We derive an original measure of dead-weight loss (DWL) in an m-sector complementary monopoly and show that with non-collusive pricing DWL may be seriously understated if demand complementarities are ignored, even when m is small. Since DWL generally increases with m and with less collusive pricing, separating monopoly into complementary monopoly (risking reduced price collusion) may be a bad static move. To illustrate, separating Microsoft into two non-collusive complementary monopolies may increase DWL from $4 billion to $7 billion (for 2002-3). However, we show that such a policy may be welfare improving with even relatively modest post-separation entry and Cournot quantity competition. Copyright © 2006 The Author; Journal compilation © Blackwell Publishing Ltd and The University of Manchester 2006.