Unprofitable Common Ownership with Asymmetric
Distribution Channels
Aika Monden
Tomomichi Mizuno
October 2023
Discussion Paper No. 2317

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANUnproﬁtable Common Ownership with Asymmetric
Distribution Channels∗
Aika Monden†aaTomomichi Mizuno‡b

Graduate School of Management, Kyoto Universityb
Graduate School of Economics, Kobe University

October 10, 2023
Abstract
Common ownership has been observed in many industries and is believed to have amoderating eﬀect on competition and tends to increase the proﬁts of ﬁrms in whichshares are held. This study challenges this common ownership characteristic. We
consider a market with one upstream and two downstream ﬁrms. One downstreamﬁrm sells its products in two independent markets, while the other sells its products
in only one of the two markets. The relationship between common ownershipand input prices changes in the presence of channel asymmetry. In other words,
an increase in the degree of common ownership can lead to an increase in input
prices. Thus, common ownership may reduce downstream ﬁrms’ proﬁts, consumer
surplus, and total surplus. We also investigate whether this result is robust to
several extensions.

JEL codes: L13, D43, G32
Keywords: supply chain management, common ownership, vertical relationship, asymmetric distribution channel∗

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‡
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

Common ownership refers to institutional investors owning shares in a large number ofﬁrms within a market and is observed in various industries. 1 For example, VanguardGroup owns shares in Apple and Microsoft, which are competitors in the technology
industry. Other examples of common ownership include the airline, pharmaceutical, andbanking industries (Azar et al. [3]). The eﬀect of common ownership has been studied
for some time, and previous theoretical studies have analyzed the eﬀect of common
ownership on competition and found situations where common ownership may reduce
competition (Chen et al. [5]; López and Vives [16]; Rotemberg [19]). 2
From the perspective of managing a ﬁrm, the existence of common ownership may bewelcomed, since less competition is desirable. However, several empirical studies showthat common ownership does not necessarily increase proﬁts. For example, Koch et al.
[11] empirically show that common ownership does not have a robust positive eﬀect on
industrial proﬁtability.3
Following the empirical ﬁndings, we focus on understanding a negative relationshipbetween common ownership and proﬁts. More speciﬁcally, we introduce a new factor thatraises input prices due to common ownership, which reduces ﬁrms’ proﬁts despite theexistence of common ownership. Our analysis provides a better understanding of input
price determination and the impact of input price changes on proﬁts in decentralized
supply chains with common ownership. Our analysis allows us to answer the following questions: (i) How should input prices change as the degree of common ownershipincreases?; and (ii) Who beneﬁts from common ownership?
To address these questions, we consider a vertically related market with one upstream
and two downstream ﬁrms. Each downstream ﬁrm buys inputs from the upstream ﬁrm
1

Torshizi and Clapp [22] report on the price eﬀects of common ownership in the seeds sector.
A recent study in the operations management literature that analyzes the eﬀects of common ownership is that of Avinadav and Shamir [1].
3
See Dennis et al. [7], Kini et al. [10], and Lewellen and Lowry [13] for studies that reexamine theanticompetitive eﬀects of common ownership. See Backus et al. [4] for a survey on common ownership.
2

2

with a linear contract and sells its products to consumers. One downstream ﬁrm sells
its products to two spatially separated markets (e.g., the U.S. and Mexico), and theother downstream ﬁrm sells its products to only one of these markets (e.g., the U.S.
only). Institutional investors partially own shares in the two downstream ﬁrms, andeach downstream ﬁrm, depending on the degree of common ownership, is concernedwith the proﬁts of its rivals.
One example of the ﬁt of our model is the supermarket industry. For example,
Costco and Dollar Tree are giant retailers, and Vanguard owns shares in both. In the
U.S., Costco and Dollar Tree compete. However, in Mexico, only Costco has stores. 4
Both retailers buy products from a common upstream ﬁrm (e.g., PepsiCo). This market
structure is consistent with our basic model.
Under the above model, we analyze the eﬀects of an increase in the degree of common ownership on equilibrium outcomes. First, we ﬁnd that whether the input priceincreases with the degree of common ownership depends on the heterogeneity in thesize of downstream markets. If the market size in which two downstream ﬁrms compete
is smaller than the market size in which one downstream ﬁrm is the sole monopolist,
input prices increase with the degree of common ownership. Since a higher input priceincreases the marginal cost to downstream ﬁrms, the price of the ﬁnal product also increases. If the eﬀect of higher input prices dominates the eﬀect of relaxed competition
among downstream ﬁrms, the proﬁts of downstream ﬁrms will decline, even as the degreeof common ownership increases. Moreover, because the reduction in competition among
downstream ﬁrms due to common ownership is undesirable for the upstream ﬁrm andconsumer, the proﬁt of the upstream ﬁrm and the consumer surplus always decrease
with the degree of common ownership.
We also investigate the robustness of our results. First, we consider the case where4

For the number of Costco warehouses, see Costco’s website (https://investor.costco.com/companyproﬁle/default.aspx).
For the number of Dollar Tree stores, see the 2022 Annual Report(https://corporate.dollartree.com/investors/ﬁnancial-information/annual-reports-proxies).

