotonically increasing (if Q0 (r∗ ) > 0, this is true), and as a result,∆Q/2 > 0; price discrimination raises aggregate output.
(b) first increases, and then after the reaching the maximum (where Q0 (r) =0), decreases until r = r∗ . In this case, price discrimination may raise or
lower aggregate output: it cannot be determined whether ∆Q/2 < 0 or
∆Q/2 > 0 without further functional and/or parametric restrictions.
00

Now, we determine the sign of Q0 (0). It follows that sign[Q0 (0)] = sign[π s (p, cs )/qs0 (p)−
00

00

00

π w (p, cw )/qw0 (p)], which implies that Q0 (0) ≤ 0 ⇔ π w (p, cw )/qw0 (p) ≥ π s (p, cs )/qs0 (p).
Note also that sign[Q0 (r∗ )] = sign[hs (p∗s , cs )−hw (p∗w , cw )], which implies that Q0 (r∗ ) >00

00

0 ⇔ π s (p∗s , cw )/qs0 (p∗s ) > π w (p∗w , cw )/qw0 (p∗w ). Because




00
2 − Lm (p, cm )αFm (p) − [1 − θm (p)] 1 − Lm (p, cm )αC(p)
π m (p, cm )
m
=
0 (p)
qm
θm (p)
holds, the following proposition obtains.
Proposition 1. Given the DIRC, if θs (p) ≥ θw (p) andαFs (p) − [1 − θs (p)]αC
αF (p) − [1 − θw (p)]αC
s (p)
w (p)
≥ w
θs (p)
θw (p)
then price discrimination lowers aggregate output. If θw (p∗w ) > θs (p∗s ) and∗∗αFw (p∗w ) − [1 − θw (p∗w )]αCαFs (p∗s ) − [1 − θs (p∗s )]αCw (pw )s (ps )≥,
θw (p∗w )θs (p∗s )then price discrimination raises aggregate output.
17This is because the modified version of Aguirre, Cowan, and Vickers’ (2010, p. 1605) Lemma
also holds in our oligopoly setting.

15

Proof. First, note that
2 − Lw (p, cw )αFw (p) − [1 − θw (p)] 1 − Lw (p, cw )αC
w (p)
θ (p)

 w


F
2 − Ls (p, cs )αs (p) − [1 − θs (p)] 1 − Ls (p, cs )αC
s (p)
≥
,
θs (p)
then price discrimination lowers aggregate output. The first part is a sufficientcondition for this inequality to hold. Next, note that




∗
2 − Ls (p∗s , cs )αFs (p∗s ) − [1 − θs (p∗s )] 1 − Ls (p∗s , cs )αC
s (ps )θs (p∗s )∗2 − Lw (p∗w , cw )αFw (p∗w ) − [1 − θw (p∗w )] 1 − Lw (p∗w , cw )αC
(p
)
w w
>
,
θw (p∗w )then price discrimination raises aggregate output. Thus, θw (p∗w ) > θs (p∗s ) andLw (p∗w , cw )

3.2

∗
∗
αFw (p∗w ) − [1 − θw (p∗w )]αCαFs (p∗s ) − [1 − θs (p∗s )]αCw (pw )s (ps )
∗
>
L
(p
,
c
)
.
s
s
s
θw (p∗w )θs (p∗s )Social Welfare

Now, we study the effects of allowing third-degree price discrimination on social
welfare. To proceed further, note that


00 00
W 0 (r)
πs πw
=
− 00
2
π s + π 00w
|
{z
}
>0


(pw (r) − cw )qw0 (pw (r)) (ps (r) − cs )qs0 (ps (r))
×
−
.
π 00w
π 00s
We follow Aguirre, Cowan, and Vickers (2010, p. 1605), who define zm (p, cm ) ≡000(p−cm )qm(p)/π m (p, cm ), which is “the ratio of the marginal effect of a price increase0
on social welfare to the second derivative of the profit function.” However, our qm
00

