Income inequality measures and the middle class

We study how the presence of the middle class in the sense of Gevorgyan-Malykhin affects the value of income inequality measures including the Gini coefficient J and the Hoover index H. It is proved that in the presence of the middle class (1) $J \leqslant \frac{1}{2}\frac{{L'\left( 0 \right)}}{2}$ (where L is the Lorenz function), (2) $H \leqslant \frac{1}{2}$, (3) the longest vertical distance between the diagonal and the Lorenz curve (which is equal to H) is attained at ${z_0} < \frac{3}{4}$ A tight upper bound for P90/P10 ratio is found assuming L′(0)>0. Tight upper and lower bounds for the differential deviation in terms of the Gini coefficient are found as well.

. The Lorenz curve ( ). The doubled area of the shaded region is the Gini coefficient.
How much wealth may belong to the middle part (i.e. the second and third quartiles) of the population?
In case of the perfect equality ( ) = so that ( . In the example below the richest 25% own more than 50% of wealth so ( Gevorgyan and Malykhin [3] gave the following mathematical definition: there is the middle class if ( , hence there is no middle class in a such a society. Similarly, there is no middle class in a society with the Lorenz curve from Figure 2, but there is a middle class in case of the perfect equality. The authors proved in [4] that in the class of two-segment polygonal Lorenz curves (with the vertices (0;0), (a;b), and (1;1)) there is the middle class if and only if a≤ 1 2 .
It is clear that the presence of the middle class does influence the distribution of wealth in a society. In order to measure such an influence numerically we shall consider several inequality indices (which can be interchangeably applied both to income and savings). The most popular one, the Gini coefficient, was defined in [5], also see [6]. Geometrically it can be described as the fraction of the area between the Lorenz curve and the diagonal AD (which is shaded in Figure 1) relative to the area of the triangle under that diagonal. The closer is the Lorenz curve to the diagonal, the smaller is the Gini index. Given a Lorenz curve L, we denote the correspondent Gini coefficient by ( ) or simply by .
The Hoover index [7] is the largest deviation of ( ) from the identity, that is, the maximal value of − ( ). The Hoover index is also known as the Pietra index and the Schutz index; it is also called the Robin Hood index because it equals to the amount of wealth needed to be redistributed from the rich to the poor in order to attain the perfect equality. We also consider the P90/P10 ratio which is the ratio of the upper bound value of income of the 90% of people with highest income (i.e. of the ninth decile) to that of the 10%. Yet another inequality measure -differential deviation -was introduced in [8 The main research question considered in this paper is: How can the presence of a middle class affect the value of the Gini coefficient and other income inequality measures? Throughout the paper we assume ( ) to be continuous.

The Gini coefficient
Gevorgyan and Malykhin proved that if there is the middle class, then ≤ is attained at ( ) which is equal to zero on [0; 0.5] (in such a society the poorest one half of the population one does not have any income whatsoever). It is unlikely that such a distribution of wealth is possible in a modern society! So we will assume that every initial segment of the population does have some income. As ′( ) is proportional to the income of person/household corresponding to , it is feasible to consider the case ′ (0) > 0.
What are the possible values of the Gini coefficient given ′ (0) > 0 in the presence of the middle class? We answer this question in the following Theorem 1:  ) ≥ 4 , we have that  ) and such that the Lorenz curve is on or above 2 . Also, the vertex D is on or above 2 since the Lorenz curve is convex. Then the upper envelope of the lines = and 2 is a two-segment polygonal chain OAB (see Figure  3) which contains the points (0; 0) and ( ) such that no point of the Lorenz curve lies below it. Hence the area between the diagonal AD and the Lorenz curve is not greater than the area of the quadrilateral OABD. This means that ( ) is not greater than twice the area of OABD which is further denoted by SOABD.  It is proved in [9] that if a triangle is cut off an angle by a line, which goes through an interior point ′ of the angle, then this triangle has the minimum area if ′ = ′ . In order to consider this case, we need to check whether belongs to the segment . We have that ′ = ′ since ′ = ′ and ′ ′ ∥ . | ′ | = | ′ ′ | = . This implies that the area of the parallelogram ′ ′ ′ is one half of , so ≥ The proof is complete. The estimate for the Gini coefficient given by Theorem 1 is sharp. Indeed, for the twosegment polygonal Lorenz curve with vertices (0; 0), ( Note that without a requirement for the middle class the Gini coefficient may reach 1-′ (0) (this value is attained at the two-segment polygonal Lorenz curve with vertices (0; 0), (1; ), and (1; 1). Therefore, the presence of the middle class cuts the Gini coefficient in half.

