Cournot vs. Bertrand Competitions in a Vertical Relationshipwith Separate Downstream Markets
Aika Monden
Shuhei Takezawa
Tomomichi Mizuno
November 2023
Discussion Paper No. 2318

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCournot vs. Bertrand Competitions in a Vertical
Relationship with Separate Downstream Markets∗
Aika Monden†aShuhei Takezawa‡b c

Tomomichi Mizuno§caGraduate School of Management, Kyoto UniversitybInstitute of Social and Economic Research, Osaka UniversitycGraduate School of Economics, Kobe University

November 1, 2023
Abstract
Contrary to conventional wisdom, our analysis of vertical relationships involving two geographically distinct downstream markets challenges the notion that
Bertrand competition yields lower proﬁts than Cournot competition. We demonstrate that when the market size where two downstream ﬁrms compete on either
quantity or price is smaller than the other downstream market, the input pricein downstream Bertrand competition is lower compared to Cournot competition.
Therefore, this may result in higher proﬁts for downstream ﬁrms under Bertrand
competition.

JEL codes: L13, D43,
Keywords: Cournot competition, Bertrand competition, vertical relationship∗

The authors gratefully acknowledge the ﬁnancial support of the Japan Society for the Promotionof Science (JSPS), KAKENHI Grant Numbers JP20K13618, JP22H00043, and JP23H00764. The usual
disclaimer applies.
†
Corresponding author: Aika Monden, Graduate School of Management, Kyoto University, Yoshida,
Sakyo-ku, Kyoto, 606-8501, Japan. E-mail: aikamonden@gmail.com‡
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 5670047, Japan. E-mail: takeshoot0216@gmail.com§
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo, 657-8501,
Japan. Phone: (81)-78-803-6802. E-mail: mizuno@econ.kobe-u.ac.jp1

1

Introduction

The literature on industrial organization has extensively examined the contrasts between
Cournot and Bertrand competitions (Singh and Vives, 1984). Conventionally, Bertrand
competition is deemed more competitive, leading to diminished proﬁts but augmented
consumer and overall surpluses. Following Singh and Vives (1984), subsequent studies,such as Arya et al. (2008) and Correa-López and Naylor (2004), analyze the outcomes of
Cournot and Bertrand competitions, exploring instances where the conventional wisdom
may not hold. Following this literature, we scrutinize the results on proﬁts.
We focus on a situation where an upstream ﬁrm supplies input to two geographically separate downstream markets; we compare the outcomes of Cournot and Bertrand
competitions in one downstream market. In this setting, increased competition in thedownstream market may lower the input price. The reason is that the input price depends on the price elasticities of demands for the input in both downstream markets,
and when competition increases in a downstream market with elastic input demand,the input price will decrease. Therefore, lower input prices can be realized in Bertrand
competition, which increases the proﬁts of downstream ﬁrms.
To incorporate the mechanism described above, we construct the following model.
An upstream ﬁrm produces input and sells it to downstream ﬁrms. There are two geographically separate downstream markets with three downstream ﬁrms. In one market,
two downstream ﬁrms compete, while in the other market, one downstream ﬁrm is amonopolist. We assume that the downstream markets have diﬀerent sizes. In a market
with two downstream ﬁrms, the downstream ﬁrms produce diﬀerentiated products andcompete on either quantity or price.
Comparing equilibrium outcomes under Bertrand and Cournot competitions, we ﬁnd
the following results. If the size of the downstream market with two downstream ﬁrms
is smaller than that with one downstream ﬁrm, the input price under Bertrand competition is smaller than that under Cournot competition. In addition, if the size of the2

downstream market with two downstream ﬁrms is suﬃciently small, the proﬁts of all
ﬁrms and consumers and total surpluses are larger under Bertrand competition thanunder Cournot competition. Therefore, Bertrand competition is potentially desirable forall economic agents.
Our study relates to the literature on the comparison of the outcomes of Bertrand andCournot competitions in vertical markets. This literature can be divided into two groups:
studies that consider Nash bargaining and studies that consider asymmetries in a downstream market. Correa-López and Naylor (2004) show that downstream ﬁrms obtain
substantial proﬁts under Bertrand competition than under Cournot competition whendownstream ﬁrms engage in Nash bargaining with their unions over wages. Subsequentstudies show that when upstream and downstream ﬁrms engage in Nash bargaining over
two-part tariﬀs, downstream ﬁrms earn substantial proﬁts under Bertrand competition(Alipranti et al., 2014; Basak and Mukherjee, 2017). Furthermore, in the frameworkof Nash bargaining and two-part tariﬀs, Liu and Wang (2020) incorporate downstream
ﬁrms’ investment and Wang and Li (2020) consider downstream ﬁrms’ relative proﬁt
maximization.
Some studies have shown that the proﬁt under Bertrand competition is higher thanunder Cournot competition by considering asymmetries in the downstream market; Aryaet al. (2008) and Fanti and Scrimitore (2019) discuss the situation where an upstreamﬁrm has direct and indirect sales channels, Matsuoka (2023) considers sequential contracts of input price, and Mukherjee et al. (2012) consider technological asymmetry
among downstream ﬁrms. Because we do not consider Nash bargaining, our study ismore related to studies that consider asymmetries between downstream ﬁrms. However,considering that our study is symmetric with downstream ﬁrms in a downstream market,
we can propose a new insight into this literature.