00

0
and π m have strategic effects. More specifically, our qm
and π m are written as0qm(pm ) =

∂xB,m
∂xA,m
(pm , pm ) +(pm , pm )∂p∂p
| A {z} | B {z}0 )<0 (ACV’s qm

16>0 (strategic)

and
00

π m (pm , cm ) = D2 π m (pm , cm ) + Gm (pm , cm )
|
{z
} |
{z
}
ACV’s π 00mstrategic

where
 


∂xA,m
d ∂xA,m∂ 2 xA,mGm (p, cm ) =(p, p) + (p − cm )(p, p) −(p, p) .
∂pB
dp
∂pA
∂p2A
As in Aguirre, Cowan, and Vickers (2010, p.1605), we can write


00 00
W 0 (r)πs πw= − 00[zw (pw (r), cw ) − zs (ps (r), cs )] ,2
π s + π 00w
|
{z
}
>0

and their lemma also holds in our case of oligopoly if we assume zm (·, cm ) is increasing(the Increasing Ratio Condition; IRC).18 Then, (1/2)W (r) behaves on [0, r∗ ] ineither manner:19
1. If W 0 (0) ≤ 0, then (1/2)W (r) is monotonically decreasing in r, and a result∆W/2 = [W (r∗ ; cs , cw )−W (0; cs , cw )]/2 < 0; price discrimination lowers social
welfare.
2. If W 0 (0) > 0, then (1/2)W (r) either(a) is monotonically increasing (if W 0 (r∗ ) > 0, this is true), and as a result,∆W/2 > 0; price discrimination raises social welfare.
(b) first increases, and then after the reaching the maximum (where W 0 (r) =0), decreases until r = r∗ . In this case, price discrimination may raiseor lower social welfare: it cannot be determined whether ∆W/2 < 0 or
∆W/2 > 0 without further functional and/or parametric restrictions.
00

0

00

0

000

00

0
Note that zm(p; cm ) = {[(p − cm )qm (p) + qm (p)]π m (p) − (p − cm )qm (p)π m (p)}/[π m (p)]2 and
0
00
00
0
000
thus, IRC is equivalent to [(p − cm )qm (p) + qm (p)]π m (p) > (p − cm )qm (p)π m (p). Appendix B
of Agguire, Cowan, and Vickers (2010) discusses sufficient conditions for the IRC in the case ofmonopoly. If hm (·, cm ) is decreasing, as we assume throughout, then zm (·, cm ) is increasing because00zm(p; cm ) = [1 − zm (p; cm )h0m (p; cm )]/hm (p; cm ) so that zm
is positive if h0m is negative. That is,
DIRC is a sufficient condition for IRC to hold.
19
This is because the modified version of Aguirre, Cowan, and Vickers’ (2010, p. 1605) Lemma
also holds in our oligopoly setting.
18

17

Now, we determine the sign of W 0 (0). First, define the markup in market m by00

µm (p, cm ) ≡ p−cm . Then, it follows that sign[W 0 (0)] = sign[µw (p, cw )qw0 (p)/π w (p, cw )00−µs (p, cs )qs0 (p)/π s (p, cs )], and thus, the following proposition is obtained.
Proposition 2. Given the IRC, if the markup in strong market relative to the weakmarket at the uniform price p is sufficiently large, i.e.,q 0 (p)q 0 (p)µs (p, cs ) s00≥ µw (p, cw ) w00,
π s (p)π w (p)(2)

then price discrimination lowers social welfare.
If there are no strategic effects (i.e., ∂xB,m /∂pA = 0 or θm (p, cm ) = 1), then000(p) = 2−Lm (p, cm )αFm (p), and inequality (2) above reduces to µs (p, cs )/µw (p, cw )π m (p, cm )/qm

