Complementary monopolies and multi-product firms

According to the classical result on complementary monopolies, a single-product firm unambiguously prefers purchasing complementary inputs from an integrated monopolistic supplier rather than from different non-integrated monopolistic suppliers. In this note, we account for the fact that firms often manufacture multiple products and show that the classical result on complementary monopolies can be reversed in such a case. Purchasing complementary inputs from non-integrated suppliers can be optimal for multi-product firms.