3the inverse demand function is nonlinear. More precisely, we assume an inverse demand
function with constant curvature (Hu et al. [9]; López and Vives [16]). Since in thebasic model, the condition for obtaining the main results is the small size of the market
in which the two downstream ﬁrms compete, we limited our analysis to a situation inwhich the equilibrium output of the market supplied by the two downstream ﬁrms is
positive but suﬃciently close to zero. Consequently, the results obtained using the basicmodel are robust.
Next, we analyzed the case in which the downstream ﬁrms have symmetric distribution channels. Speciﬁcally, we analyzed two cases: (i) both downstream ﬁrms sell theirproducts to two markets, and (ii) each downstream ﬁrm exclusively sells its products
in a diﬀerent market. The main results of the basic model are not obtained when thedownstream ﬁrms’ sales channels are symmetric. Therefore, we ﬁnd that it is important
for the sales channel to be asymmetric so that common ownership reduces the proﬁts ofdownstream ﬁrms.
Furthermore, we analyzed the case in which each downstream ﬁrm produces a diﬀerentiated product in a market in which the two downstream ﬁrms compete. The analysis
shows that, as in the basic model, when the size of the market in which the two downstream ﬁrms compete is small, the proﬁts of the downstream ﬁrms decrease with thedegree of common ownership. Thus, product diﬀerentiation does not alter the main
results qualitatively.
This study is related to two strands of literature: the literature on the relationshipbetween competition and input prices, and the literature on non-controlling ownershipin vertical markets.5
In the literature on the relationship between competition and input prices, the study
most relevant to ours is Yenipazarli [25]. He analyzed the eﬀect of downstream entry5

Our research may be related to studies analyzing the relationship between input prices and competition in the literatures on industrial organization or management. For these studies, see Cho [6],Greenhut and Ohta [8], Mukherjee [17], Pinopoulos [18], and Salinger [20].

4on wholesale price and proﬁts. Before entry, the two downstream ﬁrms sell goods exclusively to two diﬀerent markets. Yenipazarli [25] proposed a condition in which thewholesale price decreases when one of the downstream ﬁrms enters the other market.
Furthermore, he showed that if this wholesale price reduction eﬀect exceeds the eﬀect ofincreased competition due to entry, then entry increases downstream ﬁrms’ proﬁts. One
diﬀerence between Yenipazarli [25] and our study is that Yenipazarli [25] used entry tocapture changes in competition in the downstream market, whereas we used the degreeof common ownership. Furthermore, Yenipazarli [25] analyzed under a linear inverse
demand function, whereas we analyzed not only under a linear inverse demand function
but also under an inverse demand function with a constant curvature. Therefore, ourstudy complements the ﬁndings of Yenipazarli [25].
A seminal study analyzing the relationship between downstream competition andinput prices was conducted by Tyagi [23], who showed that if the elasticity of the slopeof the inverse demand function is constant, input prices are independent of the degree ofdownstream competition. Koulamas and Kyparisis [12] incorporated the eﬀect of lower
marginal costs associated with downstream entry into Tyagi’s [23] model and showed thatcompetition in downstream markets can increase input prices. These studies considereddownstream market entry, but not common ownership in the downstream market.
Our study also contributes to the literature on non-controlling ownership in verticalmarkets, which is mainly in the ﬁeld of economics.6 Li and Shuai [14] considered asituation in which one of the downstream ﬁrms owns shares in a competitor, but thecompetitor does not own shares in the downstream ﬁrm (i.e., one-way shareholding).
They showed that one-way shareholding reduces a holding ﬁrm’s input price. Shuai
et al. [21] also considered a situation with one-way shareholding. They showed thatas the degree of shareholding increases, the input prices for downstream shareholdingﬁrms decrease. Hu et al. [9] showed that cross-holdings among downstream ﬁrms raise6

One of the few studies to consider downstream shareholdings in supply chain management is thatof Aviv and Shamir [2]. However, their model is signiﬁcantly diﬀerent from ours.

5input prices when an upstream ﬁrm engages in marginal cost-reducing R&D. Lømo [15]showed that, under a general inverse demand function, when there is common ownershipamong downstream ﬁrms, input prices depend on the degree of common ownership if
the curvature of the inverse demand function is not constant. Chen et al. [5] consideredhorizontal and vertical shareholding and noted that higher input prices can be realized as
the degree of shareholding increases because the price elasticity of demand for upstreamﬁrms decreases.
The remainder of this paper is organized as follows. The basic model is described in
Section 2. Section 3 presents the primary results. Section 4 discusses the robustness of
our main results. Finally, Section 5 concludes the paper.

2

Basic Model

We consider a decentralized supply chain with one upstream and two downstream ﬁrms.
The upstream ﬁrm U produces input at no cost and sells it to the downstream ﬁrms (D1and D2) at input price w.
To produce one unit of the ﬁnal product, each downstream ﬁrm uses one unit of
input and does not bear any costs except for payments for inputs. Downstream ﬁrm
D2 supplies its product to two spatially separated markets, markets A and B, while
downstream ﬁrm D1 supplies its product only to market A. We assume that markets
A and B are independent of each other. Additionally, each downstream ﬁrm supplies ahomogeneous product in market A. For this assumption, in Section 4.3, we consider thesituation in which both downstream ﬁrms produce diﬀerentiated products in market A.
We denote downstream ﬁrm D1’s output in market A by qA1 and downstream ﬁrm
D2’s outputs in markets A and B by qA2 and qB2 , respectively. We assume that theinverse demand functions in markets A and B are pA ≡ α − qA1 − qA2 and pB ≡ β − qB2 ,respectively. We denote the ratio of the intercept of the inverse demand function ofmarket A to that of market B as r ≡ α/β. The operating proﬁts of the downstream
6

ﬁrms are as follows:
πD1 ≡ (pA − w)qA1 ,πD2 ≡ (pA − w)qA2 + (pB − w)pB2 .

