Cross-ownership and managerial delegation under vertical product differentiation

Abstract

We construct a vertical product differentiation duopoly model incorporating managerial delegation and cross-ownership. By exploring the interplay of these factors, we find a U-shaped relationship between endogenous managerial delegation coefficients and cross-ownership. The difference in managerial delegation coefficients between the two firms decreases as the cross-ownership proportion increases. In an ownership structure involving cross-ownership of firms producing different quality products, managerial delegation improves firms’ profits while reducing consumer surplus and social welfare in a vertical product differentiation market. Moreover, cross-ownership intensifies the positive impact of managerial delegation on joint profits and the negative effects on consumer surplus and social welfare. Consequently, regulating cross-ownership among firms in vertically differentiated product markets is an important policy issue for competition law.

Appendix

We assume $\alpha \in \left[0,\frac{1}{2}\right)$ and $\delta \in (\mathrm{0,1})$ throughout the following proofs.

1.1 Proof of propositions and lemmas

1.1.1 Proof of Lemma 1

$$\begin{aligned} \partial p_{1}^{e} /\partial \lambda_{1} & = - \frac{(1 + \delta )(1 + 2\delta )}{{3 + (9 - \alpha - 6\delta )\delta }} < 0,\;\partial p_{1}^{e} /\partial \lambda_{2} = - \frac{1 + \alpha + (2 + \alpha )\delta }{{(1 - \alpha )(3 + (9 - \alpha + 6\delta )\delta )}} < 0, \\ \partial p_{2}^{e} /\partial \lambda_{1} & = - \frac{{(1 + \delta )^{2} }}{3 + (9 - \alpha + 6\delta )\delta } < 0,\;\partial p_{1}^{e} /\partial \lambda_{2} = - \frac{\alpha 1 + (1 + \delta )(1 + 3\delta )}{{(1 - \alpha )(3 + (9 - \alpha + 6\delta )\delta )}} < 0. \\ \end{aligned}$$

$$\begin{aligned} \partial \left| {\partial p_{1}^{e} /\partial \lambda_{1} } \right|/\partial \alpha & = \frac{\delta (1 + \delta )(1 + 2\delta )}{{(3 + (9 - \alpha + 6\delta )\delta )^{2} }} > 0, \\ \partial \left| {\partial p_{1}^{e} /\partial \lambda_{2} } \right|/\partial \alpha & = \frac{{6 + \delta (28 - \alpha (2 + \alpha ) + 41\delta - \alpha (4 + \alpha )\delta + 18\delta^{2} )}}{{(1 - \alpha )^{2} (3 + (9 - \alpha + 6\delta )\delta )^{2} }} > 0, \\ \partial \left| {\partial p_{2}^{e} /\partial \lambda_{1} } \right|/\partial \alpha & = \frac{{\delta (1 + \delta )^{2} }}{{(3 + (9 - \alpha + 6\delta )\delta )^{2} }} > 0, \\ \partial \left| {\partial p_{2}^{e} /\partial \lambda_{2} } \right|/\partial \alpha & = \frac{{6 + \delta (31 - \alpha^{2} - 2\alpha (1 + \delta )(1 + 3\delta ) + \delta (61 + 18\delta (3 + \delta )))}}{{(1 - \alpha )^{2} (3 + (9 - \alpha + 6\delta )\delta )^{2} }} > 0. \\ \end{aligned}$$

1.1.2 Proof of Proposition 1

$$\lambda_{1}^{e} - \lambda_{2}^{e} = \frac{(1 - \alpha )\delta (8 + 3\delta (7 + 4\delta ) + \alpha (5 + 6\delta (2 + \delta )))}{{21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta ))}} > 0,$$

$$\partial (\lambda_{1}^{e} - \lambda_{2}^{e} )/\partial \alpha = - \frac{{\left( \begin{gathered} 3\delta (1 + \delta )(1 + 2\delta )(5 + 2\delta (19 + 18\delta (2 + \delta )) - \hfill \\ \alpha^{2} (10 + \delta (25 + 12\delta )) + \alpha (70 + 2\delta (115 + 6\delta (19 + 6\delta )))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} < 0.$$

1.1.3 Proof of Proposition 2

$$\partial \lambda_{1}^{e} /\partial \alpha = \frac{6(\alpha ( - 7 + \alpha - 6\delta ) - 1)(1 + \delta )(1 + 2\delta )(3 + \delta (7 + 3\delta ))}{{(6\alpha (1 + \delta )(1 + 2\delta ) - 4\delta (20 + 3\delta (8 + 3\delta )) - (21 + \alpha^{2} \delta ))^{2} }}.$$

When $\alpha$ is relatively low, i.e., $\alpha < \left( {7 + 6\delta - \sqrt {45 + 84\delta + 36\delta^{2} } } \right)/2$, $\partial \lambda_{1}^{e} /\partial \alpha < 0$;

when $\alpha$ is relatively high, i.e., $\alpha \ge \left( {7 + 6\delta - \sqrt {45 + 84\delta + 36\delta^{2} } } \right)/2$, $\partial \lambda_{1}^{e} /\partial \alpha \ge 0$.

