Positive Inequality Indices

Abstract

This chapter discusses positive inequality measures, as functions that try to provide descriptive estimates of the variability of income distributions that satisfy most of the properties presented in Chap. 2 and are compatible with our intuitions about what inequality means. There are no explicit references to social welfare even though different measures incorporate implicitly different value judgements. Indeed, one can think of those measures as particular ways of aggregating the differences between individual incomes and a reference value, most of the times the mean income. We review here four different ways of comparing income distributions from a descriptive point of view: the Lorenz curve, the Gini index, the Theil’s family of indices and the Palma ratio. We shall check the properties that those measures satisfy in order to illustrate the different ways of valuing inequality. There are many other ways of measuring inequality, but these are the most common ones.

Appendix: Decomposability of the Generalised Indices of Theil

Let us describe how the generalised index of Theil can be decomposed. The formula of this uni-parametric family of indices is give by:

$$ {I}_{\theta}\left(n,y\right)=\frac{1}{n}\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{i=1}^n\left[{\left(\frac{y_i}{\mu}\right)}^{\theta }-1\right]} $$

with θ ≠ 0,1 (the 0 and 1 cases have already been discussed). We can proceed as follows:

$$ \begin{array}{l}{I}_{\theta}\left(n,y\right)=\frac{1}{n}\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G{\displaystyle \sum_{i=1}^{n_g}\left[{\left(\frac{y_i^g}{\mu}\right)}^{\theta }-1\right]}}\hfill \\ {}=\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G\frac{1}{n}}\left[{\displaystyle \sum_{i=1}^{n_g}{\left(\frac{y_i^g}{\mu}\right)}^{\theta }-{n}_g}\right]\hfill \\ {}=\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G\frac{n_g}{n}}{\left(\frac{\mu_g}{\mu}\right)}^{\theta}\left[\frac{1}{n_g}{\displaystyle \sum_{i=1}^{n_g}\left[{\left(\frac{y_i^g}{\mu_g}\right)}^{\theta }-1\right]}+1-{\left(\frac{\mu }{\mu_g}\right)}^{\theta}\right]\hfill \\ {}={\displaystyle \sum_{g=1}^G\frac{n_g}{n}}{\left(\frac{\mu_g}{\mu}\right)}^{\theta}\underset{I_{\theta}^g}{\underbrace{\frac{1}{n_g}\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{i=1}^{n_g}\left[{\left(\frac{y_i^g}{\mu_g}\right)}^{\theta }-1\right]}}}+\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G\frac{n_g}{n}}\left[{\left(\frac{\mu_g}{\mu}\right)}^{\theta }-1\right]\hfill \end{array} $$

The first term of this expression is a weighted sum of the indices I _θ within groups. The weights are given by:

$$ {\omega}_g^G\left(\boldsymbol{\mu}, \mathtt{n}\right)=\frac{n_g}{n}{\left(\frac{\mu_g}{\mu}\right)}^{\theta } $$

which depend on the parameter θ of inequality aversion.

Note that those weights can be expressed in terms of population and income shares, that is,

$$ {\omega}_g^G\left(\boldsymbol{\mu}, \mathtt{n}\right)=\frac{n_g}{n}{\left(\frac{\mu_g}{\mu}\right)}^{\theta }={\left(\frac{n_g}{n}\right)}^{1-\theta }{\left(\frac{n_g{\mu}_g}{n\mu}\right)}^{\theta } $$

This expression is interesting because it shows that only when θ = 0 and θ = 1 (the first and second indices of Theil), those weights add up to 1.

The second term of that expression corresponds to I _θ applied to the mean values of each group, that is, it describes the between groups component.

We can thus write:

$$ {I}_{\theta}\left(n,\mathbf{y}\right)={\displaystyle \sum_{g=1}^G\frac{n_g}{n}{\left(\frac{\mu_g}{\mu}\right)}^{\theta }{I}_{\theta}^g}+\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G\frac{n_g}{n}\left[{\left(\frac{\mu_g}{\mu}\right)}^{\theta }-1\right]} $$

(3.22a)

or,

$$ {I}_{\theta }={I}_{\theta, W}+{I}_{\theta, B} $$

(3.22b)

where,

• (i)
  $$ {I}_{\theta, W}={\displaystyle \sum_{g=1}^G\frac{n_g}{n}{\left(\frac{\mu_g}{\mu}\right)}^{\theta }{I}_{\theta}^g} $$
• (ii)
  $$ {I}_{\theta, B}=\frac{1}{\theta \left(\theta -1\right)}{\displaystyle \sum_{g=1}^G\frac{n_g}{n}\left[{\left(\frac{\mu_g}{\mu}\right)}^{\theta }-1\right]} $$