Kutlu (2009, â€œPrice discrimination in Stackelberg competitionâ€, Journal of Industrial Economics) shows that the Stackelberg leader sells to the highest value consumers and only the Stackelberg follower practises price discrimination. We show that this result is not robust if the marginal cost of the leader is lower than that of the follower. In this situation, both the leader and the follower practise price discrimination.Cost asymmetry; Price discrimination; Stackelberg competition

Arijit Mukherjee

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 4 
  
Price discrimination in oligopoly with asymmetric firms 
  
 
 
Arijit Mukherjee  
University of Nottingham 
Abstract 
Kutlu (2009, “Price discrimination in Stackelberg competition”, Journal of Industrial Economics) shows that the 
Stackelberg leader sells to the highest value consumers and only the Stackelberg follower practises price 
discrimination. We show that this result is not robust if the marginal cost of the leader is lower than that of the 
follower. In this situation, both the leader and the follower practise price discrimination.
I thank an anonymous referee for valuable comments and suggestions. The usual disclaimer applies. 
Citation: Arijit Mukherjee, (2010) ''Price discrimination in oligopoly with asymmetric firms'', Economics Bulletin, Vol. 30 no.4 pp. 2668-
2670. 
Submitted: Aug 12 2010.   Published: October 10, 2010. 
 
       1
1. Introduction 
The literature on price discrimination considers different types of price discrimination. 
Stole (2007) provides a nice survey of this literature. In a recent paper, Kutlu (2009) 
considers second degree price discrimination in a Stackelberg model, where the firms 
are 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. In this framework, Kutlu (2009) shows that the 
Stackelberg leader sells to the highest value consumers and only the Stackelberg 
follower practises price discrimination. We show that the above result of Kutlu (2009) 
does not hold if the marginal cost of the leader is lower than that of the follower. In 
this situation, both the leader and the follower practise price discrimination. 
  The remainder of the paper is organized as follows. Section 2 describes the 
model and derives the results. Section 3 concludes. 
 
2. The model and the results 
Consider an economy with a Stackelberg leader (firm 1) and a Stackelberg follower 
(firm 2). We assume that the firms produce a homogeneous product with constant 
marginal costs. The marginal costs of firms 1 and 2 are respectively zero and c, with 
c > 0. Hence, firm 1 is more cost efficient than firm 2. We assume that each consumer 
buys at most one unit of the good and the consumers differ in terms of valuations.   
We adopt the price discrimination model of 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 according to their 
valuations. Since the main result of Kutlu (2009), which says that only the 
Stackelberg follower practises price discrimination, does not depend on the number of 
different prices or the number of consumer groups, we focus on two groups of 
consumers (i.e., consider two different prices) to show our result in the simplest way.  
We assume that the prices for groups 1 and 2 are respectively 
 
12
11 1 1 Pq q =− −           ( 1 )  
 
1212
21 1 2 2 1 Pq q q q =− − − − ,         ( 2 )  
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 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 the market at different times and their valuations differ. 
The airlines charge different prices to consumers with different valuations. 
  We consider the following game. At stage 1, firm 1 determines its outputs. At 
stage 2, firm 2 determines its outputs and the profits are realized. We solve the game 
through backward induction. 
  Given the demand and cost functions, firm 2 determines its outputs to 
maximize the following expression: 
 
22
12
12 2 1212 2
11 1 11 22 2
,
(1 ) (1 )
qq
Ma x qqc q qqqqc q −−− +−−−−− .                  (3)  
Straightforward calculation gives: 
 
11
2 12
1
(1 )
3
cq q
q
−− +
=                                   (4a) 
 
11
2 12
2
12
3
cq q
q
−− −
= .                        (4b)   2
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 −− +−−−− ,                   (5) 
subject to (4a) and (4b). 
  The equilibrium outputs of firm 1 can be found as 
 
1
1
1
2
q =  and 
1
2 2
c
q = .                        (6) 
Hence, the equilibrium outputs of firm 2 are: 
 
2
1
(1 )
6
c
q
−
=  and 
2
2
14
6
c
q
−
= .                      (7) 
Both firms produce positive outputs for  1
4 c < . If  1
4 c < , firm 2 serves only the high 
value consumers. If all the outputs are positive, the equilibrium prices are 
2
1 6
c P
+ =  and 
12
2 6
c P
+ = . 
  The following results are immediate from the above discussion. 
 
Proposition 1:  If  1
4 0 c <<, both firms produce positive outputs for both groups, and 
the outputs are higher for group 1, i.e., 
11
12 qq >  and 
22
12 qq > . 
 
  In contrast to Kutlu (2009), Proposition 1 shows that cost difference between 
the leader and the follower forces both firms practising price discrimination. 
The reason for our result is as follows. While choosing 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 (4a) and (4b) that higher output of firm 1 
for the low-valued consumers reduces firm 2’s output for the low-valued consumers, 
but it increases firm 2’s output for the high-valued consumers. It follows from Kutlu 
(2009) that, under symmetric costs, this strategic reaction of firm 2 eliminates firm 1’s 
incentive for serving the low-valued consumers. However, cost difference between 
the firms forces firm 2 (the less cost efficient firm) to reduce sales to the low-valued 
consumers and increase those to the high value ones, thus encouraging firm 1 (the 
more cost efficient firm) to enter the low value segment through price discrimination. 
 
3. Conclusion 
We show that the main result of Kutlu (2009), which says that the Stackelberg leader 
sells to the highest value consumers and only the Stackelberg follower practises price 
discrimination, does not hold under cost asymmetry. Under cost asymmetry, both the 
leader and the follower practise price discrimination.  
 
References 
Hazledine, T. (2006) “Price discrimination in Cournot-Nash oligopoly” Economics 
Letters 93, 413-20. 
 
Kutlu, L. (2009) “Price discrimination in Stackelberg competition” Journal of 
Industrial Economics 57, 364. 
 
Stole, L.A. (2007) “Price discrimination and competition”, in Handbook of Industrial 
Organization by M. Armstrong and R. Porter, Eds., Elsevier, Amsterdam, The 
Netherlands, Vol. 3, Ch. 34, 2221-99. 
 

Price discrimination in oligopoly with asymmetric firms

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