In a recent issue of this journal, Ralph Bradburd (1995) argues that the efficiency gains of privatising a natural monopoly are likely to outweigh any loss of allocative efficiency. However, a trade-off based on a definition of the total surplus which includes profits but not internal rents is flawed. Correcting the analysis shows that even stronger conditions have to be met before privatising a natural monopoly is beneficial. The same is true if demand schedules are iso-elastic rather than linear. Recent evidence on demand elasticity and efficiency differences suggests that privatisation would be likely to reduce welfare even according to Bradburd's approach.