Protected Income and Inequality Aversion

We discover a fundamental and previously unrecognized structure within the class of additively separable social welfare functions that makes it straightforward to fully characterize and elicit the social preferences of an inequality-averse evaluator. From this structure emerges a revealing question: if a large increment can be given to one individual in a society, what is the maximal sacrifice that another individual can be asked to bear for its sake? We show that the answer uncovers the evaluator's degree of inequality aversion. In particular, all translation-invariant evaluators would sacrifice the full income of the sacrificed individual if their income were low enough and a constant amount of their income otherwise. Scale-invariant evaluators would sacrifice the full income of the sacrificed individual at all income levels if their inequality aversion was no greater than one, and a constant fraction of their income otherwise. Motivated by these findings, we propose a class of social preferences that, starting from a minimum-income level of protection, ensure a higher fraction of the sacrificed individual's income is protected the lower their income.