The upstream ﬁrm’s proﬁt is as follows:πU ≡ w(qA1 + qA2 + qB2 ).
We assume that an institutional investor holds shares in both downstream ﬁrms.
Following the literature on common ownership (López and Vives [16]; Vives [24]), eachdownstream ﬁrm has the following objective function:
VDi ≡ πDi + θπDj ,where i, j = 1, 2 and i ̸= j and θ ∈ (0, 1) is the degree of common ownership. We assume
rmin ≡ (3 + θ)/[2(5 + θ)] < r < (11 + θ)/4 ≡ rmax , which guarantees positive outcomesin equilibrium.7
The consumer, producer, and total surpluses are, respectively, as follows:
CS ≡2(qA1 + qA2 )2 qB2+,
2
2

P S ≡ πU + πD1 + πD2 ,T S ≡ CS + P S.

The timing of the game is as follows. In the ﬁrst stage, the upstream ﬁrm U chooses
the input price w. In the second stage, the downstream ﬁrm D1 chooses qA1 and thedownstream ﬁrm D2 chooses qA2 and qB2 . We solved this game using backward induction.

3

Analysis3.1

Calculating equilibrium

In the second stage, the ﬁrst-order conditions of the downstream ﬁrms, ∂VD1 /∂qA1 = 0,
∂VD2 /∂qA2 = 0, and ∂VD2 /∂qB2 = 0, lead to the following outputs:
qA1 (w, θ) = qA2 (w, θ) =
7

α−w
,
3+θ

qB2 (w) =β−w.
2When r < rmin or r > rmax , equilibrium outputs in market A or B become zero.

7(1)

By substituting the outputs into the upstream ﬁrm’s proﬁt, solving the ﬁrst-ordercondition ∂πU /∂w = 0 and using r = α/β, we obtain the equilibrium input price.
w∗ =β(3 + θ + 4r),
2(7 + θ)where the superscript ‘∗’ denotes equilibrium outcomes. Then, by using r = α/β, we
obtain the following outcomes:
β[2r(5 + θ) − (3 + θ)]β(11 + θ − 4r)
∗
, qB2
=
,
2(3 + θ)(7 + θ)
4(7 + θ)β[3 + θ + r(11 + 8θ + θ2 )]β(17 + 3θ + 4r)p∗A =, p∗B =,
(3 + θ)(7 + θ)
4(7 + θ)β 2 (1 + θ)[3 + θ − 2r(5 + θ)]2β 2 (3 + θ + 4r)
∗
,
πU∗ =
, πD1
=
8(3 + θ)(7 + θ)
4(3 + θ)2 (7 + θ)2
β 2 (1 + θ)[3 + θ − 2r(5 + θ)]2 β 2 (11 − 4r + θ)2
∗
πD2
=
+
,
4(3 + θ)2 (7 + θ)216(7 + θ)2[](3 + θ)2 (137 + 22θ + θ2 ) + 16r2 (109 + 46θ + 5θ2 )2
β−8r(219 + 139θ + 25θ2 + θ3 )∗CS =,32(3 + θ)2 (7 + θ)2[](3 + θ)2 (171 + 50θ + 3θ2 ) + 16r2 (101 + 96θ + 25θ2 + 2θ3 )2
β−8r(33 + 65θ + 27θ2 + 3θ3 )∗PS =,16(3 + θ)2 (7 + θ)2
[
]
2
2
2
2
3
(3
+
θ)
(479
+
122θ
+
7θ
)
+
16r
(311
+
238θ
+
55θ
+
4θ
)
β2
−8r(285 + 269θ + 79θ2 + 7θ3 )∗TS =.
8r(285 + 260θ + 79θ2 + 7θ3 )
∗
∗
qA1
= qA2
=

3.2

Comparative statics

We discuss the eﬀects of common ownership on the equilibrium outcomes. First, weconsider the eﬀect of common ownership on pricing. By diﬀerentiating p∗A , p∗B , and w∗
with respect to θ, we obtain the following derivatives:
β[2r(29 + 10θ + θ2 ) − (3 + θ)2 ]
∂p∗A
=
> 0,
∂θ
(3 + θ)2 (7 + θ)2β(1 − r)
∂p∗B
=
,
∂θ
(7 + θ)22β(1 − r)
∂w∗
=
,
∂θ
(7 + θ)2

where the ﬁrst inequality is satisﬁed because we assume rmin < r < rmax . Subsequently,
we obtain the main results.
8

Proposition 1 The price in market A always increases with the degree of common ownership. An increase in the degree of common ownership increases the price in market
B and the input price if the size of market A is smaller than that of market B, that is,r < 1.
To illustrate and intuition behind Proposition 1, we explain the price elasticity ofthe demand for inputs. From the outputs in the second stage (1), we obtain the priceelasticities of demand for inputs in markets A and B.
∂[qA1 (w, θ) + qA2 (w, θ)]
w
w
·
=
,
∂w
qA1 (w, θ) + qA2 (w, θ)
α−w∂qB2 (w)ww
εB (w) ≡ −·=.
∂w
qB2 (w)
β−w
εA (w) ≡ −Since the price elasticity of the aggregate demand for inputs is equal to the weighted
average of the price elasticities for inputs, we obtain the following equation:
qB2 (w)
qA1 (w, θ) + qA2 (w, θ)· εA (w) +· εB (w)qA1 (w, θ) + qA2 (w, θ) + qB2 (w)qA1 (w, θ) + qA2 (w, θ) + qB2 (w)w(7 + θ)=.
β(3 + 4A + θ) − w(7 + θ)ε(w, θ) ≡For ε(w, θ), the coeﬃcient of εA (w) decreases with the degree of common ownershipθ and that of εB (w) increases with θ. This is because common ownership weakens
competition only in market A, such that output in market A falls, but output in market
B remains unchanged. Thus, as θ increases ε(w, θ) approaches εB (w). Therefore, if
εB (w) is greater than εA (w), ε(w, θ) increases, and vice versa.
To explain an intuition behind Proposition 1, we need only explain the change in theprice elasticity of demand for inputs as the degree of common ownership increases. Morespeciﬁcally, it is suﬃcient to explain that when the size of market A is smaller than that ofmarket B, the price elasticity of demand for input decreases with the degree of common
ownership. First, when the market is large, the price elasticity of demand for inputs
decreases. Thus, if α < β, which is equivalent to r < 1, we have εA (w) > εB (w). Since9

