Bertrand competition in vertically related markets
Tomomichi Mizuno
Kazuhiro Takauchi
April 2022
Discussion Paper No. 2208

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANBertrand competition in vertically related markets
Tomomichi Mizuno∗

Kazuhiro Takauchi†‡April 12, 2022
Abstract
We build a successive Bertrand model with homogenous good. We show that increasing the production eﬃciency of upstream industry can reduce upstream firms’ profits. We also show that
increasing the production eﬃciency of downstream industry may reduce downstream firms’ profits. Hence, an industrial policy that aims at improving production eﬃciency may be undesirable
for firms.

Key words: Successive Bertrand; Production eﬃciency; Homogenous good
JEL classification: L13∗

Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City, Hyogo 657-8501,
Japan. E-mail: mizuno@econ.kobe-u.ac.jp.
†
Corresponding author: Kazuhiro Takauchi, Faculty of Business and Commerce, Kansai University, 3-3-35
Yamate-cho, Suita, Osaka 564-8680, Japan. E-mail: kazh.takauch@gmail.com; Tel.: +81-6-6368-1817.
‡
Research Fellow, Graduate School of Economics, Kobe University.

1

Introduction

We consider Dastider (1995)-type homogenous-good Bertrand competition in both upstream anddownstream markets. We show that when upstream and downstream production is eﬃcient, increasing upstream and downstream production eﬃciency reduces upstream and downstream
firms’ profits, respectively. Because improving eﬃciency in a vertical structure does not necessarily increase firms’ profits, our results imply that aiming to improve eﬃciency is a problematic
issue in industrial policy.
According to an empirical study conducted by Flath (2012), among 70 Japanese industries,
35 engage in Bertrand competition, whereas five engage in Cournot competition. Although weassume homogenous-good Bertrand competition in both upstream and downstream markets,
empirical evidence suggests that price competition is more common than quantity competition;
hence, our model is partially consistent with the results of the empirical study.
We incorporate Dastider (1995)-type Bertrand competition into a Salinger (1988)-type vertical structure model. Salinger (1988) assumed Cournot competition in both upstream anddownstream markets.1 Dastider (1995) examined homogenous Bertrand competition with aconvex cost of firms, and demonstrated that, in homogenous Bertrand competition, equilibrium
has a range if oligopolists have a convex cost.2 To the best of our knowledge, price competition
for a homogeneous product in both upstream and downstream markets has not been considered
in a study.

2

Model

We consider a market with m upstream and n downstream firms. Each upstream and downstream firm produces homogenous input and a final product, respectively. Upstream firms
1

Salinger’s (1988) model is frequently used in industrial organization literature, for example, see Matsushima(2006) and Mukherjee (2019).
2
Dastider’s model (1995) is applied in various scenarios; see, for example, Cabon-Dhersin and Drouhin (2014),
Mizuno and Takauchi (2020), and Takauchi and Mizuno (2021).

1

have a convex cost function; we assume that the cost is γx2k if the output of upstream firmk (= 1, . . . , m) is xk . Upstream firms compete on price, and upstream firm k’s input price isdenoted by wk . Then, upstream firm k’s profit is πku ≡ wk xk − γx2k .
Each downstream firm purchases input from upstream firms, choosing the lowest input pricew ≡ min{w1 , . . . , wm }. If multiple upstream firms choose w, then downstream firms purchase anequal quantity of inputs from upstream firms. We assume that to produce one unit of output,
each downstream firm uses one unit of input. Additionally, downstream firms incur production
costs. When downstream firm i (= 1, . . . , n) produces qi output, its production cost is λqi2 .
Downstream firms compete on price and the price of downstream firm i is denoted by pi . The
demand function in the downstream market is Q ≡ (a − p)/b, where p ≡ min{p1 , . . . , pn }. Eachdownstream firm’s demand is determined by qi = Q if for any j (= 1, . . . , n) and i ̸= j, pi < pj ;qi = Q/s if pi is the lowest price and s downstream firms choose it; and qi = 0 if pi is not thelowest price. Downstream firm i’s profit is Πdi ≡ (pi − w)qi − λqi2 . The consumer surplus is
CS = bQ2 /2 and the total surplus is SW = CS +∑mu
k=1 πk +∑ndj=1 Πj .

