We generalize the analyses of Hazledine (2006, “Price discrimination in Cournot-Nash oligopoly”, Economics Letters) and Kutlu (2009, “Price discrimination in Stackelberg competition”, Journal of Industrial Economics) with asymmetric cost firms. We show that the main result of Hazledine, which shows that the average revenue is not dependent on the extent of price discrimination, remains under cost asymmetry but at the industry level. However, the main result of Kutlu, which shows that the Stackelberg leader does not price discriminate at all, does not hold under cost asymmetry. Both the leader and the follower discriminate price under cost asymmetry.Cost asymmetry; Cournot competition; Price discrimination; Stackelberg competition

Arijit Mukherjee

English

Research Papers in Economics

      
UNIVERSITY OF NOTTINGHAM 
 
 
 
Discussion Papers in Economics 
 
___________________________________________________ 
Discussion Paper 
No. 10/04 
 
Price discrimination in oligopoly with 
asymmetric firms 
 
Arijit Mukherjee 
 
April 2010 
 
__________________________________________________________________ 
 2010               DP 10/04 Price discrimination in oligopoly with asymmetric firms  
 
 
Arijit Mukherjee 
University of Nottingham, UK, and The Leverhulme Centre for Research in 
Globalisation and Economic Policy, UK 
 
 
April 2010 
 
Abstract: We generalize the analyses of Hazledine (2006, “Price discrimination in 
Cournot-Nash oligopoly”, Economics Letters) and Kutlu (2009, “Price discrimination 
in Stackelberg competition”, Journal of Industrial Economics) with asymmetric cost 
firms. We show that the main result of Hazledine, which shows that the average 
revenue is not dependent on the extent of price discrimination, remains under cost 
asymmetry but at the industry level. However, the main result of Kutlu, which shows 
that the Stackelberg leader does not price discriminate at all, does not hold under cost 
asymmetry. Both the leader and the follower discriminate price under cost 
asymmetry.  
 
 
Key Words: Cost asymmetry; Cournot competition; Price discrimination; 
Stackelberg competition 
JEL Classifications: D43 
 
 
 
 
 
Correspondence to: Arijit Mukherjee, School of Economics, University of 
Nottingham, University Park, Nottingham, NG7 2RD, UK. 
E-mail: arijit.mukherjee@nottingham.ac.uk 
Fax: +44-115-951 4159 
 
 
 
 
 
 
   1
1. Introduction 
The literature on price discrimination considers different types of price discrmination. 
Stole (2007) provides a nice of survey of this literature. In a recent paper, Hazledine 
(2006) examines second degree price discrimination in a Cournot model. Under 
second degree price discrimination, the firm is able to segment consumer demand by 
ranges of reservation price. For example, the consumers with reservation price 
between  1 r  and  2 r  pay one price, those between  2 r  and  3 r  pay another, and so on 
(Kutlu, 2009). The main result of Hazledine (2006) is to show that the average 
revenue (i.e., the output-weighted revenue) is independent of the extent of price 
discrimination. In other words, the average revenue under k different prices is the 
same to the average revenue under (k+1) different prices. 
In a framework similar to Hazledine (2006) but with a Stackelberg leader-
follower competition, Kutlu (2009) shows that the Stackelberg leader serves only the 
high-valued customers, and therefore, does not price disciminate at all. The 
Stackelberg follower does all price discrimination. While Hazledine (2006) and Kutlu 
(2009) ask different questions under different types of product market competition, 
both papers focus on symmetric cost firms. Hence, a natural qustion is to ask whether 
their results hold under cost asymmetry. 
We generalize the analyses of Hazledine (2006) and Kutlu (2009) with 
asymmetric cost firms and show that the result of Hazledine (2009) about the average 
revenue remains but at the industry level. However, the result of Kutlu (2009), which 
shows that the leader dos not price discriminate, does not hold under cost asymmetry. 
  The remainder of the paper is organized as follows. Section 2 shows the 
equilibrium outcomes under Cournot and Stackelberg competition with no price 
discrimination. Section 3 focuses on price discrimination. Section 4 concludes.   2
 
2. The results under no price discrimination 
First, we briefly mention the results of both Cournot competition and Stackelberg 
competition under no price discrimination. 
Assume that there are two firms, firm 1 and firm 2, producing a homogeneous 
product with the constant marginal costs zero and  f cc =  respectively, with c > 0. We 
assume that each consumer buys at most one unit of the good and the consumers 
differ in terms of valuations.  We assume that the price of the product is given by 
 
12 1 Pq q =− − ,         ( 1 )  
where 
i q  is the output of firm i, i = 1, 2, and P is the price. 
 