≥ [2 − Ls (p, cs )αFs (p)]/2 − Lw (p, cw )αFw (p). On the other hand, if there are no cost00

differentials (i.e., cs = cw ≡ c), then inequality (2) above reduces to π w (p, c)/qw0 (p) ≥00

π s (p, c)/qs0 (p) because the markups are the same in the two markets. Thus, if there
are no strategic effects and no cost differentials, then inequality (2) coincides with
Aguirre, Cowan, and Vickers’ (2010, p. 1605) Proposition 1 (αFs (p) ≥ αFw (p) in ournotation; in their notation, αs (p) ≥ αw (p)) because Ls (p, cs ) = Lw (p, cw ). That is,
the firm’s “direct demand function in the strong market is at least as convex as thatin the weak market at the nondiscriminatory price” (Aguirre, Cowan, and Vickers,
2010, p. 1602).
Recall that in our case of oligopoly,




00
2 − Lm (p, cm )αFm (p) − [1 − θm (p)] 1 − Lm (p, cm )αC
π m (p, cm )
m (p)
=
0 (p)
qm
θm (p)
holds, which leads to the following corollary, another expression for the sufficientcondition for price discrimination to lower social welfare in the case of no costdifferentials.
Corollary 1. Suppose there are no cost differentials across markets (cs = cw ).
Given the IRC, if θw ≥ θs andαFw − (1 − θw )αC
αFs − (1 − θs )αCws≥εIs (p)εIw (p)at p, then price discrimination lowers social welfare.
18This is because0000

π s (p, cs ) π w (p, cw )−≤0
qs0 (p)qw0 (p) F
αFs − (1 − θs )αC11
αw − (1 − θw )αC
s
w
−
+ −
≤ 0,
⇔ L·
θw
θs
θs θw
where L ≡ Ls (p, c) = Lw (p, c).
We also define the pass-through rate (under symmetric equilibrium discriminatory pricing) in market m by ρm ≡ (p∗m )0 (cm ). Then, we obtain the following
sufficient condition on welfare improvement by allowing price discrimination.
Proposition 3. Given the IRC, if the markup in strong market relative to the weakmarket under price discrimination is sufficiently small, i.e.,θs (p∗s )ρs (p∗s )µs (p∗s , cs ) ≤ θw (p∗w )ρw (p∗w )µw (p∗w , cw ),then price discrimination raises social welfare.
Proof. To prove this proposition, note first that(pm − cm )εIm (pm )qm (pm )pm π 00m (pm , cm )qm (pm )= −θm (pm ) 00
π m (pm , cm )zm (pm , cm ) = −holds. Now, defineqm (pm )F (pm , cm ) = ∂xA,m+ p m − cm
(p
,
p
)
m
m
∂pA
so that F (p∗m , cm ) = 0. Then, by applying the implicit function theorem, we have
10qmd (∂xA,m /∂pA ) /dpm
1+
− qm
∂xA,m /∂pA(∂xA,m /∂pA )2∂xA,m /∂pA

,
=
∂xA,m
qm
d
∂xA,m
0
+ qm −∂pA∂xA,m /∂pA dpm
∂pA

ρm =

19

and under the equilibrium discriminatory prices,∂xA,m (p∗m , p∗m )/∂pA


d
∂x
∂x
A,m
A,m
0 (p∗ ) +(p∗ , p∗ ) + (p∗m − cm )(p∗ , p∗ )qmm
∂pA m mdpm
∂pA m m∂xA,m (p∗m , p∗m )/∂pA=,
π 00m (p∗m , cm )ρm =which implies thatzm (p∗m , cm ) = −θm (p∗m )ρm (p∗m )qm (p∗m )∂xA,m ∗ ∗(p , p )∂pA m m= θm (p∗m )ρm (p∗m )µm (p∗m , cm )and thus
W 0 (r∗ )
=
2



00 00
πs πw
− 00
π s + π 00w
× [θw (p∗w )ρw (p∗w )µw (p∗w , cw ) − θs (p∗s )ρs (p∗s )µs (p∗s , cs )] .