Therefore, there is a U-shaped relationship between the managerial delegation coefficient of firm 1 and the cross-ownership.

$$\partial \lambda_{2}^{e} /\partial \alpha = \frac{{\left( \begin{gathered} 3(1 + \delta )(1 + 2\delta )((2 + 3\delta )(4\delta^{2} (4 + 3\delta ) - 3) - \hfill \\ \alpha^{2} (6 + \delta (24 + \delta (31 + 12\delta ))) + \hfill \\ \alpha (42 + 4\delta (51 + \delta (89 + 6\delta (11 + 3\delta ))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }}$$

When $\alpha$ is relatively low,

i.e., $\alpha < \alpha ^{\prime\prime}(\delta ) = \left( \begin{gathered} \frac{13}{4} + 3\delta + \frac{6 + 3\delta (8 + 7\delta )}{{4(6 + \delta (24 + \delta (31 + 12\delta )))}} - \hfill \\ \sqrt 3 \sqrt {\frac{(1 + \delta )(1 + 2\delta )(5 + 6\delta )(3 + \delta (7 + 3\delta ))(9 + 4\delta (8 + 3\delta (3 + \delta )))}{{(6 + \delta (24 + \delta (31 + 12\delta )))^{2} }}} \hfill \\ \end{gathered} \right)$, $\partial \lambda_{2}^{e} /\partial \alpha < 0$;

when $\alpha$ is relatively high, i.e., $\alpha \ge \alpha ^{\prime\prime}$, $\partial \lambda_{2}^{e} /\partial \alpha \ge 0$.

Therefore, there is a U-shaped relationship between the managerial delegation coefficient of firm 2 and the cross-ownership.

1.1.4 Proof of Lemma 2

$$\partial \lambda_{1}^{e} /\partial \delta = \frac{{3(1 - \alpha^{2} )(31 + \alpha^{2} (1 - \delta^{2} ) - 2\alpha (2 + \delta (6 + 5\delta )) + 6\delta (25 + 2\delta (21 + \delta (14 + 3\delta ))))}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} > 0$$

$$\partial \lambda_{2}^{e} /\partial \delta = \frac{{\left( \begin{gathered} 3(\alpha - 1)(25 + 144\delta + \alpha^{3} (1 + \delta )(\delta + 4\delta^{2} - 1) + 8\delta^{2} (38 + \delta (35 + 12\delta )) - \hfill \\ \alpha^{2} (1 + 2\delta )(7 + 2\delta (11 + \delta (13 + 6\delta ))) - 2\alpha (1 + \delta )(4 + \delta (23 + 2\delta (19 + 9\delta )))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} < 0$$

1.1.5 Proof of Lemma 3

The controlling shareholders’ profit of firm $i$, the consumer surplus, and social welfare are as follows:

$$\Pi_{1}^{e} = \frac{{\left( {1 + \delta } \right)\left( \begin{gathered} 3\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right)\left( {18 + \delta \left( {136 + 9\delta \left( {37 + 12\delta \left( {3 + \delta } \right)} \right)} \right)} \right) - \hfill \\ \alpha^{2} \left( {27 + \delta \left( {237 + \delta \left( {823 + 3\delta \left( {457 + 36\delta \left( {10 + 3\delta } \right)} \right)} \right)} \right)} \right) - \hfill \\ \alpha^{4} \delta^{2} \left( {1 + 3\delta } \right) - 3\alpha^{3} \left( {1 + \delta } \right)\left( {1 + 2\delta } \right)\left( {3 + 4\delta - 6\delta^{2} } \right) + 4\left( {2 + 3\delta } \right)\left( {3 + \delta \left( {7 + 3\delta } \right)} \right)^{2} \hfill \\ \end{gathered} \right)}}{{2\left( {21 + \alpha^{2} \delta - 6\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right) + 4\delta \left( {20 + 3\delta \left( {8 + 3\delta } \right)} \right)} \right)^{2} }},$$

$$\Pi_{2}^{e} = \frac{{9\left( {1 - \alpha } \right)\left( {2 - \alpha + 3\delta } \right)\left( {1 + \delta } \right)^{2} \left( {1 + 2\delta } \right)^{2} \left( {4 + \alpha + 11\delta + 6\delta^{2} } \right)}}{{2\left( {21 + \alpha^{2} \delta - 6\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right) + 4\delta \left( {20 + 3\delta \left( {8 + 3\delta } \right)} \right)} \right)^{2} }},$$

$$\begin{aligned} CS^{e} & = \mathop \smallint \nolimits_{{p_{1} }}^{{\frac{{p_{2} - p_{1} }}{\delta }}} \left( {x - p_{1} } \right){\text{d}}x + \mathop \smallint \nolimits_{{\frac{{p_{2} - p_{1} }}{\delta }}}^{1} \left( {\left( {1 + \delta } \right)x - p_{2} } \right){\text{d}}x = \\ & \quad \frac{{\left( {1 + \delta } \right)^{2} \left( \begin{gathered} 36\left( {2 - \alpha } \right)^{2} + 3\left( {2 - \alpha } \right)\left( {146 - \alpha \left( {63 - 4\alpha } \right)} \right)\delta + 324\delta ^{5} - \hfill \\ 3\left( {4\alpha \left( {125 - \left( {22 - \alpha } \right)\alpha } \right) - 779} \right)\delta ^{3} + 36\left( {37 - \left( {14 - \alpha } \right)\alpha } \right)\delta ^{4} + \hfill \\ \left( {2053 - \alpha \left( {1680 - \alpha \left( {403 - \left( {30 - \alpha } \right)\alpha } \right)} \right)} \right)\delta ^{2} \hfill \\ \end{gathered} \right)}}{{2\left( {21 + \alpha ^{2} \delta - 6\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right) + 4\delta \left( {20 + 3\delta \left( {8 + 3\delta } \right)} \right)} \right)^{2} }}, \\ \end{aligned}$$