an increase in θ brings ε(w, θ) closer to εB (w), a large θ decreases ε(w, θ). Therefore,
as the degree of common ownership increases, the input demand becomes less elastic,leading to a high input price: ∂w∗ /∂θ > 0 if r < 1.
Intuitively, the other results for Proposition 1 are simple. Since downstream ﬁrm
D2 is a monopolist in market B and the degree of common ownership has no eﬀect oncompetition in market B, the price of the ﬁnal product moves in the same direction as
the input price. Finally, an increase in θ reduces competition in market A. Consequently,
the total output in market A decreases, leading to higher prices for the ﬁnal product inmarket A.
Next, we consider the eﬀects of common ownership on the proﬁts of the upstream
and downstream ﬁrms.
β 2 (3 + θ + 4r)[(3 + θ) − 2r(5 + θ)]
∂πU∗
=
< 0,
∂θ
2(3 + θ)2 (7 + θ)2∗∂πD1
β 2 [2r(5 + θ) − (3 + θ)][2r(47 − 7θ − 7θ2 − θ3 ) − (5 − θ)(3 + θ)2 ]=,∂θ
4(3 + θ)3 (7 + θ)3[ 2]
4r (181 − θ4 − 14θ3 − 60θ2 − 42θ)2β∗+2r (3θ4 + 40θ3 + 174θ2 + 256θ + 39) − (3θ + 17)(θ + 3)3
∂πD2
=
,
∂θ
4(3 + θ)3 (7 + θ)3
where the sign of the ﬁrst inequality is satisﬁed because we assume rmin < r < rmax . By
showing the signs of the derivatives, we obtain the following proposition:
Proposition 2 The upstream ﬁrm’s proﬁt always decreases with the degree of common
ownership. The operating proﬁts of the downstream ﬁrms D1 and D2 decrease with thedegree of common ownership if r < rD1 and r < rD2 , respectively, where rD1 < rD2 and(5 − θ)(3 + θ)2,2(47 − 7θ − 7θ2 − θ3 )[]√(3 + θ) (7 + θ) 757 − 3θ4 − 44θ3 − 170θ2 + 36θ − 3θ3 − 31θ2 − 81θ − 13rD2 ≡.
4(181 − θ4 − 14θ3 − 60θ2 − 42θ)rD1 ≡Proof See Appendix.
10

An intuition behind this proposition is straightforward. From Proposition 1, if market
A is smaller than market B, an increase in the degree of common ownership increases the
input price. In other words, when r is small, common ownership increases the marginal
costs for downstream ﬁrms. This tendency strengthens as r decreases. Therefore, if themarginal cost-raising eﬀect dominates the competition-relaxing eﬀect of common ownership, common ownership is harmful to downstream ﬁrms. Furthermore, since market B
is a monopoly, relaxed competition due to common ownership does not occur in market
B. Thus, if the input prices increase with common ownership, the cost of increased
common ownership is greater for downstream ﬁrm D2. Therefore, if input prices increase because of common ownership, the cost of common ownership will be greater fordownstream ﬁrm D2. Therefore, in Proposition 2, the threshold value for downstreamﬁrm D2 is greater than that for downstream ﬁrm D1: rD2 > rD1 .

r
1.0 ∂πD1 ∂θ > 0∂πD2/∂θ > 0
0.8∂πD1/∂θ > 0
0.6 ∂πD2/∂θ < 0

rD2∂πD1/∂θ < 0∂πD2/∂θ < 0

rD1

0.4
0.2
θ
0.2

0.4

0.6

0.8

1.0

Figure 1: The eﬀects of common ownership on operating proﬁt of downstream ﬁrms.
Figure 1 shows that the degree of common ownership must be small if it harms
downstream ﬁrms. Furthermore, we can conﬁrm 0.479 < rD1 < 1 and 0.744 < rD2 < 1.
Thus, if r has an intermediate value for each downstream ﬁrm, then θ maximizes its
operating proﬁts. We implicitly deﬁne θDi such that r = rDi at θ = θDi . Then, we
obtain the following result:11

