Comparing Bertrand and Cournot Competition
with Product Innovation and Licensing
Ray-Yun Chang, Hong Hwang and Cheng-Hau Peng
To be presented at the IO Workshop
The University of Tokyo
April 22, 2015
Introduction
• Singh and Vives (1984) show that Bertrand
competition is more efficient but less profitable for
firms than Cournot competition when goods are
substitutes.
• This standard result has drawn considerable attention
and been challenged by sizeable theoretical literature.
Related literature
• Differentiated goods:
Singh and Vives (1984, RAND), Vives (1985, JET),
Cheng (1985, RAND) and Okuguchi (1987, JET).
• Firm’s R&D behavior:
Delbono and Denicolo (IJIO,1990), Reynolds and
Isaac (ET,1992), Qiu (JET, 1997); Bonanno and
Haworth(IJIO, 1998), Boone(IJIO, 2001), Symeonidis
(IJIO, 2003) and Mukherjee (MS, 2011).
Related literature
• Spatial context:
D'Aspremont and Motta (2000),
Liang et al (2006, RSUE)
• Number of firms:
Häckner (2000, JET)
Related literature
• Labor union:
López and Naylor (2004, EER)
• Mixed oligopoly:
Ghosh and Mitra (2010, EL)
Motivation
• Empirical evidences have shown that most of the innovations
are on product innovation.
• Qualcomm licensed its new wireless technology, which is a
product innovation, to Motorola (Mock, 2005).
• BlackBerry licensed its innovated wireless e-mail services to
Nokia (Frankel 2005).
• Biovail Corp. licensed from Depo Med, Inc. the rights to
manufacture and market a once-daily metformin product
that was undergoing Phase 3 clinical trials for Type II
diabetes.
Motivation
• There is a common feature of the above examples:
The licensor firms license its product innovation to
and compete in the output market with its licensee
firm.
• This is the first paper that compares the relative
merits of Bertrand and Cournot equilibria if one of the
firms licenses its product innovation to its rival.
Preview of our findings
• The licensor always licenses its product innovation to
the (potential) rival.
• Under the product innovation licensing:
1. the optimal royalty rate under Bertrand competition is
definitely higher than that under Cournot competition;
2. market output is smaller but industrial profit is higher
under Bertrand than Cournot competition;
3. Bertrand competition is less socially desirable than
Cournot competition.
Preview of our findings
• If the licensee is an incumbent firm, Cournot
competition, relative to Bertrand competition, results
in higher (lower) social welfare but less (more)
producer surplus if the innovation is high (low).
Preview of our findings
• If the innovator can engage in product R&D to
enhance its quality in the long run:
1. The innovator definitely does more product innovation
under Bertrand competition than Cournot competition.
2. Bertrand competition becomes more socially desirable
than Cournot competition if the R&D efficiency is high.
Outline of this paper
• Section 2 introduces our basic model, in which the licensee
firm is a potential entrant, and compares the relative merits
between under Corunot and Bertrand competition.
• Section 3 examines the case in which the licensee firm is also
an incumbent firm.
• Section 4 investigates and compares the long run equilibria
in which the licensor firm can carry out product R&D.
• Section 5 concludes the paper.
THE BASIC MODEL
Model settings
• Assume there are two firms in the market. Firm 1 is a
licensor firm who owns a new innovation and can use it to
produce product 1 to be sold in the market. Firm 1 also
licenses this know-how to a rival, firm 2, who can use the
innovation to produce a differentiated product (called
product 2) to be sold in the same market.
• The two products though developed by the same innovation,
are horizontally differentiated due to different plant
locations or brand names.
Model settings
• Following Singh and Vives (1984), the demand and the
inverse demand functions for the two products are specified
as follows:
qi  qi ( pi , p j ) 
a( f  h)  fpi  hp j
f h
2
2
, and pi  pi ( qi , q j )  a  fqi  hq j , (1)
for i, j  1, 2; i  j, where qi and pi are the outputs for firm i,
a is the price intercept and f denotes the self-price effect
which greater than h , the cross-price effect.
Model settings
• Firm 1 licenses its product innovation to firm 2 via a two-part
tariff licensing contract, i.e., an upfront fee (F) plus a per-unit
royalty ( r).
• Following Singh and Vives (1984), we assume the marginal
costs of the two firms to be nil for simplicity.
• Before licensing, firm 1 is a monopolist in the market,
earning a monopoly profit (  1M ). After licensing, the market
becomes that of differentiated duopoly.
Game structure
• The game in question consists of two stages.
• First stage: firm 1 chooses the optimal royalty and fixed fee
and firm 2 determines whether or not to accept the licensing
contract.
• Second stage: the two firms compete in either Bertrand or
Cournot fashion.
• The sub-game perfect Nash equilibrium is solved through
backward induction.
• We begin our analysis by considering the Cournot regime
first, followed by the Bertrand regime.
THE COURNOT EQUILIBRIUM
The profit functions in the output stage
• The profits of firm 1 and firm 2 under the Cournot
regime are specified respectively as follows:
 1  p1q1  rq2  F ,
(2)
 2   p2  r  q2  F ,
(3)
where variables with a superscript “C” indicate that
they are associated with the Cournot regime.
Equilibrium and comparative statics
• By routine calculus, we have:
 1
p
 p1  1 q1  0
q1
q1
,
(4)
 2
p
  p2  r   2 q2  0
.
q2
q2
(5)
• The second-order and the stability conditions are all satisfied.
• The comparative static effects are derivable from and as
follows:
q r  h (4 f  h )  0 ,
C
1
2
2
q r  2 f (4 f  h )  0.
C
2
2
2
Figure 1. The reaction functions under Cournot
q2
R1
C
C0
R2 r  r
0
C1
R2
0
D
q1
r  r1
The objective function in the first stage
• The profits of firm 1 in the first stage can be expressed as
follows:
Max  1C  q1C ( r ), q2C ( r ), F ( r ), r   p1q1  rq2  F,
r
(9)
s. t. r , F  0 .
• We assume that the licensor firm can extract the entire rent
of licensing accruing to the licensee firm. Hence, the fixed
fee charged by the licensor firm is as follows:
F (q1C (r), q2C (r ), r )   p2  r  q2   2M   p2  r  q2.
(8)
The optimal licensing contract
• By differentiating (9) with respect to r and applying
the envelope theorem, we can derive the first-order
condition for profit maximization as follows:
 q2C  p2 C  q1C
 1  1 q2C  1 F  1  p1 C
 C



