Losses from cross-holdings in a duopoly with convex cost and strategic input price determination

It is well-known that positive output externality on the outside firms is the reason for unprofitable passive cross-holding, which refers to a situation where a producer holds non-controlling shares in rival firms. Considering a final goods market with Cournot duopoly, where cross-holdings do not create positive output externality on the outside firms, we show that cross-holdings can be unprofitable under strategic input price determination by an input supplier if the final goods are produced with decreasing returns to scale technologies. Our results hold under symmetric and asymmetric cross-holdings. We show that cross-holdings can be unprofitable also under Bertrand duopoly in the final goods market. Thus, we provide a new reason for unprofitable passive cross-holdings. We also show the implications of a higher product differentiation on the profits and welfare.