Corollary 1 For 0.479 < r < 1 and 0.744 < r < 1, the operating proﬁts of downstreamﬁrms D1 and D2 are maximized at θD1 and θD2 , respectively.
An intuition behind this result is as follows. As θ approaches 1, the downstream ﬁrm’s
objective function approaches that of the merging ﬁrms. When input prices are ﬁxed,
the competition-reducing eﬀect of common ownership disappears at θ = 1 because thedownstream ﬁrm’s operating proﬁt is maximized in the monopoly. However, because theeﬀect of changing input prices persists, the eﬀect of common ownership on input prices
dominates the competition-reducing eﬀect when θ is large. Thus, at some intermediate
θ, the operating proﬁts of the downstream ﬁrms are maximized.
Finally, we provide the results of the comparative statics for consumer, producer, andtotal surpluses.
[]−4r2 (5θ3 + 69θ2 + 343θ + 607)β+r(θ4 + 40θ3 + 354θ2 + 1216θ + 1461) − (θ + 15)(θ + 3)3
∂CS ∗
=
< 0,
∂θ
4(3 + θ)3 (7 + θ)3[]−4r2 (θ4 + 15θ3 + 81θ2 + 157θ + 2)2
β−r(705 − 3θ4 − 24θ3 − 6θ2 + 352θ) + (1 − θ)(θ + 3)3∂P S ∗=< 0,∂θ
2(3 + θ)3 (7 + θ)3[]−4r2 (2θ4 + 35θ3 + 231θ2 + 657θ + 611)2β+r(7θ4 + 88θ3 + 366θ2 + 512θ + 51) − (3θ + 13)(θ + 3)3∂T S ∗=0,∂θ
4(3 + θ)3 (7 + θ)32

where the signs of the inequalities above are satisﬁed because we assume rmin < r < rmax .
Then, we obtain the following proposition:
Proposition 3 Consumer, producer, and total surpluses decrease with the degree of common ownership.
Proof See Appendix.
An intuition behind this result is straightforward. Since an increase in the degree ofcommon ownership relaxes competition in market A, the double marginalization prob12

lem worsens, leading to small total output. Therefore, consumer, producer, and total
surpluses decrease with the degree of common ownership.

4

Robustness

4.1

A large class of inverse demand functions

In the basic model, we use linear inverse demand functions. Here, we demonstrate thatthe main results hold under more general inverse demand functions. We denote theinverse demand function in market k ∈ A, B by pk (Qk ), where Qk is the total output
in market k. We assume that the inverse demand function in market k has a constantcurvature zk ; more precisely, we deﬁne zk ≡ −p′′k (Qk )Qk /p′k (Qk ), where for any Qk > 0,zk is constant. This assumption leads to the following parametric form: pk (Qk ) =ka − bQ1−z/(1 − zk ) if Qk < [a(1 − zk )/b]1/(1−zk ) ; pk (Qk ) = 0 if Qk ≥ [a(1 − zk )/b]1/(1−zk ) ,kwhere a, b > 0. This type of inverse demand function is used in the literature on industrialorganization (Hu et al. [9]; López and Vives [16]). For simplicity, we assume zk < 0.
Thus, each inverse demand function is concave. The other settings are identical to thoseused for the basic model.
The main results of the basic model, Propositions 1 and 2, are obtained under asmall r. In other words, the main results are obtained when market A is suﬃciently
smaller than market B. Therefore, we restrict our analysis to a situation in which theequilibrium total output of market A is positive but suﬃciently close to zero.
Assumption 1 The equilibrium total output in market A is positive but suﬃciently close
to zero.
Decision on output of downstream ﬁrms Since the downstream ﬁrms are symmetric in market A, both downstream ﬁrms choose the same output in market A; wedenote the output chosen by each downstream ﬁrm in market A by qA and the output
13

chosen by downstream ﬁrm D2 in market B by qB . Substituting these outputs into theﬁrst-order conditions of downstream ﬁrms, we obtain the following:∂VDi= pA (2qA ) − w + (1 + θ)qA p′A (2qA ) = 0,
∂qAi

∂VD2
= pB (qB ) − w + qB p′B (qB ) = 0. (2)∂qB2

Diﬀerentiating the ﬁrst-order conditions with respect to w or θ, we obtain the following lemma:
Lemma 1 The results of the comparative statistics are as follows:
∂qA
1
∂qA
qA
=
< 0,
=−
< 0,
′
∂w
[3 + θ(1 − zA ) − zA ]pA (2qA )∂θ
3 + θ(1 − zA ) − zA
∂qB
1
=
< 0.
∂w
(2 − zB )p′B (qB )Proof See Appendix.
An intuition behind Lemma 1 is simple. An increase in the input price raises themarginal cost of downstream ﬁrms. Thus, a large w results in small outputs: ∂qk /∂w < 0.
As the degree of common ownership increases, competition in the downstream market
A becomes more moderate. Therefore, output in market A decreases with θ.
Decision on input price By substituting the outputs in the second stage, qA andqB , into the proﬁt of the upstream ﬁrm, diﬀerentiating it with respect to w, and using
Lemma 1, we obtain the ﬁrst-order condition in the ﬁrst stage.
∂πU
2w
w= 2qA + qB +
+
= 0.
′
∂w
[3 + θ(1 − zA ) − zA ]pA (2qA ) (2 − zB )p′B (qB )(3)

By solving the ﬁrst-order condition for w, we obtain the equilibrium input price.
w(θ) =(2qA + qB )[3 + θ(1 − zA ) − zA ](2 − zB )p′A (2qA )p′B (qB ).
−[3 + θ(1 − zA ) − zA ]p′A (2qA ) − 2(2 − zB )p′B (qB )

14

(4)

