Passive forward ownership and upstream collusion
Konstantinos Charistos
Department of Economics, University of Macedonia, Thessaloniki, Greece.
e-mail: kostchar@uom.edu.gr
Ioannis N. Pinopoulos
Department of Economics, National and Kapodistrian University of Athens, Athens, Greece.
e-mail: ipinop@econ.uoa.gr
Panagiotis Skartados
Department of Economics, Athens University of Economics and Business, Athens, Greece.
e-mail: skartados@aueb.gr
September 17, 2021
Preliminary and incomplete. Please, Do Not Quote!
Abstract
We examine the effects of passive forward ownership on the sustainability of upstream
collusion. We consider a homogeneous Cournot duopoly with competing vertical chains. In
one chain, the upstream firm has non-controlling partial ownership over its downstream
exclusive client. We find that passive forward ownership hinders upstream collusion; the higher
is the degree of ownership, the more difficult it is for upstream collusion to be sustained. The
driving force behind our result is that a higher degree of passive forward ownership decreases
collusive profits of the unintegrated upstream firm.
Keywords: Tacit collusion; Passive forward ownership; Vertical chains; Cournot competition
JEL classification: D43; L13; L40; L81
1
1. Introduction
The impact of vertical relations on competition concerns antitrust authorities’
guidelines which underline the risk of mergers’ coordinated effects.1 During the last
few decades controlling vertical mergers and acquisitions, and their collusive effects on
manufacturers’ markets, have received the attention of policymakers and scholars.
However, vertical relations often involve passive or non-controlling vertical
acquisitions, the effect of which on the likelihood of the collusive outcome has been
investigated only marginally.
This paper studies whether passive vertical ownership facilitates or hinders upstream
collusion. We consider a vertically related market where firms’ interactions are
infinitely repeated. There are two upstream and two downstream firms that are locked
in exclusive relations, and in one vertical chain, the upstream firm has passive (noncontrolling) ownership over its downstream customer.2 Downstream firms offer a
homogeneous product to consumers, engaging in Cournot competition. Upstream firms
may collude on (linear) input prices, and collusion is sustained by Nash reversion
trigger strategies. In such a framework, we find that passive forward ownership hinders
upstream collusion, and the higher is the degree of ownership, the more difficult it is
for upstream collusion to be sustained.
A higher degree of passive forward ownership relaxes the incentive constraint of the
upstream shareholder whereas it tightens the incentive constraint of the unintegrated
upstream firm. On the one hand, a higher degree of ownership increases collusive,
deviation, and punishment profits of the upstream shareholder. The anti-collusive
effects of a higher degree of ownership on deviation and punishment profits are
dominated by the anti-competitive effect on collusive profits so that the upstream
shareholder is more willing to stick to collusion as the degree of its ownership rises.
On the other hand, a higher degree of ownership decreases collusive, deviation, and
punishment profits of the unintegrated upstream firm, that is, the upstream firm that
1
See, e.g., U.S. Vertical Merger Guidelines, or EU Non-Horizontal Merger Guidelines.
Exclusive relations are often observed in real-world industries. For example, in the soft drink industry
(Luco and Marshall, 2018), and the petroleum/oil industry (Milliou and Petrakis, 2007). Also, it is
common to observe non-controlling stock buyouts of suppliers in their product distributors and/or
retailers in these industries. In the soft drink industry, the Coca-Cola Company holds a share of 18% in
Coca-Cola EuroPacific Partners (Fiocco, 2016). Moreover, in 2020, Coca-Cola EuroPacific Partners
acquired 25% in Chicago-based ITS, investing in its self-pour self-pay dispenser. In 2019, the world's
second-largest liquefied natural gas company Total acquired 37% of the Indian conglomerate Adani
Group's gas distribution business.
2
2
does not possess any stakes in its downstream customer. The pro-collusive effects of a
higher degree of ownership on deviation and punishment profits are outweighed by the
pro-competitive effect on collusive profits. So, the unintegrated upstream firm is less
willing to stick to collusion as the degree of its upstream rival’s forward ownership
rises.
