Income Inequality Measures

Income distributions are commonly unimodal and skew with a heavy right tail. Different skew models, such as the lognormal and the Pareto, have been proposed as suitable descriptions of income distribution and applied in specific empirical situations. More wide-ranging tools have been introduced as measures for general comparisons. In this study, we review the income analysis methods and apply them to specific Lorenz models.


Introduction
Income distributions are commonly unimodal and skew with a heavy right tail.
Therefore, different skew models, such as the lognormal and the Pareto, have been proposed as suitable descriptions of income distribution, but they are usually applied in specific empirical situations [1]. For general studies, more wide-ranging tools have been considered. The target for them is to introduce measures that are useable for comparisons of different distributions. Primary income data yield the most exact estimates of income inequality coefficients such as Gini and Pietra. Earlier studies have shown that no method is always optimal.
Therefore, different attempts are still worth studies. In this study, we review income analysis methods based on Lorenz curves. The theory is applied to specific models.

Methods
The Lorenz curve. The most commonly used theory is based on the Lorenz curve. Lorenz [4] developed it in order to analyze the distribution of income and wealth within populations. He described the Lorenz curve, ( ) L p , for wealth within populations in the following way: "Plot along one axis accumulated per cents of the population from poorest to richest, and along the other, wealth held by these per cents of the population".
where p x is the p quantile, that is ( ) is commonly interpreted as the Lorenz curve for complete equality between income receivers, but according to [6] it is not perfectly associated with the Lorenz curve. As everyone has the same income level, strictly speaking, no one can be said to be at the lowest or highest level of the population. The associated Lorenz curve then exists only at the origin and the termination point by the definition of the curve. To overcome this problem, they adopted the convention of allocating any fraction 0 1 x < < of the population to be the lowest/highest x percent. This convention then allowed the 45 degree line through the origin to be associated with complete equality, as usually loosely taken to be so. This permitted Wang & Smyth [6] to use ( ) as a useful component in the creation of Lorenz curves.
On the other hand, increasing inequality lowers the Lorenz curve, and theoretically, it can converge towards the lower right corner of the square. A sketch of a Lorenz curve is given in Figure 1.
Variable transformations. Consider a transformed variable ( ) where ( ) g ⋅ is positive and monotone increasing. Then, the distribution of ( )  For the transformed variable Y, the p quantile p y is Hence, If the transformation is linear ( ) To every distribution ( ) We denote this relation . An example of Lorenz dominance is given in Figure 2.  Properties of Lorenz curves. The Lorenz curve has the following general properties: If the Lorenz curve is differentiable, the derivative has the following properties.
and the density function When we differentiate the equation The differentiation of ( ) ( ) and consequently, If the Lorenz curve is differentiable twice, then the second derivative is Hence, and ( ) As a consequence of (12), Consider a one-parametric class of cumulative distribution functions ( ) , F x θ , defined on the positive x-axis. If we assume that ( ) ( ) i.e. it depends only on the product x θ , then the following theorem holds [1]: and ( ) where p t and c are independent of θ . Proof. Let θ be an arbitrary, positive parameter. Then the quantile proved. The formula (14) and the statement that ( ) ( ) ( ) Furthermore, we can prove the following . Now we shall prove the sufficiency, that is, θ is a distribution whose mean is 1 µ θ = and whose Lorenz curve After We obtain The given function ( ) L p has a monotone-increasing inverse function whose mean is µ .
Using the same transformation, we obtain that the Lorenz curve ( ) and the theorem is proved.
These results have been collected in the following theorem [7] [8]:  Fellman [7] presented this result and later Fellman [8] presented the following theorem: The maximum of D implies 1 For Gini index. The most frequently used index is the Gini coefficient, G [12].
Using the Lorenz curves, this coefficient is the ratio of the area between the diagonal and the Lorenz curve and the whole area under the diagonal (cf. Figure   1). The formula is This definition yields Gini coefficients satisfying the inequalities 0 1 G < < .
The higher the G value, the lower the Lorenz curve and the stronger the inequa- There are other inequality measures defined by the Gini coefficient. Yitzhaki [14] proposed a generalized Gini coefficient Using the mean income ( µ ) and the Gini coefficient (G), Sen [15] proposed a welfare index ( ) Pietra index. The Pietra index P is defined as the maximum  Figure 4. The Pietra index can be interpreted as the income of the rich that should be redistributed to the poor in order to obtain total income equality. In other words, the value of the index approximates the share of total income that must be transferred from households above the mean to those below the mean to achieve equality in the distribution of incomes. Higher values of P indicate more inequality, and more redistribution is needed to achieve income equality. Therefore, the index is sometimes named the Robin Hood index. The Pietra index is also known as the Hoover index and it is still better known as the Schutz index [16] [17] [18].
An alternative definition of the Pietra index has also been given. It can be defined as twice the area of the largest triangle inscribed in the area between the Lorenz curve and the diagonal line [9]. In Figure 5, one observes that the triangle obtains its maximum when the corner lies on the Lorenz curve where the tangent is parallel to the diagonal. The height of the triangle is 2 P h = , and the base is the diagonal 2 b = . The double of the area is 1 2area 2 2 2 2 In comparison, the Gini index, G, is twice the area between the Lorenz curve and the diagonal, and the Pietra index is twice the area of the triangle inscribed in this area. Hence, the inequality G P ≥ holds generally [1].