$$\begin{aligned} SW^{e} & = CS^{e} + \Pi_{1}^{e} + \Pi_{2}^{e} = \\ & \quad \frac{{\left( {1 + \delta } \right)\left( \begin{gathered} 288 - 2\alpha 1^{4} \delta^{3} - 6\alpha 1^{3} \delta \left( {1 + \delta } \right)\left( {1 - 2\delta } \right)\left( {1 + 2\delta } \right) - \hfill \\ 6\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right)\left( {30 + \delta \left( {113 + 20\delta \left( {7 + 3\delta } \right)} \right)} \right) + \hfill \\ \alpha^{2} \left( {18 + \delta \left( {129 + \delta \left( {279 + 2\delta \left( {89 - 6\delta \left( {5 + 6\delta } \right)} \right)} \right)} \right)} \right) + \hfill \\ \delta \left( {2130 + \delta \left( {6480 + \delta \left( {10453 + 3\delta \left( {3173 + 12\delta \left( {130 + 27\delta } \right)} \right)} \right)} \right)} \right) \hfill \\ \end{gathered} \right)}}{{2\left( {21 + \alpha^{2} \delta - 6\alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right) + 4\delta \left( {20 + 3\delta \left( {8 + 3\delta } \right)} \right)} \right)^{2} }}. \\ \end{aligned}$$

$$\partial (\Pi_{1}^{e} + \Pi_{2}^{e} )/\partial \alpha = \frac{{\left( \begin{gathered} 3(1 - \alpha )(1 + \delta )(1 + 2\delta )(162 + \delta (1377 - \alpha^{3} \delta (1 + \delta )(1 + 3\delta ) + \hfill \\ \alpha^{2} \delta (27 + \delta (109 + 6\delta (23 + 9\delta ))) + \hfill \\ \delta (4985 + 2\delta (5011 + 6\delta (1013 + 6\delta (124 + 51\delta + 9\delta^{2} )))) - \hfill \\ \alpha (81 + \delta (558 + \delta (1511 + 12\delta (167 + 27\delta (4 + \delta ))))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} > 0$$

$$\partial CS^{e} /\partial \alpha = - \frac{{\left( \begin{gathered} 3(1 + \delta )^{2} (1 + 2\delta )(\alpha^{4} \delta^{3} - 9\alpha^{3} \delta^{2} (1 + \delta )(1 + 2\delta ) + \hfill \\ 2(2 + 3\delta )^{2} (3 + 4\delta )^{2} (3 + \delta (7 + 3\delta )) + 9\alpha^{2} \delta (1 + 2\delta )(6 + \delta (19 + 6\delta (3 + \delta ))) - \hfill \\ 3\alpha (1 + 2\delta )(36 + \delta (225 + \delta (507 + 2\delta (253 + 3\delta (37 + 6\delta )))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} < 0,$$

$$\partial SW^{e} /\partial \alpha = - \frac{{\left( \begin{gathered} (3(1 + \delta )(1 + 2\delta )^{2} (\alpha^{3} \delta^{2} (19 + 3\delta (13 + 6\delta )) - \alpha^{4} \delta^{2} (1 + \delta ) - \hfill \\ 9\alpha^{2} \delta (1 + \delta )(3 + \delta (19 + 6\delta (5 + 2\delta ))) + \hfill \\ (1 + \delta )(2 + 3\delta )(27 + 2\delta (81 + \delta (151 + 9\delta (11 + 2\delta )))) + \hfill \\ \alpha (54 + \delta (351 + \delta (1079 + 6\delta (324 + \delta (335 + 3\delta (59 + 12\delta )))))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} < 0.$$

1.1.6 Proof of Proposition 3

When the firm’s owner manages the firm by himself, i.e., ${\lambda }_{i}=0$, the profit of firm i’s controlling shareholders, the consumer surplus, and social welfare are as follows:

$$\Pi_{1}^{E} = \frac{{\left( {1 + \delta } \right)\left( {1 + 3\delta } \right)\left( {\left( {1 + \delta } \right)^{2} + \alpha \left( {1 + \delta } \right)\left( {1 + 2\delta } \right)\left( {1 + 3\delta } \right) - \alpha^{2} \delta^{2} } \right)}}{{2\left( {3 + \delta \left( {9 - \alpha + 6\delta } \right)} \right)^{2} }},$$

$$\Pi_{2}^{E} = \frac{{\left( {1 - \alpha } \right)\left( {1 + \delta } \right)^{2} \left( {1 + 2\delta } \right)\left( {1 + 3\delta } \right)^{2} }}{{2\left( {3 + \delta \left( {9 - \alpha + 6\delta } \right)} \right)^{2} }},$$