The timing of the game is as follows: In the first stage, each upstream firm chooses its input
price wk . In the second stage, each downstream firm decides where to purchase its input andchooses its final-good price pi .
Because we assume homogenous price competition with a convex cost, the equilibria of eachstage game are given by a certain closed interval (Dastider, 1995). To choose a unique equilibrium, we use the payoﬀ-dominance criterion as equilibrium refinement. This criterion was often
used in previous studies (e.g., Cabon-Dhersin and Drouhin, 2014; Mizuno and Takauchi, 2020).
Hence, the equilibrium concept is a subgame-perfect equilibrium with a payoﬀ-dominance criterion. We further assume that λ < bn/(n − 1) ≡ λ̂. Under this assumption, the downstream pricechosen through equilibrium refinement with the payoﬀ-dominance criterion does not coincidewith the price that maximizes the joint profits of downstream firms.

2

3

Equilibrium calculation

Because downstream firms are symmetric, they choose symmetric price p(w) in equilibrium.
We define the aggregate and each downstream firm’s outputs at p(w) as Q(w) = [a − p(w)]/b
and q(w) = Q(w)/n, respectively. In equilibrium, downstream firms never deviate from p(w).
Hence, the following inequalities must be satisfied:[p(w) − w]q(w) − λq(w)2 ≥ [p(w) − w]Q(w) − λQ(w)2 ,[p(w) − w]q(w) − λq(w)2 ≥ 0.

The first inequality represents the condition under which each downstream firm does not undercut the price and the second inequality indicates that each downstream firm has no incentive toincrease its price.
Solving the above inequalities for p(w), we obtain Nash equilibrium prices as the interval:p(w) ∈ [p(w), p̄(w)], wherep(w) =

bnw + aλ,bn + λp̄(w) =

bnw + a(1 + n)λ.
bn + (1 + n)λ(1)

Because we use the payoﬀ-dominance criterion as equilibrium refinement, we derive symmetric collusive price pcol (w) that maximizes the joint profit of downstream firms
pcol (w) =∑ndj=1 Πj :a(bn + 2λ) + bnw.
2(bn + λ)(2)Under the assumption λ < λ̂, we obtain p̄(w) < pcol (w). Hence, the subgame outcome is p̄(w);that is, downstream firms choose the highest price in Nash equilibria.
Next, we consider the first stage. Because upstream firms are symmetric, they choose symmetric input price w∗ in equilibrium, which leads to symmetric demand Q(w∗ )/m. As in thesecond stage, the conditions under which upstream firms do not undercut or raise prices are
expressed by the following inequalities:w∗∗ Q(w )m]
Q(w∗ ) 2−γ≥ w∗ Q(w∗ ) − γQ(w∗ )2 ,
m
[

3

w

∗
∗ Q(w )m]
Q(w∗ ) 2
−γ
≥ 0.
m
[

Then, input prices in Nash equilibria are given by the interval w∗ ∈ [w, w̄], where
w=

aγn
,
bmn + λm + λmn + γnw̄ =aγ(m + 1)n.
bmn + λm + n(γ + γm + λm)

Similarly, the price that maximizes the joint profit of upstream firms iswcol =a(bmn + λm + λmn + 2γn).
2[bmn + λm + n(γ + λm)]

Comparing the above input prices, we obtain the following lemma.
Lemma 1. (i) w̄ > w. (ii) wcol > w̄ iﬀ γ < m[bn+(1+n)λ]≡ γ̂.
n(m−1)
2aγm n(bn+λ+λn)
Proof. (i) w̄ − w = (bmn+λm+λmn+γn)(bmn+λm+γmn+λmn+γn)> 0.
am(bn+λ+λn)(bmn+λm−γmn+λmn+γn)(ii) wcol − w̄ = 2(bmn+λm+λmn+γn)(bmn+λm+γmn+λmn+γn). Hence, Lemma 1 holds.

From this lemma, the equilibrium input price is w̄ if γ < γ̂; that is, wcol if γ ≥ γ̂.

4Comparative statics

4.1

Case with upper bound pricing

First, we consider the case with γ < γ̂; that is, equilibrium upstream and downstream prices
are w = w̄ and p = p̄(w̄). Substituting these prices into the upstream and downstream profits
yieldsπ̄ u =

a2 γmn2,[bmn + λm + n(γ + γm + λm)]2

(3)

Π̄d =

a2 λm2 n.
[bmn + λm + n(γ + γm + λm)]2(4)

From (3) and (4), we establish Proposition 1.3

Proposition 1. Suppose γ < γ̂. (i) The profit of each upstream firm is a single-peaked functionof γ and takes its maximum value at γ = m[bn + (1 + n)λ]/[n(m + 1)] ≡ γ̄; the profit of3
For welfare analysis, we only have well-known results. Hence, these results are reported in the Online Appendix.

4each upstream firm decreases with λ. (ii) The profit of each downstream firm is a single-peaked
function of λ and takes its maximum value at λ = n[bm + (1 + m)γ]/[m(n + 1)] ≡ λ̄; the profitof each downstream firm decreases with γ.

