This appendix covers some of the principal concepts and results in the formal literature on inequality measurement and related topics in the analysis of income distribution. For surveys of this field, see, for example, Champernowne (1974), Cowell (1995, 1999), Foster (1985), Jenkins (1991), Lambert (1993), Sen (1973) and Sen and Foster (1997).

The axiomatic approach

Assuming that the concepts ‘income’ and ‘income receiver’ have been defined, we index the members of the population by i = 1, 2, …, n, and assume that person i's income is a non-negative scalar xi. The symbol x denotes a vector of such incomes (x1, x2, …, xn), and 1 denotes a vector of ones. For convenience we write n(x) and μ(x) respectively for the number of components and the arithmetic mean of the components of vector x. Also we let x[m]: =(x, x, …, x), a mn-vector that represents a concatenation of m identical n-vectors x.

Write N for the set of integers {1, 2, …, n} and X for the set of all possible income vectors, which we will take to be the set of all finite-dimensioned non-negative vectors excluding the zero vector. Also write X′ for the subset of X that excludes all vectors of the form a1, a>0. By an inequality comparison we mean a binary relation on the members of X. Since our questionnaire is phrased in terms of inequality, the statement ‘x≽x′’ is to be read as ‘x represents an income distribution that is at least as unequal as distribution x′’. It is possible, of course, that this does not represent a complete ordering on X. We define the strict inequality comparison ≻ and inequality equivalence ∼ in the usual way.