$$\Pi_{1}^{e} - \Pi_{1}^{E} = \frac{1}{2}(1 + \delta )\left( \begin{gathered} \frac{{(1 + 3\delta )(\alpha^{2} \delta^{2} - (1 + \delta )^{2} - \alpha (1 + \delta )(1 + 2\delta )(1 + 3\delta ))}}{{(3 + \delta (9 - \alpha + 6\delta ))^{2} }} + \hfill \\ \frac{{\left( \begin{gathered} 4(2 + 3\delta )(3 + \delta (7 + 3\delta ))^{2} - \alpha^{4} \delta^{2} (1 + 3\delta ) - 3\alpha^{3} (1 + \delta )(1 + 2\delta )(3 + 4\delta - 6\delta^{2} ) + \hfill \\ 3\alpha (1 + \delta )(1 + 2\delta )(18 + \delta (136 + 9\delta (37 + 12\delta (3 + \delta )))) - \hfill \\ \alpha^{2} (27 + \delta (237 + \delta (823 + 3\delta (457 + 36\delta (10 + 3\delta ))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} \hfill \\ \end{gathered} \right) > 0$$

$$\Pi_{2}^{e} - \Pi_{2}^{E} = \frac{1}{2}(1 - \alpha )(1 + \delta )^{2} (1 + 2\delta )\left( {\frac{{9(2 - \alpha + 3\delta )(1 + 2\delta )(4 + \alpha + 11\delta + 6\delta^{2} )}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} - \frac{{(1 + 3\delta )^{2} }}{{(3 + \delta (9 - \alpha 1 + 6\delta ))^{2} }}} \right) > 0$$

$$\begin{gathered} \partial (\Pi _{1}^{e} + \Pi _{2}^{e} - \Pi _{1}^{E} - \Pi _{2}^{E} )/\partial \alpha = \hfill \\ \frac{{\left( \begin{gathered} (1 + \delta )(9\alpha ^{5} \delta ^{3} (1 + \delta )(1 + 2\delta )(31 + \delta (181 + \delta (377 + 18\delta (18 + 5\delta )))) - \hfill \\ \alpha ^{6} \delta ^{4} (29 + \delta (173 + 6\delta (61 + \delta (53 + 15\delta )))) - 3\alpha ^{4} \delta ^{2} (261 + \delta (2922 + \hfill \\ \delta (14125 + \delta (38447 + \delta (64297 + 2\delta (33683 + 3\delta (7145 + 24\delta (104 + 15\delta )))))))) + \hfill \\ 3\alpha ^{3} \delta (1 + \delta )(1 + 2\delta )(144 + \delta (243 - \delta (5694 + \delta (33963 + \delta (85712 + \hfill \\ 3\delta (39029 + 6\delta (5023 + 12\delta (173 + 30\delta )))))))) + \hfill \\ 3\alpha ^{2} \delta (1 + \delta )(5049 + \delta (76383 + \delta (509277 + \delta (1969234 + \hfill \\ 3\delta (1628731 + 2\delta (1353842 + 9\delta (169607 + 4\delta (32018 + 9\delta (1723 + 484\delta + 60\delta ^{2} ))))))))) - \hfill \\ 3\alpha (1 + \delta )(1 + 2\delta )(4374 + \delta (73440 + \delta (546363 + \delta (2378220 + \hfill \\ \delta (6728478 + \delta (12987847 + 6\delta (2908697 + 18\delta (151211 + 3\delta (32219 + 6\delta (2237 + 548\delta + 60\delta ^{2} )))))))))) + \hfill \\ (1 + \delta )(13122 + \delta (224235 + \delta (1734345 + \delta (8024625 + \delta (24713615 + \hfill \\ 6\delta (8882357 + 2\delta (6863465 + 6\delta (1273504 + 9\delta (112307 + \delta (61757 + 12\delta (1853 + 6\delta (65 + 6\delta ))))))))))))) \hfill \\ \end{gathered} \right)}}{{(3 + \delta (9 - \alpha + 6\delta ))^{3} (21 + \alpha ^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} > 0 \hfill \\ \end{gathered}$$

$$\begin{gathered} \partial (\Pi_{1}^{e} + \Pi_{2}^{e} - \Pi_{1}^{E} - \Pi_{2}^{E} )/\partial \delta = \frac{1}{72} \hfill \\ \left( \begin{gathered} \frac{3(1 - \alpha )(3 - \alpha (11 - 2\alpha ))}{{3 + (9 - \alpha + 6\delta )\delta }} + \hfill \\ \frac{{\left( \begin{gathered} - 27(1 + \delta ) + \alpha (3\alpha (\alpha (73 - (20 - \alpha )\alpha )) - \hfill \\ 21 - (9 + \alpha (165 - \alpha (480 - \alpha (235 - (29 - \alpha )\alpha ))))\delta ) \hfill \\ \end{gathered} \right)}}{{(3 + \delta (9 - \alpha + 6\delta ))^{3} }} - \hfill \\ \frac{{\left( \begin{gathered} 6(\alpha^{2} - 1)( - 543 - 3\alpha ( - 220 + \alpha (2 + \alpha )(5 + \alpha (3 + 2\alpha ))) - \hfill \\ 1670\delta + \alpha (1978 - \alpha (61 + \alpha (104 + \alpha (52 + (11 - \alpha )\alpha ))))\delta - \hfill \\ 6(2 - \alpha )(104 - \alpha (63 + \alpha (40 + \alpha (9 + \alpha ))))\delta^{2} ) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} \hfill \\ \frac{{\left( \begin{gathered} 6( - 41 + \alpha (355 - \alpha (412 - \alpha (91 + \alpha (20 + \alpha (13 + \alpha ))))) - \hfill \\ 195\delta + \alpha (1102 - \alpha (1141 - \alpha (253 + \alpha (41 + (22 - \alpha )\alpha ))))\delta - \hfill \\ 3(28 - \alpha (242 - \alpha (283 - \alpha (62 + \alpha (23 + 2\alpha )))))\delta^{2} ) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} \hfill \\ \frac{{6( - 3(5 + 13\delta ) - \alpha (11 - 30\delta - \alpha (5 + 2\alpha 1 + \alpha^{2} - 6(4 - \alpha )\delta )))}}{{21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta ))}} + \hfill \\ \frac{9\delta + \alpha (78 + 189\delta - \alpha (96 + 345\delta + \alpha (55 - 159\delta - \alpha (20 - \alpha - 12\delta ))))}{{(3 + \delta (9 - \alpha 1 + 6\delta ))^{2} }} \hfill \\ \end{gathered} \right) < 0 \hfill \\ \end{gathered}$$

