On the Stability of Common Ownership Arrangements

Konstantinos Charistos

doi:10.1515/bejeap-2024-0236

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On the Stability of Common Ownership Arrangements

Konstantinos Charistos

Published/Copyright: June 11, 2025

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 25 Issue 3

Abstract

We investigate the stability of common ownership arrangements. In a duopolistic market, the owner of a firm acquires a non-controlling share in the rival firm. Then, the acquirer of the minority share considers reselling the share to an investor. Common ownership is stable and the acquirer retains part of the share when there is asymmetry in the production costs of the firms. Common ownership is more likely to be stable as the initially acquired share increases. The welfare effects of the stable common ownership scheme depend also on the cost-asymmetry.

Keywords: common ownership; passive investments; stability

JEL Classification: D43; L13; L41

1 Introduction

Overlapping ownership can take the form of common ownership (CO), where investors own shares of several firms in an industry. Some studies on the impact of overlapping ownership find anti-competitive effects (e.g., Azar et al. 2018; Li et al. 2015; Brito et al. 2019); other studies highlight positive effects of overlapping ownership (e.g., Bayona and Lopez 2018; Lopez and Vives 2019).^[1]

Common ownership raises concerns for anti-competitive effects, as the shares of large investors in competitors increase in several industries. The global automobile industry (Huse et al. 2024), the European banking sector (Banal-Estañol et al. 2022) and the telecommunications in Germany (Boot et al. 2022) are few examples of industries with common ownership.

The occurrence of CO in many industries raises questions in relation to the stability and the welfare implications of common ownership arrangements. In this paper we aim to identify conditions under which (i) investors with passive shares in rival firms have incentives to keep rather than to resell these shares to investors outside the market and (ii) the welfare effects of stable common ownership arrangements are positive or negative.^[2]

The present paper studies the stability of CO when firms compete in a Cournot fashion. In a duopolistic market, the owner of a firm acquires a minority share of the rival firm. Then, the acquirer can resell the minority share to an outside investor before quantities are set in the product market. The stability of CO (the acquirer’s incentive to retain part of the share) depends on the asymmetry in the production costs between the two firms and on the level of the initially acquired share.

There exist two forces at play: divestment makes the acquirer more aggressive in the product market; thus, reselling the share provides a strategic advantage for the acquirer (see Stenbacka and Van Moer 2021). On the other hand, passive ownership induces a shift in production which, in the presence of a cost-asymmetry, can be profitable for the acquirer of the minority share.

1.1 Related Literature

The present paper is related to the theoretical literature that studies the stability of cross-ownership agreements.^[3] In Stenbacka and Van Moer (2021) the controllers of firms which are engaged in cross-ownership may propose to resell the minority share to an investor before competing in quantities. They show that (a) under general assumptions regarding demand and costs, a divestment induces a firm to behave more aggressively in the product market and a firm has an incentive to resell some proportion of the minority share (b) in a symmetric Cournot competition, cross-ownership is unstable; each firm resells the minority share leading to complete divestments.

Recently, considering two vertical chains, Mukherjee and Zeng (2024) shows that the stability of downstream cross-ownership depends on the bargaining power of upstream firms: reselling the minority shares entails an increase in input prices which is significant when upstream firms exhibit high bargaining power over their downstream affiliates.

Pal and Petrakis (2024) examine the incentives of firms to divest their passive shares in their rivals considering alternative divestment mechanisms. They find that when firms’ strategies are strategic substitutes and the divestment mechanism is either private placement via independent intermediaries or competitive bidding, firms always have incentives to fully divest their passive shares.

In Moreno and Petrakis (2022) large investors hold symmetric portfolios in symmetric firms in an industry. With cost-asymmetric firms in a duopolistic market, portfolios involve larger positions on the inefficient firm.

The literature that studies the incentives of firms to engage in passive ownership (agreements must be mutually beneficial) is also related to the present work.^[4] In a setting with two downstream firms that buy their input from a common upstream supplier, Li and Shuai (2022) find that when one downstream firm considers acquiring a share in its rival, cross-ownership is more profitable (jointly) under a uniform input price as compared to input price discrimination.