The Hoover index
Let 0 be such that 0 − ( 0 ) is maximal among − ( ) for all 0≤ ≤ 1. (In other words, 0 corresponds to the person/household having the average income. If 0 is unique, then every person from [0; 0 ) has a below average income, while very person from ( 0 ;1] has an above average income.) In general, 0 can be any single number from the interval (0;1) (this observation can easily be seen from the fact that ( 0 ; ( 0 )) is the point of the Lorenz curve which is the most remote from the diagonal AD), or it can be an interval of numbers. The following Figure 5 shows three Lorenz curves corresponding to the same value of the Hoover index (and the same value of the Gini coefficient as well because for a two-segment polygonal Lorenz curves the Hoover index is equal to the Gini coefficient), but attaining their average income at different 0 . For simplicity, we assume that in the following theorem the Lorenz function is differentiable on (0;1). Theorem 2. If there is the middle class, then a. There is 0 < ). Then ′( ′ ) ≥ 1 since the denominator in the latter formula is equal to Clearly the right endpoint of this segment is the required 0 such that 0 ≤ ′ ≤ and due to convexity of ( ), the point ( ) and (1; 1) (see Figure 6).

The P90/P10 ratio
Since ′( ) is proportional to income/prosperity of a person, the P90/P10 ratio which is the ratio of the ninth decile to tenth decile is also equal to ′(0.9)/ ′(0.1) (in this section we assume ( ) to be differentiable on (0;1)). Clearly the P90/P10 ratio ≥ 1; it is well-known that the P90/P10 ratio can be any number ≥ 1. [11] noted that the presence of the middle class does not affect the P90/P10 ratio, i.e. it can still be any number ≥ 1.
whenever b≥ a. Now the proof splits in two parts. a. It then follows from (1) that (0.9) ≥ ′ (0) • 0.9 = 0.9 . Due to convexity of ( ), ′(0.9) does not exceed the slope of the secant line which goes through points (0.9; (0.9)) and (1; 1). This slope is equal to and it gets as close to 3.5−2.5 as desired as is getting closer to zero.

The differential deviation
This index was proposed in [8] as ∫ ( ′ ( ) − 1) 2 1 0 . Because defined in terms of the Lorenz function, this index does satisfy desirable properties of income inequality measures (see [12]). Also, it was shown [8] to be coherent with the natural partial order on the set of Lorenz curves. Throughout the rest of the paper the differential deviation of a society with the Gini coefficient will be denoted by ( ) or simply by . We will use the following fact from [4]: there is the middle class for a two-segment polygonal Lorenz curve if and only if its middle vertex is on or to the left of the line =  Proof of theorem 4. a. It was proved in [13] that ( ) ≥ 4 2 + 1 for a two-segment polygonal Lorenz curve whenever ≤ according to [13]. Fix . If 0 is sufficiently close to 1, then  [13]) for which there is the middle class according to the fact from [4]. The two-segment polygonal Lorenz curve with vertices (0; 0), ( ; 0), and (1; 1) at which the maximal value 1 1− is attained has the middle class due to the same fact.

Question
What are the possible values of ( ) in the presence of the middle class (if the Lorenz curve is not necessarily a two-segment polygonal chain)?

Further discussion
The study of the middle class in the sense of Gevorgyan-Malykhin can be continued in several directions. One direction is to relax the concept and allow for the difference (   − , then a sufficiently close ̃ will also satisfy it (for example, in the topology of uniform convergence).
It is possible to quantify to which extent the middle class is absent. We called in [14] the difference   for every Lorenz curve . In other words, the middle-class deficit never exceeds Also, it is possible to stipulate one or another degree of homogeneity of the middle class itself. For example, Gevorgyan and Malykhin [3] suggested to consider the P75/P25 ratio and they gave an upper estimate for the Gini coefficient provided the middle class is present and there is a perfect equality within the second and third quartiles of the population (the latter condition implies that the Lorenz curve is linear between = 0.25 and = 0.75).

Conclusions
We established that the presence of the middle class in the sense of Gevorgyan-Malykhin strongly influences various income inequality measures. The values of the Gini coefficient and the Hoover index are cut in half. The upper bound for the P90/P10 ratio (provided ′ (0) > 0) is reduced by almost three times. On the contrary, both the upper and lower bounds on the differential deviation in terms of the Gini coefficient are attained on the middle-class Lorenz curves whenever ≤ 1 2 .