1.1.7 Proof of Proposition 4

$$\begin{aligned} CS^{E} & = \mathop \smallint \nolimits_{{p_{1} }}^{{\frac{{p_{2} - p_{1} }}{\delta }}} \left( {x - p_{1} } \right){\text{d}}x + \mathop \smallint \nolimits_{{\frac{{p_{2} - p_{1} }}{\delta }}}^{1} \left( {\left( {1 + \delta } \right)x - p_{2} } \right){\text{d}}x = \\ & \quad \frac{{\left( {1 + \delta } \right)^{2} \left( {4 + \delta \left( {17 - 4\alpha + \left( {22 - \left( {8 - \alpha } \right)\alpha } \right)\delta + 9\delta^{2} } \right)} \right)}}{{2\left( {3 + \delta \left( {9 - \alpha + 6\delta } \right)} \right)^{2} }}, \\ \end{aligned}$$

$$\begin{aligned} SW^{E} & = CS^{E} + \Pi_{1}^{E} + \Pi_{2}^{E} = \\ & \quad \frac{{\left( {1 + \delta } \right)\left( {6 + \delta \left( {35 - 4\alpha + 3\left( {25 - 4\alpha } \right)\delta + \left( {73 - 2\alpha \left( {4 + \alpha } \right)} \right)\delta^{2} + 27\delta^{3} } \right)} \right)}}{{2\left( {3 + \delta \left( {9 - \alpha + 6\delta } \right)} \right)^{2} }}. \\ \end{aligned}$$

$$CS^{e} - CS^{E} = \frac{1}{2}(1 + \delta )^{2} \left( \begin{gathered} - \frac{{4 + \delta (17 - 4\alpha + (22 + ( - 8 + \alpha )\alpha )\delta + 9\delta^{2} )}}{{(3 + \delta (9 - \alpha 1 + 6\delta ))^{2} }} + \hfill \\ \frac{{\left( \begin{gathered} 36(2 - \alpha )^{2} + 3(2 - \alpha )(146 - \alpha (63 - 4\alpha ))\delta + \hfill \\ (2053 - \alpha (1680 - \alpha (403 - (30 - \alpha )\alpha )))\delta^{2} + \hfill \\ 3(779 - 4\alpha (125 + (22 + - \alpha )\alpha ))\delta^{3} + 36(37 - (14 - \alpha )\alpha )\delta^{4} + 324\delta^{5} \hfill \\ \end{gathered} \right)}}{{(21 + \alpha 1^{2} \delta - 6\alpha 1(1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} \hfill \\ \end{gathered} \right) < 0$$

$$SW^{e} - SW^{E} = \frac{1}{2}(1 + \delta )\left( \begin{gathered} - \frac{{6 + \delta (35 - 4\alpha + 3(25 - 4\alpha )\delta + (73 - 2\alpha (4 + \alpha ))\delta^{2} + 27\delta^{3} )}}{{(3 + \delta (9 - \alpha + 6\delta ))^{2} }} + \hfill \\ \frac{{\left( \begin{gathered} (288 - 2\alpha^{4} \delta^{3} + 6\alpha^{3} \delta (1 + \delta )(2\delta - 1)(1 + 2\delta ) - \hfill \\ 6\alpha (1 + \delta )(1 + 2\delta )(30 + \delta (113 + 20\delta (7 + 3\delta ))) + \hfill \\ \alpha^{2} (18 + \delta (129 + \delta (279 - 2\delta ( - 89 + 6\delta (5 + 6\delta ))))) + \hfill \\ \delta (2130 + \delta (6480 + \delta (10453 + 3\delta (3173 + 12\delta (130 + 27\delta )))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha 1(1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} \hfill \\ \end{gathered} \right) < 0$$