2.1. Cournot competition 
If firms 1 and 2 produce like Cournot duopolists, straightforward calculation shows 
that the equilibrium output of firms 1 and 2 are respectively 
 
1 1
3
c c
q
+
=  and 
2 12
3
c c
q
−
= .        ( 2 )  
Both firms produce positive outputs for 
1
2
c < . 
The industry output, i.e., the total outputs of both firms, is 
2
3
c c
q
−
=  and the 
uniform price under Cournot competition is 
1
3
c c
p
+
= .           ( 3 )  
 
 
   3
2.2. Stackelberg competition 
Now consider Stackelberg competition where firm 1 behaves like a Stackelberg leader 
and firm 2 behaves like a Stackelberg follower. Straightforward calculations shows 
that the equilibrium outputs of firms 1 and 2 are respetively 
 
1 1
2
l c
q
+
=  and 
2 13
4
f c
q
−
= .        ( 4 )  
Both firms produce positive outputs for 
1
3
c < . 
The industry output is 
3
4
s c
q
−
=  and the uniform price under Stackelberg 
competition is 
1
4
s c
p
+
= .           ( 5 )  
 
3. The case of price discimnation 
Now we consider the case of price discrimination. 
 
3.1. Cournot competition 
We adopt the model under price discrimination from Hazledine (2006) and Kutlu 
(2009). The firms know the valuations of the consumers and can prevent resale of the 
good. The firms divide the consumers into different groups accoding to their 
valuations. Since the main results of Hazledine (2006) and Kutlu (2009), which have 
been mentioned in the introduction, do not depend on the number of different prices 
or the number of consumer groups, to show our results in the simplest way, we focus 
on two groups of consumers, i.e., consider two different prices. We assume that the 
prices of the product for group 1 and the group 2 are respectively 
 
12
11 1 1 Pq q =− −           ( 6 )    4
 
1212
21 1 2 2 1 Pq q q q =− − − − ,         ( 7 )  
where  i P  is the price for group i and 
j
i q  is the output of firm j for group i, where 
,1 , 2 ij =  and ij ≠ . The firms chosse their outputs simultaneously and the profits are 
realized. 
  The justification for such a setting follows from Hazledine (2006) and Kutlu 
(2009). Consider the airline industry, where the airline tickets are purchased in unit 
quantity. Consumers come to market at different times and their valuations differ. The 
airlines charge different prices to consumers with different valuations. 
  Given the demand cost functions, firms 1 and 2 determine their outputs 
simultaneously to maximize the following expressions respectively: 
 
11
12
12 1 1212 1
11 1 11 22 2
,
(1 ) (1 )
qq
Max q q q q q q q q −− +−−−−       ( 8 )    
 
22
12
12 2 1212 2
11 1 11 22 2
,
(1 ) (1 )
qq
Ma x qqc q qqqqc q −−− +−−−−− .     (9)   
Straightforward calculation gives: 
 
1
1
2
7
c
q
−
=  and 
1
2
13
7
c
q
+
=                             (10a) 
 
2
1
2
7
c
q
−
=  and 
2
2
14
7
c
q
−
= .                            (10b) 
Both firms produce positive outputs for both groups if 
1
4
c < . The comparison with no 
price discrimination under Cournot competition shows that if 
11
42
c <<, firm 2 
(which is more cost inefficient) serves only the high-valued consumers. 
  We concentrate on those values of cost asymmetries such that the outputs of 
both firms are positive for both groups of consumers. The equilibrium prices are then: 
  1
32
7
c
P
+
=  and  2
13
7
c
P
+
= .                    (11)   5
The output-weighted revenue or the average revenue for firms 1 and 2 are 
respectively 
11 2
1 11 22
, 11
12
1
32
ca v
Pq Pq cc
P
qq c
+ ++
==
++
 and 
22 2
2 11 22
, 22
12
12
35
ca v
Pq Pq c
P
qq c
+ −
==
+−
.            (12) 
The output-weighted industry revenue or average industry revenue is 
 
