3.1 Setup
The model assumes two cost-symmetric, risk-neutral firms
\(i,j \in \lbrace 1,2\rbrace\). The majority owner of firm
i may hold a minority stake
\(\alpha _i\) in firm
j while the majority owner of firm
j may hold a minority stake
\(\alpha _{j}\) in firm
i. The stakes may be asymmetric
\((\alpha _i\ne \alpha _{j})\), and they are assumed to have been chosen prior to the game analyzed here, which is in line with Reynolds and Snapp (
1986), Malueg (
1992), and Gilo et al. (
2006).
The firms
i and
j play an infinitely repeated game with the objective of maximizing the discounted sum of their majority owners’ individual profits. The discount factors are denoted
\(\delta _i,\delta _j\in (0,1)\). In each period, the firms choose a product market strategy
\(s_i,s_j \in \lbrace c,k\rbrace\) with
c indicating best-response behavior in the product market while
k represents the collusive choice of their strategic variable. We analyze the firms’ choice of
\(s_i,s_j\) in Sect.
4.
Our analysis is conducted in the three models that were studied by Flath (
1991): Cournot competition; Bertrand competition with differentiated products; and Bertrand competition with homogeneous products. Therefore, the product market profit/operating profit
\(\pi _i\) of firm
i is a function of the firms’ strategic variables – quantities in Cournot competition, and prices in Bertrand competition – and these strategic variables are functions of firms’ endogenous choice of
\(s_i,s_j\) and the exogenous shareholdings
\(\alpha _i,\alpha _j\). The profits can be expressed as
\(\pi _i\left( q_i(q_j,s_i,\alpha _i,\alpha _j),q_j(q_i,s_j,\alpha _j,\alpha _i)\right)\) in Cournot competition and
\(\pi _i\left( p_i(p_j,s_i,\alpha _i,\alpha _j),p_j(p_i,s_j,\alpha _j,\alpha _i)\right)\) in Bertrand competition.
In Sect.
3.2, we determine the firms’ collusive strategies and their non-collusive best responses in the product market. Firms’ best responses are initially obtained under the assumption of equation (
1). Firm
i is assumed to maximize its majority owner’s payoff
\(\tilde{\pi }_i\): Firm
i’s product market profit
\(\pi _i\), minus the dividend
\(\alpha _j \pi _i\) that is paid to firm
j, plus the dividend
\(\alpha _i \pi _j\) that is received from firm
j (Reynolds and Snapp
1986; Malueg
1992):
$$\begin{aligned} max\text { }\tilde{\pi }_i=(1-\alpha _{j})\pi _i+\alpha _i\pi _{j}. \end{aligned}$$
(1)
Flath (
1991,
1992), Gilo et al. (
2006), and Shelegia and Spiegel (
2012) modeled cross ownership and used a different profit function where the firms maximize their accounting profits:
\(\hat{\pi }_i=\pi _i+\alpha _i \hat{\pi _j}\). In Sect.
6, we discuss the exact interpretation of this profit function, and we show that the effects that are presented in this article are qualitatively identical for both profit functions.
Plugging the collusive strategies and competitive best responses into the profit functions for all four combinations of
\(s_i,s_j\) yields values of the profits that are dependent on the value of the shareholdings
\(\alpha _i\) and
\(\alpha _{j}\). For reasons of conciseness, we use indices to denote the strategy combinations
\((s_i,s_j)\):
c for (
c,
c);
k for (
k,
k);
d if
i is the deviator in (
c,
k); and
\(-d\) if
i is the cheated firm in (
k,
c). Notation is further explicated in Sect.
3.2. In the models that we analyze, the inequalities that are shown in (
2) apply, which impose a prisoner’s dilemma structure on the game:
$$\begin{aligned} \pi _{i,d}>\pi _{i,k}> & {} \pi _{i,c}\ge \pi _{i,-d}\text { and }\pi _{i,d}+\pi _{i,-d}<2\pi _{i,k} \quad \text { for all } i \in \lbrace 1,2\rbrace . \end{aligned}$$
(2)
The total payoffs
\(\tilde{\pi }_{i,c}\),
\(\tilde{\pi }_{i,k}\),
\(\tilde{\pi }_{i,d}\) of majority owner
i can be expressed as in (
3)-(
5):
$$\begin{aligned} \tilde{\pi }_{i,c}= & {} (1-\alpha _{j})\pi _{i,c}+\alpha _i\pi _{j,c} ; \end{aligned}$$
(3)
$$\begin{aligned} \tilde{\pi }_{i,k}= & {} (1-\alpha _{j}) \pi _{i,k} +\alpha _i\pi _{j,k} ; \end{aligned}$$
(4)
$$\begin{aligned} \tilde{\pi }_{i,d}= & {} (1-\alpha _{j})\pi _{i,d}+\alpha _i\pi _{j,-d} . \end{aligned}$$
(5)
3.2 The Stage Game
This subsection establishes firms’ best responses and equilibrium profits in Cournot competition, Bertrand competition with differentiated products, and Bertrand competition with homogeneous products. It demonstrates the effects of \(\alpha _i,\alpha _j\) on the firms’ stage game profits.