Proof. See the Appendix.
From this proposition, we confirm that it is possible that eﬃcient technology reduces eachfirm’s profit. The intuition behind this result is as follows.
First, we consider the eﬀects of production eﬃciency in upstream and downstream markets
on upstream firms’ profits. The ineﬃciency of upstream technology, that is, an increase in γ,
has two eﬀects: production ineﬃciency eﬀect and competition mitigation eﬀect. Because theproduction cost of upstream firms is γx2k , large γ leads to ineﬃcient production. Additionally,
the equilibrium input price is determined by the condition that each upstream firm does notundercut the equilibrium input price. For large γ, if an upstream firm gains whole-market
demand by undercutting the input price, its production cost also becomes large. Therefore,
when γ is large, the equilibrium input price increases because upstream firms are less likely toundercut the input price. Thus, an increase in γ has the eﬀect of relaxing competition.
Next, consider the scenario in which each of the eﬀects dominates. As γ converges to zero,
upstream competition approaches Bertrand competition with a constant marginal cost. Then,
upstream firms’ profits converge to zero. Additionally, when γ diverges infinitely, the profits
of upstream firms converge to zero. Thus, when γ is small, the competition mitigation eﬀect
dominates, and when γ is large, the production ineﬃciency eﬀect dominates. Therefore, theprofit of each upstream firm is an inverted-U shape for γ.
When downstream production becomes less eﬃcient, that is, an increase in λ, upstream firms
face less eﬃcient trading partners. Hence, an increase in λ always decreases upstream firms’
profits.
Second, we consider the eﬀects of production eﬃciency in upstream and downstream markets
5

on downstream firms’ profits. These eﬀects are similar to those on upstream firms’ profits. An
increase in λ has production ineﬃciency and competition mitigation eﬀects. Hence, the profit ofeach downstream firm is an inverted-U shape for λ. Additionally, for large γ, downstream firms
face ineﬃcient input suppliers. Hence, downstream firms’ profits decrease with γ.
Proposition 1 directly leads to the following.
Corollary 1. Upstream and downstream eﬃciency levels that maximize upstream and downstream profits increase with λ and γ, respectively; that is, ∂γ̄/∂λ > 0 and ∂ λ̄/∂γ > 0.
As explained in the intuition behind Proposition 1, an increase in γ has a production ineﬃciency eﬀect and competition mitigation eﬀect in the upstream market. These eﬀects arebalanced at γ = γ̄, where upstream firms’ profits are maximized. When downstream market
ineﬃciency increases, that is, λ increases, upstream firms can increase their profits by choosing
a higher price that achieves less output; that is, as downstream market ineﬃciency increases,
the competition mitigation eﬀect is strengthened. Thus, at larger γ, upstream firms’ profits are
maximized, which means that γ̄ increases with λ.
The reason that λ̄ increases with γ can be explained in the same manner. Consider the casein which λ increases at γ = γ̄, where downstream firms’ profits are maximized. Then, γ thatmaximizes downstream firms’ profits increases because the eﬀect of increasing γ on mitigatingcompetition is strengthened.

6

4.2

Case with upstream collusive pricing

Next, we examine the case with γ ≥ γ̂, which yields collusive pricing in the upstream market.4
Then, the equilibrium prices are w = wcol and p = p̄(wcol ), and equilibrium profits are
u
πcol
=

a2 n
,
4[bmn + λm + n(γ + λm)](5)Πdcol =

a2 λm2 n.
4[bmn + λm + n(γ + λm)]2(6)(5) and (6) yield Proposition 2.

Proposition 2. Suppose γ ≥ γ̂. (i) The profit of each upstream firm decreases with γ and λ.
(ii) The profit of each downstream firm is a single-peaked function of λ and takes its maximum
value at λ = n(bm + γ)/[m(n + 1)] ≡ λcol ; the profit of each downstream firm decreases with γ.

Proof. See the Appendix.
In this case, only the conditions that determine the equilibrium input price diﬀer from thecase with upper bound pricing. When the equilibrium input price is collusive, the competition
mitigation eﬀect of increased production ineﬃciency in the upstream market disappears. Hence,
upstream firms’ profits decrease with γ. The other results are similar to Proposition 1, and so
are their intuitions.

Acknowledgements
This study was supported by JSPS KAKENHI (20K01678, 19H0148, 20K01646). All errors are
our own.