Although several studies consider full vertical integration and collusion (e.g., Nocke
and White, 2007; Normann, 2009; Biancini and Ettinger, 2016), to the best of our
knowledge, there is only one paper that studies partial vertical integration and
collusion, that is, Shekhar and Thomes (2020). Unlike ours, they study the effects of
passive backward ownership on the sustainability of downstream collusion.
2. The model
We consider a two-tier vertical industry, where two upstream firms, 𝑈𝑖 , 𝑖 = 1, 2,
exclusively supply an essential input to a Cournot duopoly, 𝐷𝑖 , forming competing
vertical chains. Each 𝐷𝑖 exclusively uses 𝑈𝑖 ’s input in a one-to-one proportion to
produce a homogeneous final good. Each 𝐷𝑖 faces a linear (inverse) demand function
𝑝(𝑞1 , 𝑞2 ) = 𝑎 − 𝑞1 − 𝑞2 , where 𝑞𝑖 is 𝐷𝑖 ’s output and 𝑎 > 0 is the market size.3
Each 𝐷𝑖 incurs no other cost besides the one induced by the vertical contract in
action: a linear tariff consisting of a consumption-based input price, 𝑤𝑖 > 0. On the
other hand, each 𝑈𝑖 has a constant marginal cost 𝑐 > 0.
Firm 𝑈1 possesses an exogenous minority dividend share 𝑘 ∈ (0, 1⁄2) on 𝐷1, but no
control over its production decisions, a so-called silent financial interest. In other
words, there exists a passive forward integration between 𝑈1 and 𝐷1, with the former
having (𝑘 × 100)% stakes on 𝐷1’s gross profits.4 So, each firm’s net profits are:
𝜋𝐷1 (𝑞1 , 𝑞2 , 𝑤1 ) = (1 − 𝑘) (𝑝(𝑄) − 𝑤1 ) 𝑞1
𝜋𝐷2 (𝑞1 , 𝑞2 , 𝑤2 ) = (𝑝(𝑄) − 𝑤2 ) 𝑞2
𝜋𝑈1 = (𝑤1 − 𝑐) 𝑞1 + 𝑘 (𝑝(𝑄) − 𝑤1 ) 𝑞1
𝜋𝑈2 = (𝑤2 − 𝑐) 𝑞2
We assume a unit mass of identical consumers having the same utility function 𝑢(𝑄) = 𝑎𝑄 − (𝑄2 ⁄2) +
𝑚, where 𝑄 = 𝑞1 + 𝑞2 , and 𝑚 denotes the (normalized) numeraire sector (Singh and Vives, 1984).
4
Even though a 𝑘 > 1⁄2 implies control of 𝑈1 over 𝐷1 , the opposite is not necessarily true. Competition
authorities often inspect non-controlling shareholders that are between 15% and 25% (Salop and
O’Brien, 2000). Furthermore, to avoid the free-rider problems of small shareholders (Grossman and Hart,
1980), we assume that each firm is owned by a single shareholder.
3
3
We consider an infinitely repeated game with discrete time periods. In each period,
a two-stage game is played with observable actions. In stage 1, 𝑈𝑖 makes each 𝐷𝑖 a takeit or leave-it offer 𝑤𝑖 . In stage 2, each 𝐷𝑖 chooses 𝑞𝑖 to maximize 𝜋𝐷𝑖 . The solution
concept is subgame perfection.
3. Equilibrium analysis and results
Firm 𝐷1 chooses 𝑞1 to maximize:
𝑚𝑎𝑥 𝜋𝐷1 = (1 − 𝑘)[𝑝1 (𝑞1 , 𝑞2 ) − 𝑤1 ]𝑞1
𝑞1
(1)
Firm 𝐷2 chooses 𝑞2 to maximize:
𝑚𝑎𝑥 𝜋𝐷2 = [𝑝2 (𝑞1 , 𝑞2 ) − 𝑤2 ]𝑞2
(2)
𝑞2
Solving (1) and (2) together, we obtain the final-good quantities as functions of input
prices:
𝑞𝑖 (𝐰) =
𝑎 − 2𝑤𝑖 + 𝑤𝑗
,
3
𝑖, 𝑗 = 1,2,
𝑖 ≠ 𝑗,
(3)
where 𝐰 = [𝑤𝑖 , 𝑤𝑗 ].