Shuai et al. (2023) also consider vertically related markets with three firms in the downstream market. They find that it is jointly profitable for two of the downstream firms to engage in partial ownership if the upstream market is an oligopoly but not when the upstream market is a monopoly.

Kanjilal and Muñoz-García (2020) consider symmetric firms which first select equity shares on their rival and then choose quantities. They compare the equilibrium shares to the equity shares that maximize social welfare.

In Ma, Qin, and Zeng (2024) one firm acquires shares in rivals. They characterize the conditions for jointly profitable crossholdings which are welfare-improving if the acquirer is sufficiently inefficient.^[5] In the present paper we show that CO can be stable (the acquirer retains part of the rival’s share) after two owners have agreed to engage in CO. Despite that CO is jointly profitable in a duopoly, there are divestment incentives and instability which are mitigated by asymmetries. We consider the welfare effects of a stable CO which are positive or negative depending on the cost-asymmetry.

2 Model

Two firms, firm 1 and firm 2, are engaged in quantity competition. Firm i’s operating profit is

π i q i , q j = p q i + q j − c i q i , i , j = 1 , 2

where q _i, q _j and c _i denote the production quantities of firms i and j and the constant marginal cost of firm i. The inverse demand function is p(q _i + q _j) = p(Q) = a − (q ₁ + q ₂).

Initially, investors A and B own 100 % of firm 1 and 2 respectively. Owner A sells a share λ < 1/2 to owner B, in exchange of a fee F.^[6] Then, owner B may resell any part of the share before competition takes place. We are interested in the stability of the common ownership scheme (COS); the COS is stable if B retains, at least, part of the share λ.

The timing of the game is as follows:

Stage 1: Owner B decides whether to sell the share r(≤λ) to an outside investor I. Owner B proposes a fee Φ in exchange of the share r and I decides whether to accept. The investor accepts the contract in case of indifference between accepting or not.
Stage 2: For given λ, r, owners A and B choose quantities simultaneously maximizing their own values.

We consider that owners A and B are the managers of firms 1 and 2, respectively. If 0 ≤ r < λ, Owner B derives profits from his minority share in firm 1, but cannot influence that firm’s decisions, which are under A’s control.^[7]

3 Analysis

To analyze the stability of CO we apply the backward induction.

3.1 Stage 2

Owner A maximizes its value:

u A = 1 − λ π 1 q 1 + q 2 = 1 − λ p Q − c 1 q 1

The first order condition (FOC) for profit maximization is

(1) p ′ q 1 + p − c 1 = 0

Owner B values the profit made by firm 1, apart from firm 2’s profit:

u B = π 2 Q + λ − r π 1 Q = p Q − c 2 q 2 + λ − r p Q − c 1 q 1

The maximization of B’s value gives

(2) p ′ q 2 + p − c 2 + λ − r p ′ q 1 = 0

The share δ = λ – r makes the controller of firm 2 less aggressive; CO softens competition. Denote the stage 2 quantity, price and profits equilibrium values with q _i(δ), p(δ) and π _i(δ).^[8]

Using the FOCs from (1) and (2) we obtain the best-response functions:

q 1 q 2 = a − c 1 − q 2 2 and q 2 q 1 = a − c 2 − q 1 1 + δ 2

Solving the above system gives

q 1 δ = a − 2 c 1 + c 2 3 − δ , q 2 δ = a − c 1 − 2 q 1 δ and p δ = c 1 + q 1 δ

The output for firm 2 increases with r – as expected, reselling the share on the rival’s profit makes firm 2 more aggressive in the product market. The output of firm 1, q ₁(δ), decreases with r – as the level of CO increases (δ = λ − r increases) the production transfer towards firm 1 is higher.

