Partial horizontal and vertical ownership

Abstract

Others have shown that in vertically related Cournot oligopolies, partial ownership could have no real effects on total output or price choice, and in a separate way that increasing cross-ownership among rivals leads to more collusive outcome. In a complementary manner we study the interactions between vertical and horizontal partial ownership giving no control over target. This article shows that when the choice of optimal cross-ownership profile is simultaneous, a mixed equilibrium with upward vertical and horizontal participations can be achieved, vertical and horizontal ones being strategic substitutes. We finally exhibit the significant influence of vertical shareholdings on output price as on profits and on consumer surplus, which in our model is harmful at optimum from a consumer and a social point of view.

Notes

¹The number of upstream firms has to be greater (except for and 3) to the number of downstream firms for an equilibrium of PO to be relevant in the range of intervals where noncontrolling PO should normally lie. These intervals are detailed in the working paper referenced at the end of this note.

²We use in this article the term equilibrium for the results of maximization in the second stage of our game, and we call optimum the results of the maximization in the first stage of our three-stage game. Equilibrium quantities will be annotated with a + and optimum ones with an *.

³Refer to Reynolds and Snapp (1986), Theorem 2.

⁴‘Residual rights’ refers to the theory of ‘residual rights’ partly fuelled by Grossman and Hart (1986). Here it is a partial rebate of ex-post profits of the firm target.

⁵The company is both manager and principal shareholder. Thus we avoid conflicts of interest.

⁶In exchange of a transfer price paid to the holding outside of the industry.

⁷See Appendix.

⁸ , , and thus .

⁹For detailed calculations, refer to the working paper.

¹⁰Except in the case where the number at each level of the industry is the same, that is, m = n.

¹¹The sum of all vertical shares bought must equal the sum of all vertical shares sold, this means .

¹² , if , and in the opposite case.

¹³See Appendix for the study of second-order conditions of profit functions.

¹⁴Optimal values in this situation are derived in Table 2.

¹⁵As an example, when m = 3 and n = 2, ; under this threshold we obtain a social improvement of welfare. Other numerical applications are available on request.

¹⁶For details on the PVO regulation, refer to the working paper.