This completes the proof of Proposition 3.
Here, Aguirre, Cowan, and Vickers’ (2010) IRC for zm (p, cm ) is equivalent tothat condition on θm (p)ρm (p)µm (p, cm ). Roughly speaking, if (in symmetric equilibrium) (i) the brand loyalty (θ), (ii) the pass-through (ρ), or (iii) the markup (µ)
is sufficiently small in the strong market, then social welfare is likely to be higher
under price discrimination. In particular, if these three measures are calculated (orestimated) in each separate market (and symmetry is not so far away from the reality), then it would assist one to judge whether price discrimination is desirable froma society’s viewpoint. To the best of our knowledge, Propositions 2 and 3 are themost general statements on when allowing (symmetric) oligopolisitc firms to pricediscriminate lowers or raises social welfare, allowing cost differentials that Chen and
Schwartz (2015) study in the case of monopoly, although Weyl and Fabinger (2013,
p. 565) also briefly mention the importance of θm ρm µm .
Even if there are no cost differentials (i.e., cs = cw ), this expression cannot be
further simplified. In other words, this expression is already robust to the inclusion20

of cost differentials. Now, if we further assume that there are no strategic effects (i.e.,θm = 1), then the above condition becomes (p∗s −c)/(p∗w −c) ≤ (1/ρs (p∗s ))/(1/ρw (p∗w )),which coincides with (p∗w − c)/(2 − σ Iw (qw (p∗w ))) ≥ (p∗s − c)/(2 − σ Is (qs (p∗s ))) in Proposition 2 of Aguirre, Cowan, and Vickers (2010, p. 1606), where σ Im (q) ≡ −qp00 /p0
is the curvature of the industry’s inverse demand function (in symmetric pricing),
because it is shown that ρm (p∗m ) = 1/[2 − σ ∗m (qm (p∗m ))] in our oligopoly setting as
well. Thus, price discrimination raises social welfare “if the discriminatory prices
are not far apart and the inverse demand function in the weak market is locallymore convex than that in the strong market” (Aguirre, Cowan, and Vickers 2010,
p. 1602).
Suppose that price discrimination is being conducted. Then, to evaluate it froma viewpoint of social welfare, we first compute θm , ρm and µm for each m = s, w, thenif the sufficient condition above is satisfied, then the ongoing price discrimination isjustified. Notably, to compute θm , ρm and µm in equilibrium, the cost information
is not necessary: once a specific form of demand function in market m for firm j,
qjm = xjm (pjm , p−j,m ) is provided (and if IRC is satisfied), then the three variables∗F∗are computed in the following manner: θm = 1 − εCm (pm )/εm (pm ), ρm = 1/[2 −0000(p∗m )), and µm = p∗m /εFm (p∗m ). Thus, if(p∗m )]2 − qm (p∗m )qm(p∗m )]2 /(2[qmσ ∗m (p∗m )] = [qmthe firm’s demand for each market m is estimated and the discriminatory price p∗m
is observed, then one can easily compute θm , ρm , and ρm .
It is also possible to provide coherent sufficient conditions for an increase anda decrease in social welfare by price discrimination, using θm ρm µm that appears in
Proposition 3 above. To do so, consider the case where the prevailing uniform priceis not a result from banning price discrimination: the uniform price results from aquantity transfer from the strong market to the weak market. Let the amount of thetransfer be denoted by qe > 0. Then, if per-firm quantity transfer is made from thestrong market to the weak market, the first-order condition in the strong market isqs (p) − qe + (p − cs )