Note that being active in the product market requires that firms are sufficiently efficient. For firm 1 we assume c 1 < a + c 2 2 . Firm 2 is active in the product market if

(3) q 2 λ − r > 0 ⇔ c 2 < c ̄ 2 λ − r = a 1 − λ + r + c 1 1 + λ − r 2

Note that c ̄ 2 λ − r increases with r. Henceforth we assume that firm 2 is active in the product market for every r ≥ 0, that is

(3') c 2 < c ̄ 2 λ = a 1 − λ + c 1 1 + λ 2

The equilibrium profit for firm 1 is

(4) π 1 δ = p δ − c 1 q 1 δ = a − 2 c 1 + c 2 2 3 − δ 2

The profit for firm 2 is

(5) π 2 ⁢ δ = p ⁡ δ − c 2 ⁢ q 2 ⁢ δ = a − c 1 − 2 a − 2 c 1 + c 2 3 − δ ⁢ c 1 − c 2 + a − 2 c 1 + c 2 3 − δ

The profit π ₂(δ) is increasing in r, however π ₁(δ) is decreasing in r and owner B is also interested in the rival’s profit.

3.2 Stage 1

Each owner is interested in its value given the outcome of stage 2. Owner B earns the profit of firm 2, part of firm 1’s profit and the fee received from the investor:

v B δ = π 2 δ + δ π 1 δ + Φ

We consider that owner B extracts all the investor’s rent through Φ. The maximum fee that the investor I is willing to pay is Φ = rπ ₁(δ).

The value of B from reselling r when Φ = rπ ₁(δ) is

v B ⁢ λ , r = π 2 ⁢ δ + λ π 1 ⁢ δ = a − c 1 ⁢ c 1 − c 2 + a − 2 c 1 + c 2 2 1 + r 3 + r − λ 2 + c 2 − c 1 a − 2 c 1 + c 2 3 + r − λ

Setting ∂ v B λ , r ∂ r = 0 and solving for r gives

(6) r = r * = a 1 − λ + c 1 1 + λ − 2 c 2 a − 3 c 1 + 2 c 2

Thus, r ^* maximizes v _B(λ,r) when 0 < r ^* < λ.^[9] In particular, r ^* < λ holds and the COS is viable when

(7) c 2 > c ^ 2 = a 1 − 2 λ + c 1 1 + 4 λ 2 1 + λ > c 1

or equivalently when λ > λ ^ = a − 2 c 2 + c 1 2 α + c 2 − 2 c 1 . In addition, r ^* ≤ 0 requires the violation of (3); r ^* > 0 is secured for c 2 < c ̄ 2 λ = a 1 − λ + c 1 1 + λ 2 , which is given by (3′).

Proposition 1

Owner B resells part of the share on firm 1. If c 2 > c ^ 2 > c 1 or equivalently when λ > λ ^ , owner B retains part of the share, as r ^* < λ.

The stability of CO requires cost asymmetry of commonly owned firms. The common owner is the unique owner of firm 2. The profitability of this firm increases in r providing an incentive that destabilizes the COS. The common owner is interested also in the profit made by firm 1 which increases with c ₂ − c ₁.^[10]

Owner B faces a tradeoff: reselling the minority share (i) provides a strategic advantage ( ∂ π 2 δ ∂ r > 0 ) (ii) discourages the production shift towards the efficient firm. Because some of the minority share will be sold, (partial) stability of the COS requires that the initially acquired share is sufficiently large, given the cost differential c ₂ − c ₁ > 0.

3.3 Profitability and Welfare

We are interested first in whether owners A and B have an incentive to engage in CO.^[11] As c ^ 2 > c 1 we make the following simplifying assumption that lightens the analysis:

Assumption 1:

c ₁ = 0.