Applications
In this section, we collect some examples in order to elucidate the theory. The models Pareto [19], the simplified Rao-Tam [20] and the Chotikapanich [21] contain only one parameter. Therefore, they can easily be analyzed. Rohde [22] and Fellman [2] [13] paid these models special attention and examined them in more detail. However, they are so simple that it is impossible to distinguish between the estimated length of the range for the income distribution function and   Figure 6, we present the Pareto distribution as a function of the parameter α .
Finally, the Pietra index is According to the general theory, the inequality G P ≥ holds for all parameter values, and consequently, 0 P → when α → ∞ . Let , the index inequalities D P G ≤ < hold, and for 1 α → 0 G → , and consequently, 0 P → . For increasing α values, the supremum of D p p α = − is one. This must also hold for the supremum of . Consequently, the interval 0 1 P < < cannot be shortened. In Figure 9, we present G and P for different α .
When k → ∞ , one obtains Consequently, for G, the inequalities 0 1 G < < hold and the range cannot be shortened.
The Pietra index is Hence, lim 1 k P →∞ = and the inequalities 0 1 P < < hold and cannot be shortened.
The G and P as functions of the parameter k indices are presented in Figure   11.   ). Note that the Lorenz curves for the simplified Rao-Tam and Chotikapanich models are rather similar, but the Lorenz curve for the Pareto model is markedly different.
Above, we made the general remark that different distributions can result in the same Gini index. In Figure 12, we present a simple example of this finding.
We compare a Chotikapanich model with the Gini index 0.500 (k = 3.593525) with a Pareto model (with

Discussion
In general, the step from the Lorenz curve to the income distribution starts from the formula where p x is the p-percentile and µ is the mean of the corresponding distribution ( ) Equation (19) indicates that Primary income data yield the most exact estimates of the income inequality coefficients, such as Gini and Pietra. Fellman [2] analyzed different methods for numerical estimation of Gini coefficients based on Lorenz curves. As an application of these methods, he considered Pareto distributions. Using Lorenz curves, various numerical integration attempts were applied to obtain accurate estimates.
The trapezium rule is simple, but yields a positive bias for the area under the Lorenz curve, and consequently, a negative bias for the Gini coefficient. Simpson's rule is better fitted to the Lorenz curve, but this rule demands an even number of subintervals of the same length. Lagrange polynomials of second degree can be considered as a generalization of Simpson's rule. Fellman [2] compared different methods and he also gave references concerning numerical integration. To include Simpson's rule in his study, he considered Lorenz curves with deciles. Compared with Simpson's rule, he used the trapezium rule, Lagrange polynomials and generalizations of Golden's method [23]. No method was uniformly optimal, but the trapezium rule was almost always inferior and Simpson's rule was superior. Golden's method is usually of medium quality.