Comparative statics By applying the implicit function theorem to (3) and using
Lemma 1 and the deﬁnition of zk , we obtain a comparative static result in the ﬁrst
stage.
2qA qB (2 − zB )2 p′A (2qA )p′B (qB )2 [w + qA (3 + θ − zA − θzA )p′A (2qA )]w (θ) =,
2qB wzA (2 − zB )2 p′B (qB )2 − qA (3 + θ − zA − θzA )p′A (2qA )Φ
′where Φ ≡ −4qB (2 − zB )2 p′B (qB )2 − (3 + θ − zA − θzA )p′A (2qA )[wzB + 2qB (2 − zB )p′B (qB )].
From Assumption 1, we obtain the following result:
Proposition 4 If the equilibrium total output in market A is positive but suﬃciently
close to zero, then the input price increases with the degree of common ownership:w′ (θ) > 0.
Proof See Appendix.
We ﬁnd that our main result is robust even if we allow the inverse demand function
to be nonlinear. The intuition behind this proposition is the same as that for Proposition
1.
Regarding the eﬀect of common ownership on the operating proﬁts of downstream
ﬁrms, diﬀerentiating the equilibrium operating proﬁt of downstream ﬁrm D1, denotedby πD1 (θ), with respect to θ and using Lemma 1, we obtain the following:∂πD1 (θ)[pA (2qA ) − w][w′ (θ) − qA p′A (2qA )]=− qA w′ (θ).
∂θ
[3 + θ(1 − zA ) − zA ]p′A (2qA )From Proposition 4, we have w′ (θ) > 0 if qA is positive but suﬃciently close to zero.
Therefore, under Assumption 1, the ﬁrst and second terms of ∂πD1 (θ)/∂θ are negative,
which means that the operating proﬁt of the downstream ﬁrm D1 decreases with thedegree of common ownership: ∂πD1 (θ)/∂θ < 0.
Next, we consider the eﬀect of common ownership on the operating proﬁt of thedownstream ﬁrm D2. Diﬀerentiating the equilibrium operating proﬁt of the downstream

15

ﬁrm D2, denoted by πD2 (θ), with respect to θ and applying Lemma 1, we obtain thefollowing:∂πD2 (θ) [pA (2qA ) − w][w′ (θ) − qA p′A (2qA )]=− qA w′ (θ)∂θ[3 + θ(1 − zA ) − zA ]p′A (2qA )qB (1 − zB )w′ (θ) [pB (qB ) − w]w′ (θ)−+.
2 − zB(2 − zB )p′B (qB )Thus, under Assumption 1, the operating proﬁt of downstream ﬁrm D2 also decreases
with the degree of common ownership: ∂πD2 (θ)/∂θ < 0.
Finally, we consider the eﬀect of common ownership on the upstream ﬁrm’s proﬁt.
Diﬀerentiating the equilibrium proﬁt of upstream ﬁrm U , denoted by πU (θ), and using
Lemma 1 and the equilibrium input price (4), we obtain the following:
2qA (2qA + qB )(1 − zB )p′A (2qA )p′B (qB )∂πU (θ)=−.
∂θ−[3 + θ(1 − zA ) − zA ]p′A (2qA ) − 2(2 − zB )p′B (qB )Thus, ∂πU (θ)/∂θ < 0. Summarizing the above results, we obtain the following proposition:
Proposition 5 The proﬁt of upstream ﬁrm U always decreases with an increase in the
degree of common ownership. If the equilibrium total output in market A is positive
but suﬃciently close to zero, then the operating proﬁts of downstream ﬁrms D1 and D2
decrease with the degree of common ownership: ∂πDi (θ)/∂θ < 0.
From this result, we ﬁnd that Proposition 2 is robust, even when we consider a casewith a nonlinear inverse demand function. The intuition behind Proposition 5 is thesame as that for Proposition 2.

4.2

Symmetric supply of downstream ﬁrms

In the basic model, we assume that downstream ﬁrm D1 supplies its product only tomarket A and downstream ﬁrm D2 supplies its product to both markets A and B. To
emphasize the importance of supply asymmetry, we show that our results are not robust
16

in symmetric situations. There are two types of symmetrical cases: (i) downstream ﬁrms
D1 and D2 exclusively supply to markets A and B, respectively; and (ii) downstreamﬁrms D1 and D2 supply to both markets A and B.
First, we consider case (i). Since we assume that markets A and B are independentand that the downstream ﬁrms do not compete with each other, the equilibrium output is
independent of the degree of common ownership. Therefore, the input price and proﬁts
of the downstream ﬁrms D1 and D2 are also independent of the degree of common
ownership. Therefore, our results are not robust in this case.
Next, we consider case (ii). Since the downstream ﬁrms supply both markets, we
assume that the inverse demand functions in markets A and B are pA = α − qA1 − qA2
and pB = β − qB1 − qB2 , respectively. To guarantee positive outputs in equilibrium, we
assume 1/3 < r < 3, where r = α/β. The other settings are the same as that for thebasic model.
In the second stage, the ﬁrst-order condition yields the following outputs:
qAi =

α−w
,
3+θ

qBi =

β−w
.
3+θ

In the ﬁrst stage, the upstream ﬁrm U sets the following input price:
w=

α+β
.
4

Thus, the equilibrium input price is independent of the degree of common ownership,
indicating that Proposition 1 is not robust under the symmetric supply of downstream
ﬁrms.
From these outcomes, the equilibrium proﬁts of the upstream and downstream ﬁrms
are as follows:πU =β 2 (1 + r)2,8(3 + θ)πDi =(5 − 6r + 5r2 )β 2 (1 + θ).
8(3 + θ)2

The equilibrium proﬁt of the upstream ﬁrm πU (θ) decreases with the degree of common ownership: ∂πU (θ)/∂θ < 0. By diﬀerentiating πDi with respect to θ, we show that17

the downstream ﬁrm’s’ equilibrium proﬁts increase with the degree of common ownership.
∂πDi(5 − 6r + 5r2 )β 2 (1 − θ)
=
> 0,
∂θ
8(3 + θ)3
where because of 1/3 < r < 3, the inequality is satisﬁed. Therefore, our results arerobust for upstream ﬁrm proﬁts but not for downstream ﬁrm proﬁts.