12 12
11 1 22 2
, 1212
11 22
() () 1
3
ca v
Pq q Pq q c
P
qqqq
++ + +
==
+++
.                  (13) 
 
Proposition 1: (a) Under Cournot competition, the average revenue for firm 1 (firm 
2) is lower (higher) under price discrimination than under no price discrimination, 
for any cost difference between firms 1 and 2. 
(b) Under Cournot competition, the average industry revenue is the same under no 
price discrimination and under price discrimination, irrespective of the cost 
difference between firms 1 and 2. 
Proof: The result (a) follows from the comparison between (3) and (12) and the result 
(b) follows from the comparison between (3) and (13). ■ 
 
  It is interesting to see from the above result that, under cost asymmetry, the 
average revenue of firm 1 (which is more cost efficient) is lower but that is of firm 2 
(which is more cost inefficient) is higher under price discrimination compared to no 
price discrimination. This is due to the effect of cost asymmetry on the equilibrium 
prices and outputs. 
The average revenue of firms 1 and 2 are the same under price discrimination 
and no price discrimination if there is no cost asymmetry, i.e., c = 0. However, as the 
cost asymmetry increases, i.e., c increases, under price discrimination, the prices for 
both groups increase, which tend to increase the average revenue of both firms 1 and   6
2. On the other hand, as c increases, it reduces firm 1’s relative share for the high-
valued consumers, while it increases its relative share for the low-valued consumers. 
This output composition effect tends to reduce the average revenue of firm 1. On the 
balance, the output composition effect dominates the price effect and reduces firm 1’s 
average revenue under price discrimination compared to no price discrimination.  
 In  contrast,  as  c increases, firm 2’s relative share for high-valued consumers 
increases, while its relative share for low-valued consumers reduces. As a result, firm 
2’s average revenue under price discrimination is higher compared to no price 
discrimination.  
While looking at the average industry revenue, the price effect balances with 
the output composition effect, and the average industry revenue is the same under 
price discrimination and no price discrimination. 
  It must be noted that, under cost asymmetry, even if the output-weighted 
revenue of firm 1 is lower under price discrimination compared to no price 
discrimination, the total revenue of firm 1 and that of firm 2 are higher under price 
discrimination than under no price discrimination for any 
1
(0, )
4
c∈ . Further, the total 
industry revenue is also higher under price discrimination than under no price 
discrimination. 
 
3.2. Stackelberg competition 
Now we consider price discrimination under Stackelberg competition, where firm 1 
behaves like a Stackelberg leader and firm 2 behaves like a Stackelberg follower. 
Again, we consider two groups of consumers, i.e., two different prices. 
  Given the demand functions (6) and (7), and the cost functions, firm 2 
determines its outputs to maximize the following expression:   7
 
22
12
12 2 1212 2
11 1 11 22 2
,
(1 ) (1 )
qq
Ma x qqc q qqqqc q −−− +−−−−− .                (14)  
Straightforward calculation gives: 
 
11
2 12
1
(1 )
3
cq q
q
−− +
=                                 (15a) 
 
11
2 12
2
12
3
cq q
q
−− −
= .                      (15b) 
Firm 1 maximizes the following expression to determine its outputs: 
 
11
12
12 1 1212 1
11 1 11 22 2
,
(1 ) (1 )
qq
Max q q q q q q q q −− +−−−−                   (16) 
subject to conditions (15a) and (15b). 
  The equilibrium outputs of firm 1 can be found as 
 
1
1
1
2
q =  and 
1
2 2
c
q = .                      (17) 
The equilibrium outputs of firm 2 are: 
 
2
1
(1 )
6
c
q
−
=  and 
2
2
14
6
c
q
−
= .                    (18) 
Both firms produce positive outputs for 
1
4
c < . The comparison with no price 
discrimination under Stackelberg competition shows that if 
11
43
c < < , firm 2 (which 
is more cost inefficient) serves only the high-valued consumers. 
  The following results are immediate from the above discussion. 
 