Now we consider three interactions at the upstream level: Punishment, Collusion, and
Deviation, denoted by the superscripts 𝑃, 𝐶, and 𝐷 respectively.
3.1. Punishment
First, we consider the case where 𝑈𝑖 ’s set their input prices non-collusively. Firm 𝑈1
chooses 𝑤1 to maximize:
𝑚𝑎𝑥 𝜋𝑈1 = (𝑤1 − 𝑐)𝑞1 (𝐰) + 𝑘[𝑝1 (𝑞1 (𝐰), 𝑞2 (𝐰)) − 𝑤1 ]𝑞1 (𝐰)
𝑤1
(4)
Firm 𝑈2 chooses 𝑤2 to maximize:
𝑚𝑎𝑥 𝜋𝑈2 = (𝑤2 − 𝑐)𝑞2 (𝐰)
𝑤2
Solving (4) and (5) together, we obtain the optimal input prices:
4
(5)
𝑤1𝑃 =
5𝑎(3 − 4𝑘) + 2𝑐(15 − 4𝑘)
,
45 − 28𝑘
𝑤2𝑃 =
3𝑎(5 − 4𝑘) + 2𝑐(15 − 8𝑘)
45 − 28𝑘
(6)
Substituting (6) back to (3) we get:
𝑞1𝑃 =
10(𝑎 − 𝑐)
,
45 − 28𝑘
𝑞2𝑃 =
2(𝑎 − 𝑐)(5 − 4𝑘)
45 − 28𝑘
(7)
6(𝑎 − 𝑐)2 (5 − 4𝑘)2
=
(45 − 28𝑘)2
(8)
Upstream profits are:
𝜋𝑈𝑃1
50(𝑎 − 𝑐)2 (3 − 2𝑘)
=
,
(45 − 28𝑘)2
𝜋𝑈𝑃2
Lemma 1. In the punishment phase it holds:
(i) 𝜕𝑤1𝑃 ⁄𝜕𝑘 < 0 and 𝜕𝑤2𝑃 ⁄𝜕𝑘 < 0,
(ii) 𝜕𝑞1𝑃 ⁄𝜕𝑘 > 0 and 𝜕𝑞2𝑃 ⁄𝜕𝑘 < 0,
(iii) 𝜕𝜋𝑈𝑃1 ⁄𝜕𝑘 > 0 and 𝜕𝜋𝑈𝑃2 ⁄𝜕𝑘 < 0.
Under vertical separation, 𝑘 = 0, it can be easily verified from (6) to (8) that 𝑈𝑖 ’s
charge the same input price, sell the same input quantity and make the same net profit.
Passive forward integration by 𝑈1 into 𝐷1, 𝑘 > 0, eliminates in part the double-margin
problem and hence decreases the input price charged by 𝑈1 . The reduction in 𝑤1 leads
to an increase in 𝑞1 (a lower marginal cost for 𝐷1) and a reduction in 𝑞2 (strategic
substitutability) for any given 𝑤2 .
In other words, the reduction in 𝑤1 due to a higher 𝑘 decreases 𝐷2’s derived demand
for the input and hence decreases the input price charged by 𝐷2. A lower 𝑤2 leads to an
increase in 𝑞2 and a decrease in 𝑞1 . First-order effects are of higher importance than
second-order ones. So, a higher 𝑘 increases the optimal final-good output of 𝐷1 and
decreases the optimal final-good output of 𝐷2.
Passive forward integration, as compared to vertical separation, increases
punishment profits of 𝑈1 and decreases punishment profits of 𝑈2 , and these effects are
more pronounced the higher is 𝑘.
3.2. Collusion
Next, consider the case where 𝑈𝑖 ’s collude in setting input prices (i.e., maximize
joint profits).
