A note on Sen's representation of the Gini coefficient: Revision and repercussions

Sen (1973 and 1997) presents the Gini coefficient of income inequality in a population as follows. "In any pair-wise comparison the man with the lower income can be thought to be suffering from some depression on finding his income to be lower. Let this depression be proportional to the difference in income. The sum total of all such depressions in all possible pair-wise comparisons takes us to the Gini coefficient." (This citation is from Sen 1973, p. 8.) Sen's verbal account is accompanied by a formula (Sen 1997, p. 31, eq. 2.8.1), which is replicated in the text of this note as equation (1). The formula yields a coefficient bounded from above by a number smaller than 1. This creates a difficulty, because the "mission" of a measure of inequality defined on the unit interval is to accord 0 to perfect equality (maximal equality) and 1 to perfect inequality (maximal inequality). In this note we show that when the Gini coefficient is elicited from a neat measure of the aggregate income-related depression of the population that consists of the people who experience income-related depression, then the obtained Gini coefficient is "well behaved" in the sense that it is bounded from above by 1. We conjecture a reason for a drawback of Sen's definition, and we present repercussions of the usage of the "well-behaved" Gini coefficient.