21

∂xA,s
(p, p) = 0,
∂pA

while that in the weak market isqw (p) + qe + (p − cw )∂xA,w(p, p) = 0.
∂pA

It is possible to find a unique qe = q such that ps = pw = p because summing theseequations yields the first-order condition under uniform pricing. As qe, starting at q,
approaches to zero, pm (eq ) moves from the uniform price p to the discriminatory pricep∗m . In this sense, the role of qe is similar to considering the constraint ps − pw = r as
above. Now, let the corresponding equilibrium (per-firm) output be defined by (withan abuse of notation) qm (eq ) ≡ qm (pm (eq )). Then, dead-weight loss from oligopoly inmarket m can be written as a function of qe, DW Lm (eq ), and as Weyl and Fabinger(2013, p.538) show,
dDW Lm= −θm (eq )ρm (eq )µm (eq ).
deqThus price discrimination raises social welfare (∆W > 0) if and only if
Z 0
θw (e
q )ρw (eq )µw (eq )deq|∆DW Lw | =−q

is greater than
Z q|∆DW Ls | =θs (eq )ρs (eq )µs (e
q )de
q,
0

and vice versa. Notice that the IRC for zm (p, cm ) is equivalent to θm (eq )ρm (eq )µm (eq)
decreasing. Then, as Figure 1 shows, the following proposition holds.
Proposition 4. Given the IRC, if θw (eq )ρw (eq )µw (eq ) > θs (eq )ρs (eq )µs (eq ) for all qe ∈[0, q], then price discrimination raises social welfare.
Note that Proposition 4 is stronger than Proposition 3. Similarly, as Figure 2
shows, the following proposition also holds.
Proposition 5. Given the IRC, if θs ρs µs > θw ρw µw , then price discrimination
lowers social welfare.
Again, Proposition 5 is stronger than Proposition 2. To see this, note that

p − cspqs0 (p)θ s ρs =−ρsp
qs (p)
22

Figure 1: Sufficient condition for price discrimination to raise social welfare:θw ρw µw > θs ρs µs for all qe ∈ [0, q]. (Weaker sufficient condition (Prop 3):θ∗w ρ∗w µ∗w > θ∗s ρ∗s µ∗s )23Figure 2: Sufficient condition for price discrimination to lower social welfare:θs ρs µs > θw ρw µw

24

 0 q (p)= (p − cs ) − sρs .
qs (p)
Now, by applying the implicit function theorem to F (p; q, cs ) ≡ qs (p) − q + (p −cs )∂qsA (p, p)/∂ps , one obtains∂p(q, cs )∂cs∂F/∂cs= −∂F/∂p∂xA,s (p, p)/∂ps=
π 00s (p)1 qs (p) − q= − 00,π s (p) p − csρs =which leads to
θs ρs

 0 
1 qs (p) − qq (p)= − 00(p − cs ) − s
π s (p) p − cs
qs (p)
0q (p) qs (p) − q= s00.
π s (p) qs (p)00

Thus, it is shown that µs (qs0 (p)/π s (p)) > θs ρs µs . Similarly, it is verified that θw ρw >00

qw0 (p)/π w (p) becauseθ w ρw =qw0 (p) qw (p) + q.
π 00w (p) qw (p)0000

In summary, if θs ρs µs > θw ρw µw holds, then µs · (qs0 (p)/π s (p)) > µw · (qw0 (p)/π w (p)).

3.3

Consumer Surplus

One can extend the analysis above to consumer surplus. First, consumer surplus isdefined by replacing cm in W (r) by pm (r) to defineCS(r; cs , cw ) = Us (qs (ps (r))) + Uw (qw (pw (r))) − 2ps (r) · qs (ps (r)) − 2pw (r) · qw (pw (r)),
which implies that
CS 0 (r)= ps (r) · qs0 · p0s (r) + pw (r) · qw0 · p0w (r)2−p0s (r) [ps (r) · qs0 + qs ] − p0w (r) [pw (r) · qw0 + qw ]25

= − [p0s (r)qs + p0w (r)qw ]



00 00
πs πw
qs
qw
=
− 00
− 00 .
π s + π 00w
π 00s
πw
|
{z
}
>0


00 00
πs πw
=
− 00
[gs (ps (r), cs ) − gw (pw (r), cw )]π s + π 00w
{z
}
|
>0

00

where gm (p, cm ) ≡ qm (p)/π m (p, cm ). If gm (·, cm ) is assumed to be decreasing, then
one can use a similar argument.