q1  r 

q2 
r q2 r
F r
r  q2
 r  q1  r
 (hq  r )(q
C
1
C
2
r )  hq (q
C
2
C
1
r )=0 .
The Cournot equilibrium
• The optimal royalty rate is:
C
C
C
C
h
[
q
(

q

r
)+
q
(

q
C
1
2
2
1 r )]
r 
0
C
(q2 r )
.
(10)
• In addition, by comparing the profits of firm 1 before and after
licensing with Cournot competition, we can derive that
 1C   1M  a 2 ( f  h )2 f (4 f 2  3h 2 )  0 .
(11)
• Thus, product licensing necessarily occurs under Cournot.
Figure 2. The equilibria under Cournot and Bertrand
q2
q1  q2
B
C
G
C*
B*
0
E
D
q1
THE BERTRAND EQUILIBRIUM
Equilibrium of the output stage
• By substituting the demand functions into (1) and (2), then
differentiating (1) and (2) with respect to p1 and p2
respectively, we have:
 1
q
q
 q1  p1 1  r 2  0 .
p1
p1
p1
(11)
 2
q
 q2   p2  r  2  0 .
p2
p2
(12)
• The comparative static effects are as follows:
p1B r  3hf (4 f 2  h 2 )  0
p2B r  (2 f 2  h 2 ) (4 f 2  h 2 )  0
Figure 3. The reaction functions under Bertrand
q2
R1 r  r R1 r  r0
1
B
B0
R2
r  r0
R2
r  r1
B1
0
E
q1
Equilibrium in the first stage
• The object function for firm 1 is specified as follows:
Max  1B  p1B ( r ), p2B ( r ), F ( r ), r   p1q1  rq2  F ,
r
(11)
s. t. r , F  0 .
• By routine calculus, we can derive that:
 p1B  q1 p2   p2B r   p2B  q2 p1   p1B r 
 0
rB  
.
B
B
 q2 p1   p1 r    q2 p2  p2 r 
(12)
Equilibrium in the first stage
• By comparing the profits of firm 1 before and after licensing
under Bertrand competition, we can derive that:
 