Cournot competition In Cournot competition, the payoff
\(\tilde{\pi }(q_i(q_j,s_i,\alpha _i,\alpha _j), q_{j}(q_i,s_j,\alpha _j,\alpha _i))\) of firm
i’s majority owner, which will be abbreviated as
\(\tilde{\pi }(q_i,q_{j})\), is a function of the outputs
\(q_i\) and
\(q_{j}\). The term
\(\tilde{q}_{i}^R(q_{j},\alpha _i,\alpha _j)\) denotes firm
i’s reaction function after choosing
\(s_i=c\) if it maximizes payoff function
\(\tilde{\pi _i}\). The arguments
\(q_{j}\),
\(\alpha _i\), and
\(\alpha _j\) of the best-response function
\(\tilde{q}_{i}^R\) are sometimes dropped for reasons of conciseness. The firms are said to compete if both play their best responses –
\((s_i,s_j)=(c,c)\) – which generates the payoff
\(\tilde{\pi }_i\left( \tilde{q}_i^R\left( \tilde{q}_j^R,\alpha _i,\alpha _j\right) ,\tilde{q}_{j}^R\left( \tilde{q}_i^R,\alpha _{j},\alpha _i\right) \right)\), which we abbreviate as
\({\tilde{\pi }_{i,c}\left( \alpha _i,\alpha _{j}\right) }\). The discussion below relies on a result that is summarized in Lemma
1:
Lemma
1 implies that the product-market profits
\(\pi _{i,c}\) of firm
i in competition
fall if the majority owner of firm
i holds a higher share
\(\alpha _i\) in firm
j. This occurs if quantities are strategic substitutes:
\(\partial q_i^R/\partial q_j<0\), with
\(q_{i}^R(q_{j})=\tilde{q}_i^R(q_j,0,0)\) denoting firm
i’s reaction function when
\(\alpha _i,\alpha _j=0\); when maximizing product market profits
\(\pi _i\) only. Firm
i finds it optimal to reduce both its own output (
\(\partial \tilde{q}_i^R/\partial \alpha _i<0\)) and its product-market profits
\(\pi _{i,c}\) in order to raise the total payoff
\(\tilde{\pi }_{i,c}\) of its majority owner by raising the other firm’s profit
\(\pi _{j,c}\), so that the majority owner of firm
i receives a higher dividend from firm
j (Reynolds and Snapp
1986). Conversely, a higher value of
\(\alpha _{j}\) raises
\(\pi _{i,c}\): The product-market profits
\(\pi _{i,c}\) of firm
i in competition rise if the majority owner of firm
j holds a higher share
\(\alpha _{j}\) in firm
i.
As in Malueg (
1992) and Gilo et al. (
2006), the firms are assumed to collude in the product market by setting a 50%-share of the monopoly output:
\(q_{i,k}=q_{j,k}=Q_k/2\). Because the collusive output is independent of
\(\alpha _i\) and
\(\alpha _{j}\), the collusive profits are also independent of the value of shareholdings:
\(\partial \pi _{i,k}/\partial \alpha _i=0\),
\(\partial \pi _{i,k}/\partial \alpha _{j}=0\).
Deviation profits are defined as
\({\tilde{\pi }_{i,d}(\alpha _i,\alpha _j)=\tilde{\pi }_{i}\left( \tilde{q}_i^R(q_{j,k},\alpha _i,\alpha _j),q_{j,k}\right) }\) and
\({\tilde{\pi }_{j,-d}(\alpha _i,\alpha _j)=\tilde{\pi }_{j}\left( q_{j,k},\tilde{q}_i^R(q_{j,k},\alpha _i,\alpha _j)\right) }\): Firm
i plays its best response while firm
j sets the agreed-upon output. Lemma
2 establishes the effect of
\(\alpha _i\) on the product market profits in a deviation period:
If firm i deviates from collusion, its majority owner receives a lower dividend from firm j as compared to continued collusion (\(\alpha _i\pi _{j,-d}<\alpha _i\pi _{j,k}\)). The higher is the value of \(\alpha _i\) the stronger is this effect, and the lower is the payoff \(\tilde{\pi }_{i,d}\) that the majority owner of firm i earns after the payment of dividends. Accordingly, minority shareholdings \(\alpha _i>0\) induce the deviating firm i to set a lower deviation quantity than with \(\alpha _i=0\) and, thus, to earn lower deviation profits in the product market. This leaves higher profits for firm j: \(\partial \pi _{j,-d}/\partial \alpha _i>0\).
