Potential welfare and the sum of individual compensating or equivalent variations (Q1819683)

We investigate under which conditions the sign of the sum of both the individual compensating and equivalent variations correctly indicates changes in potential welfare. Our results reproduce exactly those of \textit{J. S. Chipman} and \textit{J. C. Moore} [Econometrica 48, 401-422 (1980)] for the National Income Test, which did not rely on individual measures of welfare directly related to consumers' preferences: (i) a necessary and sufficient condition for the new test to be valid is that individual preferences are identical and homothetic; and (ii) if the distribution of income is constant and preferences are homothetic, then the new test is valid only if preferences are also identical.