4

Concluding Remarks

This paper provides theoretical implications of oligopolistic third-degree price discrimination with general nonlinear demands, allowing cost differentials across separate markets. In this sense, this paper, with the help of Weyl and Fabinger(2013), synthesizes Aguirre, Cowan, and Vickers’ (2010) analysis of monopolistic
third-degree price discrimination with general demands and Chen and Schwartz’(2015) analysis of monopolistic differential pricing, to extend them to the case ofsymmetrically differentiated oligopoly.
If some of price-discriminating oligopolists merge into a single firm, what happens? Price discrimination is often neglected in a merger analysis. Traditionally, inmerger analyses, it has been considered as important to estimate own- and crossprice elasticities. However, our theoretical analysis suggests that own- and crossprice elasticities per se may not be so important in welfare evaluation. We conjecturethat our main thrust obtained under symmetric oligipoly, namely the fundamental
importance of the conduct index and the pass-through rate in welfare evaluation,
remains valid if asymmetric firms are allowed.

26

Appendix: Equivalence of Holmes’ (1989) and Our
Expressions for Q0(r)Holmes (1989, p. 247), who assumes no cost differentials (c ≡ cs = cw ) as in most of
the papers on third-degree price discrimination, also derives a necessary and sufficient condition for Q0 (r) > 0 under symmetric oligopoly. It is (using our notation)
written as:
ps − c d·
qs0 (ps ) dps|+

∂xA,s (ps , ps )∂pA

pw − cd− 0·q (p ) dpw{z w w

∂xA,w (pw , pw )∂pAadjusted-concavity condition (Robinson 1933)εCεCs (ps )w (pw )−Iε (p )εI (p )
| s s {z w w }


}

> 0.

elasticity-ratio condition (Holmes 1989)
I
C
I
Recall that 1/θs − 1/θw = εCs /εs − εw /εw . The first and the second terms in theleft hand side of Holmes’ (1989) inequality is rewritten as:ps − c d∂xA,s (ps , ps )pw − cd∂xA,w (pw , pw )·− 0·
qs0 (ps ) dps∂pA
qw (pw ) dpw
∂pA




pw
d
∂xA,w (pw , pw )= Lw (pw ) · − 0
q (p ) dp∂pA w w  w psd∂xA,s (ps , ps )−Ls (ps ) · − 0.
qs (ps ) dps∂pA
Now, it is also observed thatαFm − (1 − θm )αCmθm
αFm ∂xB,m /∂pA
pm
∂ 2 xA,m=−θm
q0∂qB,m /∂pA ∂pA ∂pB
 0 mFpm
qm α m∂ 2 xA,m= − 0 −+.
qmpm θ m∂pA ∂pB
This shows that inequality (1) is another expression for Holmes’ (1989, p. 247)
inequality (9). To see this, note thatd∂xA,m (pm , pm )∂ 2 xA,m∂ 2 xA,m
=
(p
,
p
)
+
(pm , pm )
m m
dpm
∂pA
∂p2A
∂pA ∂pB27

0/pm )(αFm /θm ) + ∂ 2 xA,m /(∂pA ∂pB )in Holmes’ (1989) expression is equivalent to −(qmbecause

 2
0
0
qm
αFm
qm
pm
∂ xA,m−= −−pm θ mpm θ m∂xA,m /∂pA
∂p2A
1 ∂xA,m /∂pA + ∂xB,m /∂pA ∂ 2 xA,m=
θm
∂xA,m /∂pA
∂p2A
∂ 2 xA,m
1(1 − Am )=θm∂p2A∂ 2 xA,m=.
∂p2A

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