Bertrand competition with differentiated products Similar effects are found when we assume Bertrand competition with differentiated products. The payoff
\(\tilde{\pi }(p_i,p_{j})\) of firm
i’s majority shareholder is a function of the prices
\(p_i\) and
\(p_{j}\) of the two firms. Let
\(\tilde{p}_i^R(p_{j},\alpha _i,\alpha _j)\) denote the best-response function of firm
i after choosing
\(s_i=c\). The firms compete when both play their best responses, making profits
\({\tilde{\pi }_{i,c}(\alpha _i,\alpha _{j})=\tilde{\pi }_i\left( \tilde{p}_i^R\left( \tilde{p}_j^R,\alpha _i,\alpha _j\right) ,\tilde{p}_{j}^R\left( \tilde{p}_i^R,\alpha _{j},\alpha _i\right) \right) }\). Our analysis relies on Lemma
3:
As compared to Cournot competition, firm
i’s competitive profit rises even for unilateral increases of its share
\(\alpha _i\) in firm
j as long as the shares are not too asymmetric:
\({\alpha _i< \alpha _{j}+A}\). The exact value of the threshold
A depends on the functional form of demand as is shown in the proof of Lemma
3. This effect emerges because in a Bertrand model with differentiated products prices are strategic complements. Shareholdings
\(\alpha _i\) induce firm
i to raise its price, and firm
j follows suit. Therefore, even a somewhat asymmetric increase in
\(\alpha _i\) may cause an increase in both firms’ profits. However, for
\(\alpha _i\ge \alpha _j+A\) the reduction in output that results from the higher prices becomes the dominant force, which leads to
\(\partial \pi _{i,c}/\partial \alpha _i\le 0\).
The firms are assumed to collude in the product market by setting the same prices
\(p_{i,k}\) and
\(p_{j,k}\) that a jointly profit-maximizing monopolist would set. These prices are independent of
\(\alpha _i\) and
\(\alpha _{j}\), which implies
\(\partial \pi _{i,k}/\partial \alpha _i=0\) and
\(\partial \pi _{i,k}/\partial \alpha _{j}=0\). The deviation payoffs are defined as
\(\tilde{\pi }_{i,d}(\alpha _i,\alpha _j)=\tilde{\pi }_{i}\left( \hat{p}_i^R(p_{j,k},\alpha _i,\alpha _j),p_{j,k}\right)\) and
\(\tilde{\pi }_{j,-d}(\alpha _i,\alpha _j)=\tilde{\pi }_{j}\left( p_{j,k},\tilde{p}_i^R(p_{j,k},\alpha _i,\alpha _j)\right)\). “
Appendix 1” shows that Lemma
2 (
\({\partial \pi _{i,d}/\partial \alpha _i<0}\) and
\({\partial \pi _{j,-d}/\partial \alpha _i>0}\)) applies also in Bertrand competition with differentiated products.
Bertrand competition with homogeneous products In Bertrand competition with homogeneous products “both firms set prices equal to marginal cost regardless of the state of any partial cross shareholding” (Flath
1991): The firms make zero profits (
\({\pi _{i,c}=0}\),
\({\pi _{j,c}=0}\),
\({\partial \pi _{i,c}/\partial \alpha _i=0}\), and
\({\partial \pi _{j,c}/\partial \alpha _i=0}\)), and minority shareholdings have no effect on firms’ profits in competition. Similarly, the collusive and the deviation profits are also independent of the value of minority shareholdings:
\({\partial \pi _{i,k}/\partial \alpha _i=0}\);
\({\partial \pi _{j,k}/\partial \alpha _i=0}\);
\({\partial \pi _{i,d}/\partial \alpha _i=0}\);
\({\partial \pi _{j,-d}/\partial \alpha _i=0}\). A deviating firm would cut the collusive price marginally and earn
\({\pi _{i,d}=2\pi _{i,k}}\), while the betrayed firm would earn
\(\pi _{j,-d}=0\).