Proposition 2: (a) If 
1
0
4
c << , both firms produce positive outputs for both groups, 
i.e., both firms discriminate price for both groups. In this situation, both firms 
produce more for group 1, i.e., 
11
12 qq >  and 
22
12 qq > . 
   8
  Proposition 2(a) is in contrast to Kutlu (2009), which shows that the 
Stackelberg leader sells only to the high-valued concumers. However, we show that it 
is not the case with cost asymmetries, i.e., for c > 0. 
  The reason for Proposition 2(a) follows from Kutlu (2009). As in Kutlu 
(2009), while choosing the output for the low-valued consumers, firm 1 considers the 
implication it has on the output of firm 2 for the high-valued consumers. It is clear 
from (15a) and (15b) that a higher output of firm 1 for the low-valued consumers, i.e., 
1
2 q , will reduce firm 2’s output for the low-valued consumers, i.e., 
2
2 q , but it will 
increase firm 2’s output for the high-valued consumers, i.e., 
2
1 q . This reaction of firm 
2 eliminates firm 1’s incentive for serving the low-valued consumers in Kutlu (2009). 
However, if firm 2’s cost inefficiency compared to firm 1 increases, firm 1’s loss of 
profit in the high-valued consumer group reduces following firm 2’s higher output for 
the high-valued consumers, thus inducing firm 1 to serve also the low-valued 
consumers. However, firm 1 prefers to sell more outputs to the high-valued 
consumers. 
If there is no cost asymmetry between the firms, firm 2 sells the same amount 
to both groups of consumers. However, cost asymmetry encourages firm 1 to sell to 
the low-valued consumers also, which induces firm 2 to reduce its output for the low-
valued consumers and to produce more for the high-valued consumers. As a result, in 
the case of cost asymmetry, firm 2 also sells higher outputs to the high-valued 
consumers.  
  Now consider the average revenue under Stackelberg competition. Under price 
discrimination, the average revenue for firms 1 and 2 are respectively 
11 2
1 11 22
, 11
12
1
3(1 )
sa v
Pq Pq cc
P
qq c
+ ++
==
++
 and 
22 2
2 11 22
, 22
12
12
35
sa v
Pq Pq c
P
qq c
+ −
==
+−
.            (19)   9
  The average industry revenue under Stackelberg competition with price 
discrimination is 
 
12 12 2
11 1 22 2
, 1212
11 22
() () 3
2(5 2 )
sa v
Pq q Pq q cc
P
qqqq c
++ + + −
==
+++ −
.                  (20) 
 
Proposition 3: Under Stackelberg competition, the average revenues of firm 1, firm 2 
and the industry are higher under price discrimination than under no price 
discrimination, for any cost asymmetry between firms 1 and 2. 
Proof: The result follows from the comparison of the expressions in (5) and (19) and 
(20). ■ 
 
4. Conclusion 
We generalize the analyses of Hazledine (2006) and Kutlu (2009) to show the 
implications of cost asymmetries between the firms. We show that, under Cournot 
competition, the industry average revenue is the same under price discrimiation and 
no price discrimination, for any cost asymmetries between the firms. However, the 
average revenue of firm 1 (firm 2) is lower (higher) under price discrimination than 
under no price discrimination. Under Stackelberg competition, the average revenues 
of both firms and the industry are higher under price discrimination than under no 
price discrimination. We also show that both the Stackelberg leader and the 
Stackelberg follower price discriminate for any cost difference between the firms.   
 
 
 
   10
References 
Hazledine, T., 2006, ‘Price discrimination in Cournot-Nash oligopoly’, Ecoonmics 
Letters, 93: 413-20. 
Kutlu, L., 209, ‘Price discrimination in Stackelberg competition’, Journal of 
Industrial Economics, 57: 364. 
Stole, L.A., 2007, ‘Price discrimination and competition’, in M. Armstrong and R. 
Porter (Eds.), Handbook of Industrial Organization, Elsevier, vol. 3, ch. 34: 2221-99. 
 
 

Price discrimination and competition’,

Price discrimination in Cournot-Nash oligopoly’,

Price discrimination in oligopoly with asymmetric firms

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Price discrimination in oligopoly with asymmetric firms

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