In determining the optimal redistribution of a given population's income, we ask which factor is more important: the social planner's aversion to inequality, embedded in an isoelastic social welfare function indexed by a parameter alpha, or the individuals' concern at having a low relative income, indexed by a parameter beta in a utility function that is a convex combination of (absolute) income and low relative income. Assuming that the redistribution comes at a cost (because only a fraction of a taxed income can be transferred), we find that there exists a critical level of beta below which different isoelastic social planners choose different optimal allocations of incomes. However, if beta is above that critical level, all isoelastic social planners choose the same allocation of incomes because they then find that an equal distribution of incomes maximizes social welfare regardless of the magnitude of alpha.