5
𝑚𝑎𝑥 (𝜋𝑈1 + 𝜋𝑈2 )
𝑤1 ,𝑤2
= (𝑤1 − 𝑐)𝑞1 (𝐰) + (𝑤2 − 𝑐)𝑞2 (𝐰)
+ 𝑘[𝑝1 (𝑞1 (𝐰), 𝑞2 (𝐰)) − 𝑤1 ]𝑞1 (𝐰)
The optimal input prices are:
𝑤1𝐶 =
3𝑎(1 − 𝑘) + 𝑐(3 − 𝑘)
,
2(3 − 2𝑘)
𝑎+𝑐
2
(9)
(𝑎 − 𝑐)(1 − 𝑘)
2(3 − 2𝑘)
(10)
𝑤2𝐶 =
Substituting (9) into (3), we get:
𝑞1𝐶 =
(𝑎 − 𝑐)
,
2(3 − 2𝑘)
𝑞2𝐶 =
while
𝑞1𝐶
1
𝐶 =1−𝑘 >1
𝑞2
(11)
Upstream profits are:
𝜋𝑈𝐶1 =
(𝑎 − 𝑐)2
,
4(3 − 2𝑘)
𝜋𝑈𝐶2 =
(𝑎 − 𝑐)2 (1 − 𝑘)
4(3 − 2𝑘)
(12)
Lemma 2. In the collusion phase it holds:
(i) 𝜕𝑤1𝐶 ⁄𝜕𝑘 < 0 and 𝜕𝑤2𝐶 ⁄𝜕𝑘 = 0,
(ii) 𝜕𝑞1𝐶 ⁄𝜕𝑘 > 0 and 𝜕𝑞2𝐶 ⁄𝜕𝑘 < 0,
(iii) 𝜕𝜋𝑈𝐶1 ⁄𝜕𝑘 > 0 and 𝜕𝜋𝑈𝐶2 ⁄𝜕𝑘 < 0.
Under vertical separation, 𝑘 = 0, it can be easily verified from (9) and (12) that the
colluding 𝑈𝑖 ’s charge the same monopoly input price and make the same net profit.5
When 𝑈1 ’s stake on 𝐷1’s gross profit is positive, 𝑘 > 0, the colluding 𝑈𝑖 ’s charge 𝐷1
an input price that is lower than the monopoly input price offered to 𝐷2, thereby creating
a cost-advantage in favor of 𝐷1.6 This advantage is larger, the higher is 𝑘. Defining
For 𝑘 = 0: 𝑞1𝐶 = 𝑞2𝐶 and 𝜋𝑈𝐶1 = 𝜋𝑈𝐶2 .
6
So, 𝐷1 is indirectly raising rival cost instrumentalizing his stakes in the upstream market.
5
6
𝑤
̃1 = 𝑘𝑝1 + (1 − 𝑘)𝑤1 as the effective input price that 𝑈1 receives from each unit of
input sold to 𝐷1, we obtain 𝑤
̃1𝐶 = (𝑎 + 𝑐)⁄2 = 𝑤2𝐶 . In other words, the colluding firms
equalize the effective input prices across downstream firms, 𝑤
̃1𝐶 = 𝑤2𝐶 , but charge
distinct input prices 𝑤1𝐶 < 𝑤2𝐶 .
When 𝑘 > 0, unlike 𝑈2 , firm 𝑈1 ’s rents come from two sources: profits from input
sales to 𝐷1 and profits originating from the gross profits of 𝐷1, that is, from the
downstream firm on which it has stakes. A lower 𝑤1, on the one hand, decreases 𝑈1 ’s
profits from input sales (input sales effect), whereas, on the other hand, it increases 𝑈1 ’s
profits originating from gross profits of 𝐷1 (dividend share effect). The higher is 𝑘, the
stronger is the dividend share effect: the optimal input price of 𝑈1 decreases and output
is shifted from 𝐷2 to 𝐷1 (see (11)).
Passive forward integration, as compared to vertical separation, increases 𝑈1 ’s
collusive profits and decreases 𝑈2 ’s collusive profits, and these effects are more
pronounced the higher is 𝑘.
Comparing the punishment and collusion phases, we obtain the following results.