$$\partial \left( {CS^{E} - CS^{e} } \right)/\partial \alpha = \frac{1}{2}(1 + \delta )^{2} \left( \begin{gathered} \frac{2\delta (2 + (4 - \alpha )\delta )}{{(3 + \delta (9 - \alpha + 6\delta ))^{2} }} - \frac{{2\delta (4 + \delta (17 - 4\alpha + (22 + ( - 8 + \alpha )\alpha )\delta + 9\delta^{2} ))}}{{(3 + \delta (9 - \alpha + 6\delta ))^{3} }} + \hfill \\ \frac{{\left( \begin{gathered} 4(36(2 - \alpha )^{2} + 3(2 - \alpha )(146 - \alpha (63 - 4\alpha ))\delta + \hfill \\ (2053 - \alpha (1680 - \alpha (403 - (30 - \alpha )\alpha )))\delta^{2} - \hfill \\ 3( - 779 + 4\alpha (125 - (22 - \alpha )\alpha ))\delta^{3} + \hfill \\ 36(37 - (14 - \alpha )\alpha )\delta^{4} + 324\delta^{5} )(3 + \delta (9 - \alpha + 6\delta )) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} + \hfill \\ \frac{{\left( \begin{gathered} 4\alpha^{3} \delta^{2} - 18\alpha^{2} \delta (2 + \delta )(1 + 2\delta ) + \alpha (72 + 2\delta (213 + \delta (13 + 6\delta )(31 + 6\delta ))) - \hfill \\ 12(1 + 2\delta )(12 + \delta (44 + \delta (52 + 21\delta ))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha 1^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} \hfill \\ \end{gathered} \right) > 0$$

$$\partial \left( {SW^{E} - SW^{e} } \right)/\partial \alpha = \frac{1}{2}(1 + \delta )\left( \begin{gathered} \frac{4\delta (1 + \delta (3 + (2 + \alpha 1)\delta ))}{{(3 + \delta (9 - \alpha + 6\delta ))^{2} }} - \frac{{2\delta (6 + \delta (35 - 4\alpha + 3(25 - 4\alpha )\delta + (73 - 2\alpha (4 + \alpha ))\delta^{2} + 27\delta^{3} ))}}{{(3 + \delta (9 - \alpha + 6\delta ))^{3} }} + \hfill \\ \frac{{\left( \begin{gathered} - 8\alpha^{3} \delta^{3} - 18\alpha^{2} \delta (1 + \delta )(1 - 2\delta )(1 + 2\delta ) - 6(1 + \delta )(1 + 2\delta )(30 + \delta (113 + 20\delta (7 + 3\delta ))) + \hfill \\ 2\alpha (18 + \delta (129 + \delta (279 + 2\delta (89 - 6\delta (5 + 6\delta ))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha 1^{2} \delta - 6\alpha 1(1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }} + \hfill \\ \frac{{\left( \begin{gathered} (4(3 + \delta (9 - \alpha + 6\delta ))(288 - 2\alpha^{4} \delta^{3} - 6\alpha^{3} \delta (1 + \delta )(1 - 2\delta )(1 + 2\delta ) - \hfill \\ 6\alpha (1 + \delta )(1 + 2\delta )(30 + \delta (113 + 20\delta (7 + 3\delta ))) + \hfill \\ \alpha^{2} (18 + \delta (129 + \delta (279 - 2\delta ( - 89 + 6\delta (5 + 6\delta ))))) + \hfill \\ \delta (2130 + \delta (6480 + \delta (10453 + 3\delta (3173 + 12\delta (130 + 27\delta ))))))) \hfill \\ \end{gathered} \right)}}{{(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{3} }} \hfill \\ \end{gathered} \right) > 0$$

1.2 The scenario where firm 2 holds the shares of firm 1

In the second stage, the manager of firm $i$ determines the price based on the managerial contract:

$$\left\{ \begin{gathered} \mathop {M{\text{ax}}}\limits_{{p_{1} }} M_{1} = (1 - \alpha )\pi_{1} + \lambda_{1} D_{1} , \hfill \\ \mathop {M{\text{ax}}}\limits_{{p_{2} }} M_{2} = \pi_{2} + \alpha \pi_{1} + \lambda_{2} D_{2} . \hfill \\ \end{gathered} \right.$$

(6)

By solving the above equations, we obtain the equilibrium prices charged by the two firms:

$$\left\{ \begin{aligned} &p_{1}^{\prime } = \frac{{(1 + \delta )((1 + 2\delta )(1 - \lambda_{1} ) - \alpha (1 + 2\delta + \lambda_{1} )) - (1 - \alpha )(1 + 2\delta )\lambda_{2} }}{(1 - \alpha )(3 + (9 - \alpha + 6\delta )\delta )}, \hfill \\ &p_{2}^{\prime } = \frac{{\left( \begin{gathered} (1 + \delta )(1 - \lambda_{1} - \alpha (1 + \lambda_{1} + \delta (4 + 3\delta + 2\lambda_{1} - 3\lambda_{2} ) - \lambda_{2} ) - \hfill \\ \lambda_{2} + \delta (4 + 3\delta - \lambda_{1} - 3\lambda_{2} )) \hfill \\ \end{gathered} \right)}}{(1 - \alpha )(3 + (9 - \alpha + 6\delta )\delta )}. \hfill \\ \end{aligned} \right.$$

(7)

Similar to Lemma 1, the equilibrium price of firm ${\text{i}}$ decreases as the managerial delegation coefficients of firm ${\text{i}}$ and firm ${\text{j}}$ increase, i.e., $\partial {{\text{p}}}_{{\text{i}}}^{\mathrm{^{\prime}}}/\partial {\uplambda }_{{\text{i}}}<0$, $\partial {{\text{p}}}_{{\text{i}}}^{\mathrm{^{\prime}}}/\partial {\uplambda }_{{\text{j}}}<0$. And the cross-ownership increases the impact of managerial delegation on the equilibrium price, i.e., $\frac{\partial \left|\partial {{\text{p}}}_{{\text{i}}}^{\mathrm{^{\prime}}}/\partial {\uplambda }_{{\text{i}}}\right|}{\partial \mathrm{\alpha }}>0$,$\frac{\partial \left|\partial {{\text{p}}}_{{\text{i}}}^{\mathrm{^{\prime}}}/\partial {\uplambda }_{{\text{j}}}\right|}{\partial \mathrm{\alpha }}>0$.

