Does less inequality among households mean less inequality among individuals?

Consider an income distribution among households of the same size and suppose that the individuals, equally needy from the point of view of an ethical observer, are treated unfairly within each household. We look for conditions under which welfare and inequality quasi-orders are preserved from the household level at the individual one. A necessary and sufficient condition for the Generalized Lorenz test is that the income of disadvantaged individuals is a concave function of the household income. The richer are the households, the higher is intra-household inequality. This property also proves to be crucial to the preservation of the Relative and Absolute Lorenz criteria.