Proposition 1. For 𝑘 > 0, it holds:
(i) 𝑤𝑖𝐶 > 𝑤𝑖𝑃 and 𝜕[𝑤𝑖𝐶 − 𝑤𝑖𝑃 ]⁄𝜕𝑘 > 0,
(ii) 𝑞𝑖𝐶 < 𝑞𝑖𝑃 and 𝜕[𝑞𝑖𝑃 − 𝑞𝑖𝐶 ]⁄𝜕𝑘 > 0,
(iii) 𝜋𝑈𝐶𝑖 > 𝜋𝑈𝑃𝑖 , 𝜕[𝜋𝑈𝐶1 − 𝜋𝑈𝑃1 ]⁄𝜕𝑘 > 0 and 𝜕[𝜋𝑈𝐶2 − 𝜋𝑈𝑃2 ]⁄𝜕𝑘 < 0,
with 𝑖 = 1,2.
It is straightforward that the collusive input prices and profits for both 𝑈𝑖 s are higher
than the corresponding prices and profits in the punishment (competition) stage. In
addition, it is also straightforward that input quantities (and hence final-good quantities)
for both firms are lower under collusion than under competition.
A higher 𝑘 decreases 𝑈1 ’s input price by more under competition than under
collusion. Under competition, when the stake of 𝑈1 on 𝐷1’s profits increases, 𝑈1 lowers
𝑤1 (double marginalization becomes less severe) and induces 𝐷1 to increase its input
demand and become more aggressive in the downstream market. Under collusion, 𝑈1
also lowers 𝑤1 when 𝑘 increases but to a lesser extent: upstream firms maximize their
joint profits, and hence 𝑈1 does not want 𝐷1 to be too aggressive in the downstream
market. Hence, 𝜕[𝑤1𝐶 − 𝑤1𝑃 ]⁄𝜕𝑘 > 0. Under punishment, 𝑈2 reacts in a lower 𝑤1 by
reducing its input price. The higher is 𝑘, the greater the reduction in 𝑤1, the greater the
7
reduction in 𝐷2’s derived demand and therefore the greater the reduction in 𝑤2 (Lemma
1). Under collusion, variations in 𝑘 do not affect 𝑈2 ’s input price (Lemma 2). Hence,
𝜕[𝑤2𝐶 − 𝑤2𝑃 ]⁄𝜕𝑘 > 0.
On the one hand, under both punishment and collusion, a reduction in 𝑤1 leads to an
increase in 𝑞1 and a reduction in 𝑞2 (for any given 𝑤2 ). Since the optimal input price of
𝑈1 falls by more under punishment than under collusion when 𝑘 increases, we have that
the increase in 𝑞1 and the decrease in 𝑞2 are more pronounced under punishment than
under collusion. That is, regarding the effect of a higher 𝑘 on final-good quantities that
works through 𝑤1, we have that 𝑞1𝑃 − 𝑞1𝐶 increases and 𝑞2𝑃 − 𝑞2𝐶 decreases as 𝑘 rises.
On the other hand, a higher 𝑘 does not affect 𝑤2 under collusion, however, it leads
to a reduction in 𝑤2 under punishment, which in turn increases 𝑞2 and decreases 𝑞1 : the
higher is 𝑘, the greater is the reduction in 𝑤2 , and thus the greater is the increase in 𝑞2
and the reduction in 𝑞1 . That is, regarding the effect of a higher 𝑘 on final-good
quantities that works through 𝑤2 , we have that 𝑞1𝑃 − 𝑞1𝐶 decreases and 𝑞2𝑃 − 𝑞2𝐶
increases as 𝑘 rises.
Regarding the final-good quantity of 𝐷1 (and hence the input quantity of 𝑈1 ), the
effect of higher 𝑘 that works through 𝑤1 dominates the effect that works through 𝑤2 ,
so that the optimal quantity increases by more under punishment than under collusion,
i.e., 𝜕[𝑞1𝑃 − 𝑞1𝐶 ]⁄𝜕𝑘 > 0. A higher 𝑘 decreases 𝑈1 ’s profits from input sales by more
under punishment than under collusion, but it also increases 𝑈1 ’s profits originating
from the gross profits of 𝐷1 by more under punishment than under collusion. The higher
is 𝑘, the stronger is the latter effect, so that a higher 𝑘 increases 𝑈1 ’s collusive profits
by more than it increases 𝑈1 ’s punishment profits.