Thus, c 2 ∈ 0 , a 1 − λ 2 measures the cost asymmetry between the two firms. Recall from (3′) that for c 2 < a 1 − λ 2 both firms’ quantities are positive. From (7), the minimum λ above which r ^* < λ holds, is λ ^ = a − 2 c 2 2 α + c 2 . Thus, for 0 < λ < λ ^ CO is not viable (r ^* = λ) while for λ ^ < λ < λ ̄ = a − 2 c 2 a CO is partially stable, as r ^* < λ. For λ = λ ̄ , firm 2’s quantity is zero, that is from (3′), q 2 λ = a 1 − λ + 2 c 2 2 − λ > 0 holds when λ < λ ̄ .^[12]

The value for the owners A and B depends on whether λ is greater or lower than λ ^ . From (4) and (5), the profit for firms 1 and 2 when λ < λ ^ are π 1 0 = a + c 2 2 9 and π 2 0 = a − c 2 2 9 respectively. For λ > λ ^ firm 2 retains λ − r ^* > 0 in stage 1 and the stand-alone profit of each firm depends on λ.

The value for owner A is

v A λ = 1 − λ π 1 λ + F i f λ ^ < λ < λ ̄ 1 − λ π 1 0 + F i f 0 ≤ λ ≤ λ ^

For owner B, its value is

v B λ = λ π 1 λ + π 2 λ − F i f λ ^ < λ < λ ̄ λ π 1 0 + π 2 0 − F i f 0 ≤ λ ≤ λ ^

Lemma 1

The stand-alone profit for firm 1, π ₁(λ), increases with λ while π ₂(λ) decreases with λ. The value for owner A (B) decreases (increases) with λ.

Proof

See the Appendix.□

Owner B should reimburse owner A for the share λ. CO reduces the quantity volume and the profit of firm 2. However, owner B’s gain is increasing in λ, as firm 1’s profit raises when λ increases. The total gain is π 1 λ + π 2 λ if λ ^ < λ < λ ̄ and π 1 0 + π 2 0 if 0 ≤ λ ≤ λ ^ .

Proposition 2

Owners can agree on a share λ ∈ λ ^ , ⁡ min 1 2 , λ ̄ acquired by owner B. Owner B resells r ^*(<λ) in stage 1.

Proof

See the Appendix.□

The owner of firm 2 can offer a positive fee to owner A for the share λ. Note that for λ ̄ < 1 2 c 2 > a 4 , owners can agree on λ = λ ̄ , inducing firm 2 to produce zero. We focus on λ < λ ̄ , so that both firms remain active even if the cost-difference is large. If firm 2 is not too inefficient compared to 1 c 2 < a 4 , firm 2 sells positive quantities for all λ.^[13]

The owners A and B agree on λ ∈ λ ^ , ⁡ min 1 2 , λ ̄ . Then, part of λ will be sold while CO is present as owner B has 100 % of firm 2 and (λ − r ^*) of 1. The price for consumers and the income for both owners increase with the degree of CO.^[14] In addition, the inefficient firm produces less under the COS; production volume is transferred to firm 1. Proposition 3 summarizes the effects of CO on total welfare:

Proposition 3

For c 2 < a 5 , the minority share is λ ∈ λ ^ , 1 2 , total welfare decreases with λ. For c 2 ∈ a 5 , a 4 Common Ownership reduces or increases welfare. F o r c 2 > a 4 , the minority share λ ∈ λ ^ , λ ̄ and total welfare increases with λ.

Proof

See the Appendix.□

4 Concluding Remarks

We investigate the stability of common ownership in a quantity-setting duopoly. The owner of a firm acquires a share in a rival. The common ownership scheme is stable and the common owner retains part of the share if firms are cost-asymmetric: the common owner acquires a non-controlling share of the efficient firm. Given that common ownership can be stable, the fee that the common owner must pay makes both owners better-off.

There are two forces at play: divesting the minority stake to an outsider provides a strategic advantage for the acquirer. In the presence of a cost-asymmetry, passive ownership induces a shift in production towards the efficient firm which can be profitable for the acquirer of the minority share.

When the asymmetry in costs between firms is small, the stability of common ownership requires a large proportion of the efficient firm acquired by the common owner. In this case welfare deteriorates with common ownership. If firms are sufficiently asymmetric, the quantity volume produced by the inefficient firm is low, the condition for the stability of the common ownership scheme loosens and welfare improves with common ownership.

