Optimal Cross Holding with Externalities and Strategic Interactions

We analyze a two period setting where firms first choose equity positions in each other's non-voting stock and second engage in operating activities that cause externalities. We find that firms facing positive externalities optimally choose long equity positions to increase their profits. In contrast, firms facing negative externalities encounter a Prisoner's Dilemma problem: Both firms optimally choose short positions in the first stage, but this commits the firms to be more aggressive in the operating stage resulting in an overall decrease in profits. For example, when firms decide simultaneously how much quantity to deliver to an imperfect product market, firms' profits are higher (lower) with cross holdings when products are complements (substitutes). Restrictions on cross holdings tend to reduce consumer surplus and economic welfare. The implication for regulators differs qualitatively from the prior literature which suggests restricting cross holdings. Further, we demonstrate that allowing cross holdings deters entry when products are substitutes and encourages entry when products are complements. Finally, cross holdings in voting stock have strategic effects as long as some investors are less than fully diversified