Regarding the final-good quantity of 𝐷2 (and hence the input quantity of 𝑈2 ), the
effect of higher 𝑘 that works through 𝑤2 dominates the effect that works through 𝑤1,
so that the optimal quantity decreases by more under collusion than under punishment,
i.e., 𝜕[𝑞2𝑃 − 𝑞2𝐶 ]⁄𝜕𝑘 > 0. On the one hand, the input price of 𝑈2 remains unchanged
under collusion but falls under punishment when 𝑘 increases, whereas on the other
hand, the input quantity falls by more under collusion than under punishment. As it
turns out, the latter effect outweighs the former so that a higher 𝑘 decreases 𝑈2 ’s
collusive profits by more than it decreases 𝑈2 ’s punishment profits.
8
3.3. Deviation
Finally, we consider the case where an upstream firm deviates from the collusive
path, whereas the other upstream firm sets the collusive price. We must consider two
cases here. First, the case where 𝑈1 deviates from the collusive agreement. Firm 𝑈1 ,
taking as given that 𝑈2 sets the collusive price 𝑤2𝐶 , chooses 𝑤1 to maximize:
𝑚𝑎𝑥 𝜋𝑈1 = (𝑤1 − 𝑐)𝑞1 (𝑤1 , 𝑤𝐶2 ) + 𝑘[𝑝1 (𝑞1 (𝑤1 , 𝑤𝐶2 ), 𝑞2 (𝑤𝐶2 , 𝑤1 )) − 𝑤1 ]𝑞1 (𝑤1 , 𝑤𝐶2 )
𝑤1
The deviation input price is:
𝑤1𝐷 =
3𝑎 + 𝑐
9(𝑎 − 𝑐)
−
4
8(3 − 2𝑘)
(13)
The deviation final-good output and profits of 𝑈1 are:
𝑞1𝐷 =
3(𝑎 − 𝑐)
,
4(3 − 2𝑘)
𝜋𝑈𝐷1 =
9(𝑎 − 𝑐)2
32(3 − 2𝑘)
(14)
Now we consider the case where firm 𝑈2 deviates from the collusive agreement.
Firm 𝑈2 , taking as given that 𝑈1 sets the collusive price 𝑤1𝐶 , chooses 𝑤2 to maximize:
𝑚𝑎𝑥 𝜋𝑈2 = (𝑤2 − 𝑐)𝑞2 (𝑤2 , 𝑤𝐶1 )
𝑤2
The deviation input price is:
𝑤2𝐷 =
𝑎(9 − 7𝑘) + 𝑐(15 − 9𝑘)
8(3 − 2𝑘)
(15)
The deviation final-good output and profits of 𝑈2 are:
𝑞2𝐷
(𝑎 − 𝑐)(9 − 7𝑘)
=
,
12(3 − 2𝑘)
𝜋𝑈𝐷2
(𝑎 − 𝑐)2 [9 − 7𝑘]2
=
96(3 − 2𝑘)2
Lemma 3. In the deviation phase it holds:
(i) 𝜕𝑤1𝐷 ⁄𝜕𝑘 < 0 and 𝜕𝑤2𝐷 ⁄𝜕𝑘 < 0,
(ii) 𝜕𝑞1𝐷 ⁄𝜕𝑘 > 0 and 𝜕𝑞2𝐷 ⁄𝜕𝑘 < 0,
(ii) 𝜕𝜋𝑈𝐷1 ⁄𝜕𝑘 > 0 and 𝜕𝜋𝑈𝐷2 ⁄𝜕𝑘 < 0.