B
1
a ( f  h)(h  f )

0
2
2
.
f ( f  h)(5h  4 f )
2
M
1
2
2
Figure 2. The equilibria under Cournot and Bertrand
q2
q1  q2
B
C
G
C*
B*
0
E
D
q1
Proposition 1
The licensor firm always licenses its product innovation to a
potential rival.
Intuition:
1. As the two products are differentiated, the market profit
increases if both products are available.
2. The licensor firm can use the royalty to reduce the
competition from the licensee firm and the fix fee to
extract the rent accruing to the licensee firm.
Comparison on the optimal royalty rates
• By comparing and , we can derive that
r  r  4ah
B
C
2
 f  h 4 f
2
h
2
 f  4f
2
+5h
2
 4f
2
 3h   0 .
2
Proposition 2
The optimal royalty rate under Bertrand competition is definitely
higher than that under Cournot competition.
Intuition:
1. The objective of firm 1 in the first stage of the game is to
maximize the market profits.
2. Relative to the output which maximizes the market profit, the
output under the Bertrand (Cournot) equilibrium is much too
high (too high). As a result, the licensor firm would set a high
(low) royalty.
Comparison on the output and profit levels
• By comparing the equilibrium outputs under the two
regimes, it is found that
2
2
2
ah
(
f

h
)(4
f

8
hf

h
)
C
B
Q Q 
0
2
2
2
2
f (4 f  5h )(4 f  3h )( f  h)
.
• By substituting the equilibrium outputs, prices and licensing
contracts into the corresponding profits of the licensor under
the two regimes, we can derive that
2 4
2
a
h ( f  h)
B
C
1  1 
0.
2
2
2
2
f (4 f  5h )(4 f  3h )( f  h)
Proposition 3
With new product licensing, market output is smaller but
market profit is higher under Bertrand than Cournot
competition.
Intuition:
The higher royalty rate under Bertrand decreases the market
output, increasing the market profit.
Proposition 4
If the licensee is a potential entrant, Cournot competition is
socially more desirable than Bertrand competition.
Intuition:
The market outputs are lower under Bertrand competition,
leading to higher market prices and lower social welfare.
PRODUCT INNOVATION AND
LICENSING UNDER DUOPOLY
The licensee firm is an incumbent
• In this section, we assume that the licensee firm (i.e., firm 2),
also being an incumbent.
• Firm 2 has an incentive to acquire the technology from the
licensor firm as it can raise the demand for its product. We will
investigate whether our results remain robust in this context.
• Before licensing, firm 2 produces a differentiated product. The
demand and inverse demand functions of firm 1 and firm 2 are
the same as those in , except that the price intercept of the
2
2
afh
(2
f

