Randomization, revelation, and redistribution in a Lerner world

If total social income is fixed and a social planner is uninformed of the utility representations of different individuals, then Lerner showed that the social optimum is to equally distribute income across individuals. We show that the planner by the use of randomization can in some circumstances induce individuals to reveal information about the curvature of their utility functions and then use the information to move away from equality on average. However, whether this is optimal depends in part on unobservable beliefs of the planner. These may be viewed as an aspect of the planner's ethical judgements or as something entirely arbitrary.