3.4. Indices Based on Socioeconomic Levels

We now introduce an alternative way to define the individual weights. Instead of letting the weights depend exclusively on the ranks of individuals, we define them in terms of the levels of the socioeconomic variable. In other words, we work with a weighting function, the arguments of which are levels, not ranks. What we are looking for is a set of individual weights which satisfy the conditions implied by the five properties mentioned earlier.

Looking at (12), it is obvious that the weights w_i must be proportional to the deviations of the socioeconomic levels y_i from the mean socioeconomic level μ_y, i.e., proportional to d_i = y_i − μ_y. What remains to be decided is the shape of the proportionality function β(μ_y). The property of Unit consistency requires that this function is homogeneous of degree −1 in μ_y. In fact, if all y_i are multiplied by the factor λ > 0, all deviations are also multiplied by the factor λ. That implies that the weights remain unchanged only if β(μ_y) changes by the factor λ⁻¹. The simplest function of this type is:

This functional form leads to the following ‘socioeconomic level’ weighting scheme:

Observe that it is possible to rewrite (15) as the product of the two terms: the ratio of the deviation d_i to the absolute mean deviation μd=1n∑i=1ndi, and the ratio of the absolute mean deviation μ_|d| to the mean μ_y. This decomposes the weight into two components: one part which measures the relative position of individual i by the normalised deviation (d_i/μ_|d|) of this person, and another which represents the degree of inequality in distribution y as measured by the relative mean deviation μ_|d|/μ_y. An alternative expression for the weighting function of the socioeconomic level-dependent index SL is therefore:

Note that the negative weights sum to −(n/2)(μ_|d|/μ_y), and the positive weights to (n/2)(μ_|d|/μ_y).