3

2

Model

We consider a vertical market with one upstream ﬁrm (U ) and three downstream ﬁrms
(Di, i ∈ {1, 2, 3}). In the downstream sector, two markets exist: market X and market
Y . Upstream ﬁrm U produces input and sells it to the downstream ﬁrms at input pricew. To produce one unit of the ﬁnal product, each downstream ﬁrm uses one unit of
input. Downstream ﬁrms D1 and D2 supply their products to market X; downstreamﬁrm D3 supplies its product to market Y . We assume that the production costs of all
ﬁrms are zero.
We assume that markets X and Y are geographically separated, and these markets
are independent. We denote the output and price of Di by qi and pi , respectively.
In market X, products produced by D1 and D2 are diﬀerentiated. We assume that
consumer surpluses in markets X and market Y are
1
CSX = aX (q1 + q2 ) − (q12 + q22 ) − γq1 q2 − p1 q1 − p2 q2 ,2
1 2
CSY = aY q3 − q3 − p3 q3 ,2
where aX , aY > 0 and γ is the degree of product substitutability, which satisﬁes 0 < γ <1. From the consumer surpluses, demand functions are q1 = [aX (1−γ)−p1 +γp2 ]/(1−γ 2 ),q2 = [aX (1 − γ) − p2 + γp1 ]/(1 − γ 2 ), and q3 = aY − p3 ; inverse demand functions are
p1 = aX −q1 −γq2 , p2 = aX −q2 −γq1 , and p3 = aY −q3 . To guarantee positive outputs inequilibrium, we assume that rmin < r < rmax , where r = aX /aY , rmin = (2 + γ)/(8 + 2γ),
and rmax = (10 + γ − γ 2 )/4.
From the above setting, the proﬁts of upstream and downstream ﬁrm Di are as
follows.
πU = w(q1 + q2 + q3 ),πDi = (pi − w)qi .

Consumer, producer, and total surpluses are CS = CSX + CSY , P S = πU + πD1 + πD2 +
πD3 , and T S = CS + P S, respectively.
4

Using the above setting, we compare outcomes under Cournot competition with thoseunder Bertrand competition. The timing of the game is as follows: In the ﬁrst stage,
the upstream ﬁrm U sets the input price. In the second stage, each downstream ﬁrmchooses its output or price. Using the backward induction, we solve this game.

3

Analysis

Cournot competition First, we consider the case of Cournot competition in themarket X. Solving the ﬁrst-order condition in the second stage, each downstream ﬁrmchooses the following output.
q 1 = q2 =aX − waY − w, q3 =
.
2+γ
2

(1)

Solving the ﬁrst-order condition in the ﬁrst stage leads to an equilibrium input price.
wC =4aX + aY (2 + γ),
2(6 + γ)where the superscript ‘C’ denotes the case under Cournot competition.
The equilibrium outcomes are as follows.
[4aX − aY (10 + γ)]2[2aX (4 + γ) − aY (2 + γ)]2
C
,
π
,
=
D3
4(2 + γ)2 (6 + γ)216(6 + γ)2[4aX + aY (2 + γ)]2πUC =,8(2 + γ)(6 + γ)
(1 + γ)[2aX (4 + γ) − aY (2 + γ)] [4aX − aY (10 + γ)]2
CS C =+,
4(2 + γ)2 (6 + γ)232(6 + γ)2CCC
T S C = πD1
+ πD2
+ πD3
+ πUC + CS C .
C
C
πD1
= πD2
=

Bertrand competition Next, we consider the case of Bertrand competition. Bysolving the ﬁrst-order conditions, we obtain ﬁnal product prices.
p1 = p2 =aY + waX (1 − γ) + w, p3 =
.
2−γ
2
5

(2)