In the first stage, the owners decide the intensity of managerial delegation according to the profit maximization of the controlling shareholders:

$$\left\{ \begin{gathered} \mathop {M{\text{ax}}}\limits_{{\lambda_{1} }} \Pi_{1} = (1 - \alpha )\pi_{1} , \hfill \\ \mathop {M{\text{ax}}}\limits_{{\lambda_{2} }} \Pi_{2} = \pi_{2} + \alpha \pi_{1} , \hfill \\ \end{gathered} \right.$$

(8)

Solving the equation system of the first-order conditions yields:

$$\left\{ \begin{aligned} &\lambda_{1}^{\prime } = \frac{(\alpha - 1)(1 + \alpha + \delta + 2\alpha \delta )(3 + \delta (7 + 3\delta ))}{{(1 + \delta )(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))}}, \hfill \\ &\lambda_{2}^{\prime } = \frac{{(1 + 2\delta )(2\alpha \delta - 3(1 + \delta )(1 + 2\delta ) + \alpha^{2} (3 + 2\delta ))}}{{21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta ))}}. \hfill \\ \end{aligned} \right.$$

(9)

Same as Proposition 1, the optimal managerial delegation coefficient of the firm producing low-quality products is higher than that of the firm producing high-quality products, that is, ${\uplambda }_{1}^{\mathrm{^{\prime}}}\ge {\uplambda }_{2}^{\mathrm{^{\prime}}}$. The level of cross-ownership tends to reduce the disparity in managerial delegation coefficients between the two firms, i.e., $\partial ({\lambda }_{1}{\mathrm{^{\prime}}}-{\lambda }_{2}{\mathrm{^{\prime}}})/\partial \alpha <0$.

Similar to Proposition 2, when firm 2 holds the shares of firm 1, there is a U-shaped relationship between managerial delegation coefficients and cross-ownership.

When $\alpha > \alpha^{\prime } (\delta ) = \frac{29}{8} + 3\delta - \left( \begin{gathered} \frac{3(2 + \delta (10 + 9\delta ))}{{8(6 + \delta (30 + \delta (47 + 24\delta )))}} + \hfill \\ \sqrt 3 \sqrt {\frac{(1 + \delta )(1 + 2\delta )(3 + 4\delta )(5 + 6\delta )(9 + \delta (55 + 3\delta (40 + \delta (37 + 12\delta ))))}{{(6 + \delta (30 + \delta (47 + 24\delta )))^{2} }}} \hfill \\ \end{gathered} \right)$, $\partial \lambda_{1}^{\prime } /\partial \alpha > 0$; and when $\alpha \le \alpha^{\prime } (\delta )$, $\partial \lambda_{1}^{\prime } /\partial \alpha \le 0$.

When $\alpha > \alpha^{\prime \prime } (\delta ) = \frac{1}{2}(7 + 6\delta - \sqrt {45 + 84\delta + 36\delta^{2} } )$, $\partial \lambda_{2}^{\prime } /\partial \alpha > 0$, and when $\alpha \le \alpha^{\prime \prime } (\delta )$, $\partial \lambda_{2}^{\prime } /\partial \alpha \le 0$.

The controlling shareholders’ profit of firm $i$, the consumer surplus, and social welfare are as follows:

$$\Pi_{1}^{\prime } = \frac{{(1 - \alpha )(4 + \alpha + 6\delta )(2(1 + \delta ) - \alpha )(3 + \delta (7 + 3\delta ))^{2} }}{{2(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }},$$

$$\Pi_{2}^{\prime } = \frac{{\left( \begin{gathered} 9(1 + \delta )^{2} (1 + 2\delta )^{3} (2 + 3\delta )(4 + 3\delta ) - 3\alpha^{2} (1 + 2\delta )(9 + \delta (36 + 37\delta - 9\delta^{3} )) - \alpha^{4} \delta^{2} - \hfill \\ \alpha^{3} (9 + \delta (4 + 3\delta )(9 + \delta (6 + 5\delta ))) + 2\alpha (1 + \delta )(27 + \delta (117 + \delta (91 - 6\delta (38 + 3\delta (22 + 9\delta ))))) \hfill \\ \end{gathered} \right)}}{{2(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }},$$

$$\begin{aligned} CS^{\prime } & = \mathop \smallint \nolimits_{{p_{1} }}^{{\frac{{p_{2} - p_{1} }}{\delta }}} \left( {x - p_{1} } \right){\text{d}}x + \mathop \smallint \nolimits_{{\frac{{p_{2} - p_{1} }}{\delta }}}^{1} \left( {\left( {1 + \delta } \right)x - p_{2} } \right){\text{d}}x = \\ & \quad \frac{{\left( \begin{gathered} \alpha^{4} \delta^{2} (1 + \delta ) - 2\alpha^{3} \delta (6 + \delta (21 + \delta (23 + 9\delta ))) - \hfill \\ 6\alpha (1 + \delta )(1 + 2\delta )(24 + \delta (100 + \delta (149 + 9\delta (11 + 3\delta )))) + \hfill \\ \alpha^{2} (36 + \delta (249 + \delta (640 + \delta (779 + 117\delta (4 + \delta ))))) + \hfill \\ (1 + \delta )^{2} (144 + \delta (876 + \delta (2053 + 3\delta (779 + 12\delta (37 + 9\delta ))))) \hfill \\ \end{gathered} \right)}}{{2(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }}, \\ \end{aligned}$$