9
(16)
Under vertical separation, 𝑘 = 0, it can be easily verified from (13) to (16) that the
deviating input price, output, and profit are the same for upstream firms. When 𝑘 > 0,
the deviating input price of 𝑈1 is lower, and the higher is 𝑘 the lower is 𝑤1𝐷 . Note that
the deviating final-good output is 𝑞1𝐷 (𝑤1𝐷 , 𝑤2𝐶 ). A higher 𝑘 decreases 𝑤1𝐷 and leaves
unchanged 𝑤2𝐶 (Lemma 2), so that 𝑞1𝐷 increases with 𝑘. Moreover, the higher is 𝑘 the
lower is 𝑤2𝐷 . The deviating final-good output is 𝑞2𝐷 (𝑤1𝐶 , 𝑤2𝐷 ). A higher 𝑘 decreases 𝑤2𝐷 ,
which tends to increase 𝑞2𝐷 , and decreases 𝑤1𝐶 (Lemma 2), which tends to decrease 𝑞2𝐷 .
The latter effect outweighs the former so that ultimately 𝑞2𝐷 decreases with 𝑘.
Passive forward integration, as compared to vertical separation, increases 𝑈1 ’s
deviation profits and decreases 𝑈2 ’s deviation profits and these effects are more
pronounced the higher is 𝑘.
By comparing the collusion and deviation phases, we obtain the following results.
Proposition 2. For 𝑘 > 0, it holds:
(i) 𝑤𝑖𝐶 > 𝑤𝑖𝐷 and 𝜕[𝑤𝑖𝐶 − 𝑤𝑖𝐷 ]⁄𝜕𝑘 > 0,
(ii) 𝑞𝑖𝐶 < 𝑞𝑖𝐷 and 𝜕[𝑞𝑖𝐷 − 𝑞𝑖𝐶 ]⁄𝜕𝑘 > 0,
(ii) 𝜋𝑈𝐷𝑖 > 𝜋𝑈𝐶𝑖 𝜕[𝜋𝑈𝐷1 − 𝜋𝑈𝐶1 ]⁄𝜕𝑘 > 0 and 𝜕[𝜋𝑈𝐷2 − 𝜋𝑈𝐶2 ]⁄𝜕𝑘 < 0,
with 𝑖 = 1,2.
The deviating input price for both firms is lower than the collusive input price, which
implies that the final-good output for each firm is higher under deviation than under
collusion. In addition, deviation profits are higher than collusive profits for each firm.
Consider firm 𝑈1 . A higher 𝑘 decreases 𝑤1 under both collusion and deviation
phases. It is straightforward that the reduction in 𝑤1 is less pronounced under collusion
(𝑈1 wants to induce a less aggressive behavior downstream when colluding upstream),
𝜕[𝑤1𝐶 − 𝑤1𝐷 ]⁄𝜕𝑘 > 0. Under both collusion and deviation, a reduction in 𝑤1 leads to an
increase in 𝑞1 , and since the deviating input price of 𝑈1 falls by more than the collusive
input price when 𝑘 increases, we have that the increase in 𝑞1 is more pronounced under
punishment than under collusion, that is, 𝜕[𝑞1𝐷 − 𝑞1𝐶 ]⁄𝜕𝑘 > 0. A higher 𝑘 decreases 𝑈1 ’s
profits from input sales by more under deviation than under collusion, but it also
increases 𝑈1 ’s profits originating from the gross profits of 𝐷1 by more under deviation
than under collusion. The higher is 𝑘, the stronger is the latter effect, so that a higher 𝑘
increases 𝑈1 ’s deviation profits by more than it increases 𝑈1 ’s collusive profits.
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Consider now firm 𝑈2 . The colluding input price of 𝑈2 is unaffected by a change in
𝑘 (Lemma 2), whereas the deviating input price decreases with 𝑘. Hence,
𝜕[𝑤2𝐶 − 𝑤2𝐷 ]⁄𝜕𝑘 > 0. Under both collusion and deviation, a higher 𝑘 decreases 𝑞2 , and
the reduction is more pronounced under collusion than under deviation,
𝜕[𝑞2𝐷 − 𝑞2𝐶 ]⁄𝜕𝑘 > 0. On the one hand, the input price of 𝑈2 remains unchanged under
collusion but falls under deviation when 𝑘 increases, whereas on the other hand, input
quantity falls by more under collusion than under deviation. As it turns out, the latter
effect outweighs the former so that a higher 𝑘 decreases 𝑈2 ’s collusive profits by more
than it decreases 𝑈2 ’s deviation profits.