To maximize the proﬁt of the upstream ﬁrm, we determine the equilibrium input price.
wB =4aX + aY (2 + γ − γ 2 ),2(6 + γ − γ 2 )where the superscript ‘B’ denotes the case under Bertrand competition.
Subsequently, the equilibrium outcomes are as follows.
(1 − γ)[2aX (4 + γ − γ 2 ) − aY (2 + γ − γ 2 )]2,4(2 − γ)2 (1 + γ)(6 + γ − γ 2 )2[4aX − aY (10 + γ − γ 2 )]2
B
πD3
=
,
16(6 + γ − γ 2 )2[4aX + aY (2 + γ − γ 2 )]2πUB =,8(2 − γ)(1 + γ)(6 + γ − γ 2 )[2aX (4 + γ − γ 2 ) − aY (2 + γ − γ 2 )] [4aX − aY (10 + γ − γ 2 )]2CS B =+,
4(2 − γ)2 (1 + γ)(6 + γ − γ 2 )232(6 + γ − γ 2 )2B
B
B
T S B = πD1
+ πD2
+ πD3
+ πUB + CS B .
B
B
πD1
= πD2
=

Comparison of input prices First, we analyze the diﬀerence in input prices between
Cournot and Bertrand competitions. By comparing wB with wC , we obtain the following.
wB − wC =2(aX − aY )γ 2.
(6 + γ)(6 + γ − γ 2 )This result directly leads to Lemma 1.
Lemma 1 Upstream ﬁrm U selects a lower input price under Bertrand competition thanunder Cournot competition if the size of market X is smaller than that of market Y .
Speciﬁcally, wB < wC if aX < aY .
To provide insight into Lemma 1, we examine the price elasticity of input demand.
From the outcomes in the second stage, (1) and (2), the price elasticities of input demands
in markets X and Y are expressed as εX = w/(aX −w) and εY = w/(aY −w), respectively.
Notably, these values remain consistent across both Cournot and Bertrand competitions.
Hence, if the size of market X is smaller than that of market Y , aX < aY , the priceelasticity of input demand in market X surpasses that in market Y .
6Next, we analyze the price elasticity of total input demand, derived from the weighted
average of price elasticities in markets X and Y . Denoting the second-stage outputsof downstream ﬁrm Di in Cournot and Bertrand competitions as qiC (w) and qiB (w)respectively, we proceed to present the price elasticities of total input demand for Cournot
and Bertrand competitions as follows.
q1C (w) + q2C (w)q3C (w)
ε = C· εX + C· εY
q1 (w) + q2C (w) + q3C (w)q1 (w) + q2C (w) + q3C (w)w(6 + γ)=,
4aX + aY (2 + γ) − w(6 + γ)
q B (w) + q2B (w)q3B (w)
εB = B 1
·
ε
+
· εY
X
q1 (w) + q2B (w) + q3B (w)q1B (w) + q2B (w) + q3B (w)w(3 − γ)(2 + γ)=.
4aX + aY (2 + γ − γ 2 ) − w(6 + γ − γ 2 )CThe coeﬃcient of εX is larger under Bertrand competition than under Cournot competition because intense competition results in larger outputs. Thus, if εX > εY , whichis equivalent to aX < aY , the price elasticities of total input demand are greater under
Bertrand competition compared to Cournot competition: εB > εC . Therefore, in thiscase, the upstream ﬁrm U sets a lower input price under Bertrand competition thanunder Cournot competition.
Comparison of proﬁts First, we consider the proﬁt of downstream ﬁrms Di, wherei = 1, 2. Using r = aX /aY , we obtain the following.
B
C
πDi
− πDi
=

a2Y γ 2 (Ψ2 r2 + Ψ1 r + Ψ0 ),2(3 − γ)2 (2 − γ)2 (1 + γ)(2 + γ)2 (6 + γ)2

where Ψ0 = −(2 − γ)2 (12 + 17γ + 6γ 2 + γ 3 ), Ψ1 = 4(60 + 40γ − 33γ 2 − 11γ 3 + 3γ 4 + γ 5 ),BCand Ψ2 = −4(48 + 116γ − 52γ 2 − 17γ 3 + 4γ 4 + γ 5 ). Solving πDi− πDi> 0 for r, we obtainr < rD , where
60 + 40γ − 33γ 2 − 11γ 3 + 3γ 4 + γ 5 + (36 − 24γ + γ 2 + γ 3 )rD =2(48 + 116γ − 52γ 2 − 17γ 3 + 4γ 4 + γ 5 )7√1 − γ2.

B
C
Next, we compare πD3with πD3
.
B
C
πD3
− πD3
=

a2Y (1 − r)γ 2 [60 + 16γ − 7γ 2 − γ 3 − r(24 + 4γ − 2γ 2 )].
2(3 − γ)2 (2 + γ)2 (6 + γ)2

BCSolving πD3− πD3> 0 for r, we obtain r < 1.

