It was conjectured by Pigou that an increase in real national income, as reckoned in the prices of either the initial or the terminal period, would always correctly indicate an improvement in national welfare provided the increase referred to the aggregate income of a given group of persons with fixed preferences and a fixed proportional distribution of income among them. We show that if the individual preferences are assumed to be homothetic, and if by a welfare improvement one means respectively a potential improvement (in which losers can be compensated by gainers) or an actual improvement (in which all are gainers), then on either of these respective criteria Pigou's conjecture holds true under these conditions if and only if individual preferences are identical.