h
)  b  a.
demand of firm 2 is b where
Proposition 5
If the licensee is an incumbent firm, Cournot competition is more efficient
but less profitable for firms than Bertrand competition, if the innovation
level is high. The converse is true if the innovation degree is low.
Intuition:
1. If is b equal to a , firm 2 has no incentive to buy the technology. Our
result is the same as that in Singh and Vives (1984).
2. If b is equal to zero (i.e., there is no demand for product 2 before
licensing), the model degenerates to the case with a potential entrant.
3. There exists a critical value of b, below which Bertrand competition is
less socially desirable but more profitable than Cournot competition.
INNOVATION AND WELFARE
Endogenous innovation
• In the long run, a licensor can determine its product innovation
endogenously which increases the price intercept of its demand from
a to a  a . Thus, the demand and the inverse demand functions for
the two products are re-written as follows:
pi  pi ( qi , q j ,  ; a )  a    fqi  hq j, and
qi  qi ( pi , p j ) 
(a   )( f  h)  fpi  hp j
f 2  h2
, for i , j  1, 2; i  j .
Endogenous innovation
• The product R&D cost function is specified by v 2,
where v reflects the R&D efficiency and a higher
indicates lower R&D efficiency.
Game structure
• The game in question now encompasses three stages.
• The last two stages are the same as those in the previous section.
• We need to work out only the first-stage game: Firm 1
determines its optimal product innovation. We will compare the
optimal product R&D levels and the resulting welfare levels
under the two competition modes.
The objective functions in the first stage
• In the first stage, the profit functions of firm 1 under
Cournot and Bertrand competition can be specified
respectively as follows:
Max 1C  q1C ( ), q2C ( ), r ( ), F ( ),    (a    fq1C  hq2C )q1C  rq2C  F  v 2

Max  1B  p1B ( ), p2B ( ), r ( ), F ( ),  

 p1B (
(a   )( f  h)  fp1B  hp2B
f 2  h2
)r
(a   )( f  h)  fp2B  hp1B
f 2  h2
 F  v 2 .
The optimal product investments and comparisons
• The optimal product innovations under Cournot and
Bertrand competition as follows:
2
2
a
(8
f

8
fh

h
)
C
 
4 fv(4 f 2  3h 2 )  (8 f 2  8 fh  h 2 )
2
2
3
a
(8
f

9
fh

h
)
B
 
4 fv( f  h)(4 f 2  5h 2 )  (8 f 2  9 fh 2  h 3 )
• Thus, we derive that
4
32
avfh
( f  h)
 B  C 
 0.
2
2
2
2
2
2
2
2
3
 4 fv(4 f  3h )  (8 f  8 fh  h )  4 fv( f  h)(4 f  5h )  (8 f  9 fh  h ) 
Proposition 6
• The licensor firm will do more product innovation
under Bertrand than Cournot competition.
This finding is contrary to that in Qiu (1997), Bonanno and Haworth (1998) and
Symeonidis (2003). Qiu (1997) and Bonanno and Haworth (1998) consider costreducing R&D whereas Symeonidis (2003) considers product R&D; they all
conclude that Cournot competition induces a higher R&D expenditure than
Bertrand competition.
Proposition 6
Intuition:
1. For a given technology, firm 1 makes more profits under
Bertrand competition. This implies that the marginal
benefit from product innovation is higher.
2. Given the same innovation cost function, the innovation
level is necessarily higher under Bertrand than Cournot
competition.
Welfare comparison
• We can calculate social welfare under the two regimes and
derive that
sw  sw  ()0 if v  ( )v
C
B
Proposition 7
In the long run, Bertrand competition is socially more (less)
desirable than Cournot competition if the R&D efficiency is
high (low).
Intuition:
By Proposition 6, firm 1 always invests more on innovation under
Bertrand competition which benefits social welfare. If the R&D
efficiency is high, this beneficial effect becomes significant, making
Bertrand competition socially more desirable than Cournot competition.
Summary
Short run
Potential Entrant
Long run
Duopoly
Innovation levels
Royalty rates
Producer surplus
Social welfare
Potential Entrant
 B  C
r r
B
ps  ps
B
B
C
ps B  () ps C
C
sw  sw
C
r r
C
B
swC  () swB
r B  rC
ps B  ps C
swC  () swB
Thank you
Comments and suggestions are welcome