4
For a more detailed argument on collusive pricing, for example, see Dastider (2001) and Takauchi and Mizuno(2021).

7

Appendix A. Proofs
Proof of Proposition 1. Diﬀerentiating the profit of an upstream firm with respect to γ or λyields22 [m(bn+λ+nλ)−n(1+m)γ]∂ π̄ u= a mn,3[bmn+λm+n(γ+γm+λm)]∂γ∂ π̄ u2a2 γm2 n2 (n+1)= − [bmn+λm+n(γ+γm+λm)]
3 < 0.
∂λHence, ∂ π̄ u /∂γ > 0 if γ < m[bn+(1+n)λ]≡ γ̄. This threshold is undoubtedly smaller than γ̂:n(m+1)γ̄ < γ̂. Additionally, ∂ π̄ u /∂γ ≤ 0 if γ̄ ≤ γ < γ̂. Thus, π̄ u is a single-peaked function of γ.
Diﬀerentiating the profit of the downstream firm with respect to γ or λ yields∂ Π̄d2a2 λm2 (m+1)n2= − [bmn+λm+n(γ+γm+λm)]
3 < 0,∂γ

2 2 n[n(bm+γ+mγ)−m(1+n)λ]
∂ Π̄d
=am
.
[bmn+λm+n(γ+γm+λm)]3∂λHence, ∂ Π̄d /∂λ > 0 if λ < n[bm + (1 + m)γ]/[m(n + 1)] ≡ λ̄. Hence, λ̄ < λ̂. Additionally,∂ Π̄d /∂λ ≤ 0 if λ ≥ λ̄. Therefore, Π̄d is a single-peaked function of λ.

Proof of Proposition 2. Diﬀerentiating profits yieldsu∂πcola2 mn(n+1)= − 4[bmn+λm+n(γ+λm)]
2 < 0,
∂λ

u
∂πcol
a2 n2
= − 4[bmn+λm+n(γ+λm)]
2 < 0,∂γ∂Πdcola2 λm2 n2= − 2[bmn+λm+n(γ+λm)]
3 < 0,∂γ

2 m2 n[n(bm+γ)−m(1+n)λ]∂Πdcol= a 4[bmn+λm+n(γ+λm)].
3∂λHence, ∂Πdcol /∂λ > 0 if λ < n(bm + γ)/[m(1 + n)] ≡ λcol . Additionally, ∂Πdcol /∂λ ≤ 0 if λ ≥ λcol .
Therefore, Πdcol is a single-peaked function of λ and it takes its maximum value at λ = λcol .

8

References[1] Cabon-Dhersin, M. L., Drouhin, N., 2014. Tacit collusion in a one-shot game of price competition with soft capacity constraints. J. Econ. Manage. Strat. 23, 427-442.
[2] Dastider, K. G., 1995. On the existence of pure strategy Bertrand equilibrium. Econ. Theor.
5, 19-32.
[3] Dastider, K. G., 2001. Collusive outcomes in price competition. J. Econ. 73, 81-93.
[4] Flath, D., 2012. Are there any Cournot industries in Japan? Jpn. Econ. 35, 3-36.
[5] Matsushima, N., 2006. Industry profits and free entry in input markets. Econom. Lett. 93,
329-336.
[6] Mizuno, T., Takauchi, K., 2020. Optimal export policy with upstream price competition.
Manch. Sch. 88, 324-348.
[7] Mukherjee, A., 2019. Profit raising entry in a vertical structure. Econom. Lett. 183, 108543.
[8] Salinger, M. A., 1988. Vertical mergers and market foreclosure. Q. J. Econ. 103, 345-356.
[9] Takauchi, K., Mizuno, T., 2021. Endogenous transport price, R&D spillovers, and trade.
World Econ. (forthcoming)

9

Online Appendix of “Bertrand competitionin vertically related markets”.
by Tomomichi Mizuno and Kazuhiro Takauchi