3.4. Sustainability of collusion
We assume that 𝑈𝑖 ’s use grim-trigger strategies, meaning that they collude if no firm
has deviated from the collusive path in previous periods. Should such deviation occur,
firms revert to competition forever. The use of grim-trigger strategies leads to the
critical discount factors for 𝑈1 and 𝑈2 respectively:
𝛿1 =
𝛿2 =
𝜋𝑈𝐷1 − 𝜋𝑈𝐶1
𝜋𝑈𝐷1 − 𝜋𝑈𝑃1
𝜋𝑈𝐷2 − 𝜋𝑈𝐶2
𝜋𝑈𝐷2 − 𝜋𝑈𝑃2
=
(45 − 28𝑘)2
3825 − 3480𝑘 + 656𝑘 2
(45 − 28𝑘)2 (3 − 𝑘)2
=
(45 − 39𝑘 + 4𝑘 2 )(765 − 1095𝑘 + 388𝑘 2 )
Proposition 3. It holds 𝜕𝛿1 ⁄𝜕𝑘 < 0 and 𝜕𝛿2 ⁄𝜕𝑘 > 0 with 𝛿1 = 𝛿2 for 𝑘 = 0.
Passive forward integration hinders collusion. Moreover, the higher is 𝑘, that is,
the higher is 𝑈1 ’s stake in 𝐷1’s profits, the more difficult it is for collusion to be
sustained.
Under vertical separation, 𝑘 = 0, the two upstream firms are symmetric and thus
have the same discount factor.
When 𝑘 > 0, that is, when 𝑈1 has stakes in 𝐷1’s profit, we know from Lemmata 1-3
that a higher 𝑘 increases 𝑈1 ’s collusive profits, implying a lower critical discount factor
for 𝑈1 , whereas it also increases 𝑈1 ’s deviation and punishment profits, implying a
higher critical discount factor for 𝑈1 . On the one hand, a higher 𝑘 raises 𝑈1 ’s deviation
profits by more it increases its collusive profits (Proposition 2), but on the other hand,
a higher 𝑘 increases 𝑈1 ’s collusive profits by more it raises its punishment profits
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(Proposition 1). As it turns out, the latter effect outweighs the former so that a higher 𝑘
decreases the critical discount factor for 𝑈1 .
Regarding firm 𝑈2 , we know from Lemmata 1-3 that a higher 𝑘 decreases 𝑈2 ’s
collusive profits, implying a higher critical discount factor for 𝑈2 , and it also reduces
𝑈2 ’s deviation and punishment profits, implying a lower critical discount factor for 𝑈2 .
We know from Proposition 1 that a higher 𝑘 decreases 𝑈2 ’s collusive profits by more it
decreases its punishment profits, whereas we know from Proposition 2 that a higher 𝑘
decreases 𝑈2 ’s collusive profits by more it decreases its deviation profits. Hence, a
higher 𝑘 increases the critical discount factor for 𝑈2 .
The pro-competitive effects of a higher 𝑘 on 𝑈1 ’s deviation & punishment profits are
dominated by the anti-competitive effect on 𝑈1 ’s collusive profits. The anti-competitive
effects of a higher 𝑘 on 𝑈2 ’s deviation, as well as punishment profits, are outweighed
by the pro-competitive effect on 𝑈2 ’s collusive profits. Hence, the driving force is the
pro-competitive effect of a higher 𝑘 on 𝑈2 ’s collusive profits.
4. Conclusion
In this paper, we have investigated the effects of passive forward ownership on the
sustainability of upstream collusion. We consider a homogeneous Cournot duopoly
with competing vertical chains, where in one chain, the upstream firm has noncontrolling partial ownership over its downstream exclusive client. We show that
passive forward ownership impedes upstream collusion and the higher is the degree of
ownership, the more difficult it is for upstream collusion to be sustained. We identify
as a driving force behind our finding the fact that a higher degree of passive forward
ownership decreases collusive profits of the unintegrated upstream firm.
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