$$\begin{aligned} SW^{\prime } & = CS^{\prime } + \Pi_{1}^{\prime } + \Pi_{2}^{\prime } = \\ & \quad \frac{{\left( \begin{gathered} \alpha^{4} \delta^{3} - 2\alpha^{3} \delta (3 + 4\delta )(1 + 3\delta (1 + \delta )) - \hfill \\ 2\alpha (1 + 2\delta )(90 + \delta (510 + \delta (1156 + 3\delta (443 + 9\delta (29 + 7\delta ))))) + \hfill \\ \alpha^{2} (18 + \delta (165 + \delta (548 + 3\delta (289 + \delta (224 + 69\delta ))))) + \hfill \\ (1 + \delta )(288 + \delta (2130 + \delta (6480 + \delta (10453 + 3\delta (3173 + 12\delta (130 + 27\delta )))))) \hfill \\ \end{gathered} \right)}}{{2(21 + \alpha^{2} \delta - 6\alpha (1 + \delta )(1 + 2\delta ) + 4\delta (20 + 3\delta (8 + 3\delta )))^{2} }}. \\ \end{aligned}$$

When the firm’s owner manages the firm by himself, i.e., ${\lambda }_{i}=0$, the profit of firm i’s controlling shareholders, the consumer surplus, and social welfare are as follows:

$$\Pi_{1}^{\prime \prime } = \frac{{(1 - \alpha )(1 + \delta )^{3} (1 + 3\delta )}}{{2(3 + \delta (9 - \alpha + 6\delta ))^{2} }},$$

$$\Pi_{2}^{\prime \prime } = \frac{(\alpha + (1 + \delta )(1 + 3\delta ))(1 + \delta (6 + (11 - \alpha + 6\delta )\delta ))}{{2(3 + \delta (9 - \alpha + 6\delta ))^{2} }},$$

$$\begin{aligned} CS^{\prime \prime } & = \mathop \smallint \nolimits_{{p_{1} }}^{{\frac{{p_{2} - p_{1} }}{\delta }}} \left( {x - p_{1} } \right){\text{d}}x + \mathop \smallint \nolimits_{{\frac{{p_{2} - p_{1} }}{\delta }}}^{1} \left( {\left( {1 + \delta } \right)x - p_{2} } \right){\text{d}}x = \\ & \quad \frac{{(1 + \delta )(4 + \delta (21 + \alpha^{2} \delta - 2\alpha (1 + \delta )(2 + 3\delta ) + \delta (39 + \delta (31 + 9\delta ))))}}{{2(3 + \delta (9 - \alpha + 6\delta ))^{2} }}, \\ \end{aligned}$$

$$\begin{aligned} SW^{\prime \prime } & = CS^{\prime \prime } + \Pi_{1}^{\prime \prime } + \Pi_{2}^{\prime \prime } = \\ & \quad \frac{{6 + \delta (41 + \alpha^{2} \delta^{2} - 4\alpha (1 + \delta )(1 + 3\delta (1 + \delta )) + \delta (110 + \delta (148 + \delta (100 + 27\delta ))))}}{{2(3 + \delta (9 - \alpha + 6\delta ))^{2} }}. \\ \end{aligned}$$

Same as Proposition 3, managerial delegation improves controlling shareholders’ profits for firm 1 and firm 2, i.e., ${\Pi }_{{\text{i}}}^{\mathrm{^{\prime}}}>{\Pi }_{{\text{i}}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}$ for ${\text{i}}=\mathrm{1,2}$. As the proportion of cross-ownership rises, the joint profit difference between the scenarios with and without managerial delegation increases, i.e., $\partial [({\Pi }_{1}^{\mathrm{^{\prime}}}+{\Pi }_{2}^{\mathrm{^{\prime}}})-({\Pi }_{1}^{{\prime}{\prime}}+{\Pi }_{2}^{\mathrm{^{\prime}}\mathrm{^{\prime}}})]/\partial \mathrm{\alpha }>0$.

Same as Proposition 4, managerial delegation reduces consumer surplus and social welfare with vertical product differentiation, i.e., ${{\text{CS}}}^{\mathrm{^{\prime}}}<{{\text{CS}}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}},\mathrm{ S}{{\text{W}}}^{\mathrm{^{\prime}}}<{{\text{SW}}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}$. When the cross-ownership proportion increases, the disparity in consumer surplus between the scenarios with and without managerial delegation widens. This effect is also observed in social welfare, i.e., $\partial ({{\text{CS}}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}-{{\text{CS}}}^{\mathrm{^{\prime}}})/\partial \mathrm{\alpha }>0,\partial ({{\text{SW}}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}-{{\text{SW}}}^{\mathrm{^{\prime}}})/\partial \mathrm{\alpha }>0$.