This appendix presents results of comparative statics of our model. For example, a change infirm numbers, m and n, and production eﬃciencies, γ and λ, brings standard and well-known
welfare results.
Input and final-good prices: The diﬀerentiation with respect to firm numbers is:
∂ w̄
∂ p̄
aγn(bn+λ+λn)abγn2= − (bmn+λm+n(γ+γm+λm))= − (bmn+λm+n(γ+γm+λm))
2 < 0,
2 < 0,
∂m
∂m
∂ w̄
∂ p̄
aγλm(m+1)
abλm2= (bmn+λm+n(γ+γm+λm))= − (bmn+λm+n(γ+γm+λm))
2 > 0,
2 < 0.
∂n∂n
The diﬀerentiation with respect to γ and λ is:
∂ w̄
∂ p̄
am(m+1)n(bn+λ+λn)abm(m+1)n2= [bmn+λm+n(γ+γm+λm)]= (bmn+λm+n(γ+γm+λm))
2 > 0,
2 > 0,
∂γ
∂γ
∂ w̄
∂ p̄
aγm(m+1)n(n+1)
abm2 n(n+1)= − (bmn+λm+n(γ+γm+λm))= (bmn+λm+n(γ+γm+λm))
2 < 0,
2 > 0.
∂λ∂λProfits: The diﬀerentiation with respect to firm numbers is:2
2 (bmn+λm+n(−γ+γm+λm))∂ π̄ku∂ Π̄di2a2 γλmn2= − a γn<0,
= (bmn+λm+n(γ+γm+λm))
3
3 > 0,(bmn+λm+n(γ+γm+λm))∂m∂m
22 (bmn−λm+n(γ+γm+λm))∂ π̄ku∂ Π̄di2a2 γλm2 n= (bmn+λm+n(γ+γm+λm))= − a λm
>
0,
< 0.
3
3
(bmn+λm+n(γ+γm+λm))∂n∂n¯ and total surplus, SW¯ , become:Welfare: Consumer surplus, CS,¯ =CS

a2 m2 n2 (b + 2(γ + λ))a2 bm2 n2
¯ =
,
SW
.
2[bmn + λm + n(γ + γm + λm)]22[bmn + λm + n(γ + γm + λm)]2

The diﬀerentiation with respect to firm numbers is:
¯
¯
∂ CS
∂ SW
a2 γmn3 (b+2(γ+λ))a2 bγmn3= (bmn+λm+n(γ+γm+λm))= (bmn+λm+n(γ+γm+λm))
3 > 0,
3 > 0,
∂m
∂m
¯
¯
∂ CS
∂ SW
a2 λm3 n(b+2(γ+λ))
a2 bλm3 n= (bmn+λm+n(γ+γm+λm))= (bmn+λm+n(γ+γm+λm))
3 > 0,
3 > 0.
∂n∂n

10The diﬀerentiation with respect to γ and λ is:¯¯
2 2 n2 (bn−λm+n(γ+2λ+γm+λm))
∂ CS
∂ SW
a2 bm2 (m+1)n3= − (bmn+λm+n(γ+γm+λm))= − a m (bmn+λm+n(γ+γm+λm))< 0,
3 < 0,3∂γ∂γ¯¯
2 2 n2 (bm+m(2γ+λ+γn+λn)−γn)
∂ CS
∂ SW
a2 bm3 n2 (n+1)= − (bmn+λm+n(γ+γm+λm))= − a m(bmn+λm+n(γ+γm+λm))< 0.
3 < 0,3∂λ∂λUpstream collusive pricing
Profits: The diﬀerentiation with respect to m and n is:u∂Πdcol∂πcola2 n(bn + λ)a2 γmn(bn + λ)
=−
<
0,
=
> 0,
∂m
2(2bmn + 2λm + γn)2∂m2(2bmn + 2λm + γn)3()
u
a2 m2 2b2 mn + 2bλm + bγn + 2γλ∂πcol∂Πdcol
a2 λm
=
> 0,
=−
< 0.
∂n
2(2bmn + 2λm + γn)2∂n4(2bmn + 2λm + γn)3

Welfare: Consumer surplus, CScol , and total surplus, SWcol , become:
CScol =

a2 mn(7bmn + 6λm + 2γn)a2 bm2 n2
,
SW
=
.
col
8(2bmn + 2λm + γn)28(2bmn + 2λm + γn)2

The diﬀerentiation with respect to firm numbers is:∂CScol
a2 bγmn3∂SWcola2 γn2 (5bmn + 4λm + γn)
=
>
0,
=
> 0,
∂m
4(2bmn + 2λm + γn)3∂m4(2bmn + 2λm + γn)3a2 bλm3 n∂SWcola2 λm2 (8bmn + 6λm + γn)∂CScol
=
>
0,
=
> 0.
∂n
2(2bmn + 2λm + γn)3∂n4(2bmn + 2λm + γn)3
The diﬀerentiation with respect to γ and λ is:∂CScol
a2 bm2 n3∂SWcola2 mn2 (5bmn + 4λm + γn)
=−
<
0,
=
−
< 0,
∂γ
4(2bmn + 2λm + γn)3∂γ4(2bmn + 2λm + γn)3∂CScol∂SWcol
a2 bm3 n2
a2 m2 n(8bmn + 6λm + γn)
<
0,
< 0.
=−
=
−
∂λ
2(2bmn + 2λm + γn)3∂λ4(2bmn + 2λm + γn)311