4.3

Diﬀerentiated products in market A

In the basic model, we assume that each downstream ﬁrm produces a homogenous product. Here, we consider the case of diﬀerentiated products in market A. For simplicity,
we use linear inverse demand functions: pA1 = α − qA1 − γqA2 , pA2 = α − qA2 − γqA1 , andpB = β − qB2 , where pAi is the price of downstream ﬁrm Di in market A and γ ∈ (0, 1)
is the degree of product substitutability. To guarantee positive outputs in equilibrium,
we assume that r̂min ≡ (2 + γ + γθ)/(8 + 2γ + 2γθ) < r < (10 + γ + γθ)/4 ≡ r̂max . The
remaining settings are identical to that for the basic model.
In the second stage, from the ﬁrst-order conditions, the following outputs are obtained:
qA1 = qA2 =α−w,
2 + γ + γθ

qB2 =

β−w
.
2

In the ﬁrst stage, the upstream ﬁrm chooses the following input price:w(θ) =

4α + β(2 + γ + γθ).
2(6 + γ + γθ)By substituting these outcomes, we obtain the equilibrium proﬁts of the upstream anddownstream ﬁrms.
β 2 (1 + γθ)[2 + γ + γθ − 2r(4 + γ + γθ)]2β 2 (2 + 4r + γ + γθ)2
, πD1 (θ) =,8(2 + γγθ)(6 + γ + γθ)4(2 + γ + γθ)2 (6 + γ + γθ)2β 2 (1 + γθ)[2 + γ + γθ − 2r(4 + γ + γθ)]2 β 2 (10 − 4rγ + γθ)2πD2 (θ) =+.
4(2 + γ + γθ)2 (6 + γ + γθ)216(6 + γ + γθ)2πU (θ) =By diﬀerentiating the equilibrium outcomes with respect to θ, we obtain the following
proposition:
18

Proposition 6 When the degree of common ownership increases, (i) input price increases if r < 1, (ii) the proﬁt of the upstream ﬁrm decreases, (iii) the operating proﬁtof downstream ﬁrm D1 decreases if r < r̂D1 , and (iv) the operating proﬁt of downstreamﬁrm D2 decreases if r < r̂D2 where[4 + γ(1 − θ)](γθ + γ + 2)2,
2[γ 3 (1 − θ)(θ + 1)2 + 2γ 2 (5 + 2θ − 3θ2 ) + 4γ(7 − 3θ) + 8]√
−Φ̂1 + Φ̂21 − 4Φ̂2 Φ̂0,r̂D2 =2Φ̂2
Φ̂2 ≡ 64 + 4γ(γθ + γ + 4)[γ(1 − θ2 )(γθ + γ + 8) + 8(3 − 2θ)],r̂D1 =

Φ̂1 ≡ 2γ 4 (θ + 1)3 (3θ − 1) + 8γ 3 (θ + 1)2 (8θ − 1) + 8γ 2 (27θ2 + 26θ − 1) + 32γ(8θ + 1) + 64,Φ̂0 ≡ −(γθ + γ + 2)3 (3γθ + γ + 16).
Proof See Appendix.
From Proposition 6, we ﬁnd that our main results are robust, even when we introduce
product diﬀerentiation in market A. Thus, the intuition behind this proposition is thesame as that for the basic model.

5Conclusions

We consider a market with one upstream and two downstream ﬁrms whose shares are heldby an institutional investor; that is, downstream common ownership. One downstreamﬁrm can supply its product to all markets, whereas the other can supply its product toone of the markets. This channel asymmetry may reverse the eﬀect of common ownershipon input price. That is, an increase in the degree of common ownership may increase
the input price and decrease downstream ﬁrms’ proﬁts. In this case, common ownershipworsens the double marginalization problem, resulting in small consumer, producer, andtotal surpluses.
We consider only quantity competition in a downstream market. However, it is19worth considering price competition between downstream ﬁrms. We also assume that
downstream markets are independent. However, if each market is expressed as a productdistinction rather than a spatial distinction, substitutability arises for the products soldin each market. Therefore, it is worthwhile to explore this possibility. These extensions
should be the subject of future research.

20

Appendix
A1. Proof of Proposition 2First, we consider the eﬀect of θ on an upstream ﬁrm’s operating proﬁt. By solving∂πU∗ /∂θ < 0 for r, we have r > (3 + θ)/(10 + 2θ). Assuming that rmin < r < rmax andrmin = (3 + θ)/(10 + 2θ), we obtain ∂πU∗ /∂θ < 0.
∗∗
Next, we consider the sign of ∂πD1/∂θ. By solving ∂πD1/∂θ < 0 for r, we obtain thefollowing condition:r<(5 − θ)(3 + θ)2≡ rD1 .
2(47 − 7θ − 7θ2 − θ3 )A comparison of rD1 with rmin and rmax yields that rmin < rD1 < rmax . Thus, we obtain∗∂πD1/∂θ < 0 if r < rD1 .
∗∗/∂θ < 0/∂θ < 0. By solving ∂πD2
Finally, we show the condition under which ∂πD2

for r, we have rD2′ < r < rD1 , where[]√(3 + θ) (7 + θ) 757 − 3θ4 − 44θ3 − 170θ2 + 36θ − 3θ3 − 31θ2 − 81θ − 13rD2 ≡,4(181 − θ4 − 14θ3 − 60θ2 − 42θ)[]√
(3 + θ) −(7 + θ) 757 − 3θ4 − 44θ3 − 170θ2 + 36θ − 3θ3 − 31θ2 − 81θ − 13.
rD2′ ≡
4(181 − θ4 − 14θ3 − 60θ2 − 42θ)Furthermore, for any θ ∈ (0, 1), we can numerically conﬁrm rD2′ < rmin < rD2 < rmax .
Hence, we complete the proof of Proposition 2.2