Corresponding author: Konstantinos Charistos, Department of Economics, University of Ioannina, Ioannina, Greece, E-mail: kchar@uoi.gr

Funding source: University of Crete

Acknowledgements

I acknowledge the financial support by the University of Crete.

Research funding: This work was funded by University of Crete.

Appendix

Proof of Lemma 1

The derivative of π 1 λ = a + 2 c 2 2 4 2 − λ 2 with respect to λ is ∂ π 1 λ ∂ λ = a + 2 c 2 2 2 2 − λ 3 > 0 . Similarly, π 2 λ = a 1 − λ − 2 c 2 a − 2 c 2 1 − λ 2 2 − λ 2 and ∂ π 2 λ ∂ λ = − λ a + 2 c 2 2 2 2 − λ 3 < 0 . In addition,

∂ v A λ ∂ λ = − λ a + 2 c 2 2 4 2 − λ 3 < 0 , ∂ v B λ ∂ λ = a + 2 c 2 2 4 2 − λ 2 > 0

∂ 1 − λ π 1 0 ∂ λ < 0 , ∂ λ π 1 0 + π 2 0 ∂ λ > 0

and 1 − λ π 1 0 = v A λ ^ , λ π 2 0 + π 2 0 = v B λ ^ . Thus, the value of A (B) decreases (increases) with λ and there is a kink in v _A, v _B at λ = λ ^ .

Proof of Proposition 2

As (a) ∂ π 1 λ + π 2 λ ∂ λ = 1 − λ a + 2 c 2 2 2 2 − λ 3 > 0 and (b) π 1 0 + π 2 0 = π 1 λ ^ + π 2 λ ^ , the total industry value is not affected by λ for λ < λ ^ ; then it increases with λ.

Proof of Proposition 3

The consumer’s surplus is calculated as follows

C S = 1 2 q 1 λ + q 2 λ a − p λ if λ ^ < λ < λ ̄ 1 2 q 1 0 + q 2 0 a − p 0 if 0 < λ < λ ^

C S = C S λ = a 3 − 2 λ − 2 c 2 2 8 2 − λ 2 if λ ^ < λ < λ ̄ C S 0 = 2 a − c 2 2 18 if 0 < λ < λ ^

The total surplus summarizes the profits of firms and CS:

T S = T S λ = C S λ + π 1 λ + π 2 λ if λ ^ < λ < λ ̄ C S 0 + π 1 0 + π 2 0 = 8 a a − c 2 + 11 c 2 2 18 if 0 < λ < λ ^

For λ ^ < λ < λ ̄ :

T S λ = 4 c 2 2 7 − 4 λ + a 2 5 − 2 λ 3 − 2 λ − 4 a c 2 5 − 2 λ 3 − λ 8 2 − λ 2

∂ T S λ ∂ λ = − a + 2 c 2 a − 6 c 2 + 4 c 2 λ 8 2 − λ 3

Observe that for λ ≤ 3 2 − a 4 c 2 , the total surplus increases with λ, ∂ T S λ ∂ λ ≥ 0 . We distinguish three cases for c ₂:

for 3 2 − α 4 c 2 > 1 2 ⟺ c 2 > a 4 , TS(λ) increases with λ, ∂ T S λ ∂ λ > 0 in λ ^ , λ ̄ .
for 3 2 − a 4 c 2 < λ ^ ⟺ c 2 < a 5 , TS(λ) reduces with λ, ∂ T S λ ∂ λ < 0 .
for c 2 ∈ a 5 , a 4 , CO reduces welfare when c 2 < 2 a 9 and λ > 7 a − 38 c 2 2 a − 11 c 2 :

The inequality T S λ < 8 a a − c 2 + 11 c 2 2 18 = C S 0 + π 1 0 + π 2 0 holds for c 2 < 2 a 9 and λ > 7 a − 38 c 2 2 a − 11 c 2 . Otherwise T S λ > 8 a a − c 2 + 11 c 2 2 18 .

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Received: 2024-07-12

Accepted: 2025-05-22

Published Online: 2025-06-11

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common ownership; passive investments; stability

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