Inequality aversion and risk aversion

This note shows that for two inequality averse social welfare functions, if one is more inequality averse than the other, the household preference induced by optimally allocating aggregate bundles according to this social welfare function is more risk averse than the other. We present examples showing that this comparative static can be reversed if inequality aversion is dropped. We show that the utilitarian rule always induces the least risk averse household preference among all social welfare functions (this corresponds to the sum of certainty equivalents).