A2. Proof of Proposition 3
First, the sign of ∂CS ∗ /∂θ depends on the terms in square brackets of the numerator.
This is a quadratic function of r and the discriminant is −(21+10θ +θ2 )2 (4079+2612θ +458θ2 + 20θ3 − θ4 ) < 0. Thus, we obtain ∂CS ∗ /∂θ < 0.
Next, we demonstrate that ∂P S ∗ /∂θ < 0. By solving ∂P S ∗ /∂θ < 0 for r, we obtainr < rP S ′ or r > rP S , where rP S ′ and rP S are the roots of ∂P S ∗ /∂θ = 0 and rP S ′ < rP S .
We can numerically show that rP S < rmin . Thus, for any r ∈ (rmin , rmax ), we have
∂P S ∗ /∂θ < 0.
21

Finally, we consider the sign of ∂T S ∗ /∂θ. The sign of the derivative depends on theterms in the square brackets of the numerator. This is a quadratic function of r and thediscriminant is −(21 + 10θ + θ2 )2 (7775 + 10420θ + 4522θ2 + 788θ3 + 47θ4 ) < 0. Therefore,
we obtain ∂T S ∗ /∂θ < 0.2

A3. Proof of Lemma 1
By diﬀerentiating the ﬁrst-order conditions (2) with respect to w or θ and substituting
pk′′ (2qk ) = −zk p′k (2qk )/(2qk ), we obtain the following equations:[]∂ ∂VDi∂qA= [3 + θ(1 − zA ) − zA ]p′A (2qA )− 1 = 0,∂w ∂qAi∂w[][]∂ ∂VDi∂qA′= pA (2qA ) qA + (3 + θ − θzA − zA )= 0,∂θ ∂qAi∂θ[]∂ ∂VD2∂qB= 0.
= 1 − (2 − zB )p′B (qB )∂w ∂qB2∂w
By solving the equations for ∂qA /∂w, ∂qA /∂θ, and ∂qB /∂w, we complete the proof.2

A4. Proof of Proposition 4First, we consider the sign of the denominator in w′ (θ). The ﬁrst term 2qB wzA (2 −zB )2 p′B (qB )2 is negative. When qA is positive but suﬃciently small, the second term−qA (3 + θ − zA − θzA )p′A (2qA )Φ is suﬃciently close to zero. This is because, for any
qk ≥ 0, pk (2qk ) and p′k (2qk ) are bounded; note that we use the inverse demand function
with a constant curvature. Therefore, under Assumption 1, the denominator is negative.
Next, we consider the numerator. The term outside the square brackets 2qA qB (2 −zB )2 p′A (2qA )p′B (qB )2 is negative. Owing to Assumption 1, the terms in square brackets
are positive. Thus, the numerator is also negative. Therefore, when the total output inthe downstream market A is suﬃciently small, we obtain w′ (θ) > 0.2

22

A4. Proof of Proposition 6
First, we consider result (i). The ﬁrst derivative of w(θ) leads to w′ (θ) = 2(1 − r)βγ/(6 +γ + γθ)2 . Therefore, we obtain (i) from Proposition 6.
Second, we demonstrate that ∂πU (θ)/∂θ < 0. By diﬀerentiating πU (θ) with respectto θ, we obtain the following:∂πU (θ)β 2 γ(2 + 4r + γ + γθ)[r(8 + 2γ + 2γθ) − 2 − γ − γθ)]
=−
< 0,
∂θ
2(2 + γ + γθ)2 (6 + γ + γθ)2where the terms in the square brackets of the numerator are positive because we assumer̂min < r < r̂max .
Third, we consider the eﬀect of θ on πD1 (θ). By solving the ﬁrst derivative ∂πD1 (θ)/∂θ <0 for r, we obtain the following condition:r<[4 + γ(1 − θ)](γθ + γ + 2)2≡ r̂D1 .
2[γ 3 (1 − θ)(θ + 1)2 + 2γ 2 (5 + 2θ − 3θ2 ) + 4γ(7 − 3θ) + 8]

Furthermore, we show that r̂min < r̂D1 < r̂max . Thus, we obtain (iii) from Proposition
6.
Finally, we consider the sign of ∂πD2 (θ)/∂θ.
β 2 γ(Φ̂2 r2 + Φ̂1 r + Φ̂0 )∂πD2 (θ)=,∂θ
4(2 + γ + γθ)3 (6 + γ + γθ)3where Φ̂2 ≡ 64 + 4γ(γθ + γ + 4)[γ(1 − θ2 )(γθ + γ + 8) + 8(3 − 2θ)], Φ̂1 ≡ 2γ 4 (θ +1)3 (3θ − 1) + 8γ 3 (θ + 1)2 (8θ − 1) + 8γ 2 (27θ2 + 26θ − 1) + 32γ(8θ + 1) + 64, and Φ̂0 ≡−(γθ + γ + 2)3 (3γθ + γ + 16).
Owing to Φ̂2 > 0, solving ∂πD2 (θ)/∂θ < 0 for r, we obtain r̂D2′ < r < r̂D2 , where r̂D2′
and r̂D2 are the roots for the equation ∂πD2 (θ)/∂θ = 0. Additionally, we numericallyshow that r̂D2′ < r̂min < r̂D2 < r̂max . This result leads to (iv) in Proposition 6.2

23

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