Finally, comparing πUB with πUC , we obtain the following.
πUB − πUC =a2Y γ 2 [4r2 (4 − γ) + r(4 + 2γ − 2γ 2 ) − 2 − γ + γ 2 ]> 0,2(3 − γ)(2 − γ)(1 + γ)(2 + γ)(6 + γ)

where from rmin < r < rmax , the aforementioned inequality is resolve. By summarizingthese results, we obtain the following proposition.
Proposition 1 (i) Downstream ﬁrms D1 and D2 earn larger proﬁts under Bertrand
competition than under Cournot competition if r < rD . (ii) The proﬁts of downstreamﬁrm D3 under Bertrand competition is larger than that under Cournot competition if
r < 1. (iii) The proﬁt of upstream ﬁrm U under Bertrand competition is larger than that
under Cournot competition.
From Proposition 1, under Bertrand competition, all ﬁrms obtain larger proﬁts thanunder Cournot competition, if the size of market X is suﬃciently smaller than that ofmarket Y .
We explain the intuition behind Proposition 1. In a price competition between downstream ﬁrms D1 and D2, their proﬁts are small because of heightened competition. We
call this the competition eﬀect. Additionally, as deduced from Lemma 1, in the scenario
of aX < aY , or equivalently, r < 1, the upstream ﬁrm opts for a lower input price under
Bertrand competition than under Cournot competition. We refer to this eﬀect as the
input price eﬀect. The input price eﬀect becomes robust even if r is small. Thus, whenr is suﬃciently small, the input price eﬀect dominates the competition eﬀect, leading
to higher proﬁts for downstream ﬁrms D1 and D2 under Bertrand competition thanunder Cournot competition. Hence we obtain (i) in Proposition 1. Next, we consider
8

(ii) in Proposition 1. When the input price eﬀect lowers the input price, the proﬁt ofdownstream ﬁrm D3 increases because markets X and Y are independent. Finally, we
consider (iii) in Proposition 1. Because heightened competition due to Bertrand competition partially resolved the double marginalization problem, the proﬁt of the upstreamﬁrm U increased.
Comparison of surpluses We compare consumer and total surpluses between Cournot
and Bertrand competition.
 2
2r (1008 + 312γ − 576γ 2 − 14γ 3 + 67γ 4 − γ 5 − 2γ 6 )−r(1008 + 528γ − 744γ 2 − 132γ 3 + 139γ 4 + 2γ 5 − 5γ 6 ) 
a2Y γ 2 
+2(2 − γ)2 (36 + 51γ + 12γ 2 − 4γ 3 − γ 4 )CS B − CS C => 0,
4(3 − γ)2 (2 − γ)2 (1 + γ)(2 + γ)2 (6 + γ)2 2

2r (1872 − 1080γ − 544γ 2 + 270γ 3 + 49γ 4 − 15γ 5 − 2γ 6 )−r(144 − 144γ − 280γ 2 + 100γ 3 + 77γ 4 − 18γ 5 − 3γ 6 ) 
a2Y γ 2 
+2(2 − γ)2 (36 + 45γ + 2γ 2 − 8γ 3 − γ 4 )BC> 0,TS − TS =4(3 − γ)2 (2 − γ)2 (1 + γ)(2 + γ)2 (6 + γ)2
where because of rmin < r < rmax , the above inequalities satisﬁed. Thus, we obtain thefollowing proposition.
Proposition 2 Consumer and total surpluses under Bertrand competition are larger
than those under Cournot competition.
The intuition behind this result is similar to that behind (iii) in Proposition 1. Specifically, because Bertrand competition mitigates the double marginalization problem, consumer and total surpluses are larger under Bertrand competition than under Cournot
competition.

4

Conclusions

We consider a vertical market with one upstream and three downstream ﬁrms. Two of thedownstream ﬁrms compete in a downstream market and one of them is a monopolist in
9the other downstream market. By considering these two distinct downstream markets,
we show that an increase in output in a competing downstream market increases theprice elasticity of inputs, which in turn decreases input prices. Thus, although Bertrand
competition increases competition, it can beneﬁt downstream ﬁrms by lowering input
prices. Our results add a new insight that proﬁts in Bertrand competition are larger
than those in Cournot competition, even when the competing ﬁrms are symmetric.
Our study has several limitations. First, we assume uniform linear contracts between
upstream and downstream ﬁrms. If an upstream ﬁrm discriminates input prices or uses
non-linear contracts, such as a two-part tariﬀ, the impact of downstream market competition on input prices will be weaker and our results may not be obtained. Moreover,even if the monopolistic downstream market becomes oligopolistic, the impact of downstream market competition on input prices would be weakened. We acknowledge thesigniﬁcance of these considerations and propose them as areas for future research.

10

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