Economics Letters 118 (2013) 370–374
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A bottom poor sensitive Gini coefficient and maximum entropy estimation of income distributions Hang Keun Ryu ∗ Department of Economics, Chung Ang University, 221 Heuk Seok Dong, Dong Jak Ju, Seoul, 156-756, South Korea
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Article history: Received 17 July 2012 Received in revised form 2 November 2012 Accepted 16 November 2012 Available online 13 December 2012
abstract A bottom poor sensitive Gini coefficient (pgini) is defined by replacing income observations with their reciprocal values in the Gini coefficient. The underlying true income share function can be derived approximately using the maximum entropy method given the pgini coefficient. © 2012 Elsevier B.V. All rights reserved.
JEL classification: D31 D63 Keywords: Gini coefficient Bonferroni index Wolfson polarization index Maximum entropy method
1. Introduction The original Gini is known to be less sensitive to the small income changes of the poorest group. A decrease in the Gini may not necessarily be an improvement for the poorest group. Bonferroni (1930) and Ryu (2008) modified the definition of the Gini to give heavier weight to the poor group. Atkinson (1970) suggested an index of inequality to measure inequality under different value judgments where the zero value represents the indifference to inequality and infinity represents the Rawlsian criterion. Yitzhaki (1983) extended the Gini to reflect a preference for inequality where the aversion to inequality rises as ν goes from 0 to infinity. Kakwani (1980) and Donaldson and Weymark (1983) developed versions of the extended Gini that depend on social welfare functions. Wolfson (1994) introduced a polarization index to check the collapse of the middle class and their movements to either the poor class or rich class. Small income changes of the poorest group may (or may not) be well described by the above indices. This paper defines an inconvenience level with the reciprocal value of income. Higher income provides more convenience and a smaller income gives a severe inconvenience. The rich can hire
∗
help and use superior facilities to save time and effort; in addition, a society has a total amount of inconveniences that are unevenly distributed to individuals. The inequality of inconvenience levels is measured with the Gini coefficient and is found to be highly correlated with the income shares of the bottom 5% poorest group. The correlation coefficient is 0.994 for U.S. family income shares. This inequality of inconvenience level measurement is called the bottom poor sensitive Gini coefficient (pgini) in this paper. The inconvenience Lorenz curve is defined with the sum of inconvenience shares and the pgini can be derived with the ellipse area above the Lorenz curve. Other explanations based on the original Gini coefficient and the Lorenz curve can be replicated for the pgini case. Dagum (1997) showed that this could be decomposed into several parts, the contribution of inequality within groups and the contribution of inequality between groups. The poverty line can also be drawn in the inconvenience Lorenz curve. Section 2 defines the pgini. Section 3 derives the unknown income share function approximately. Section 4 compares the performance of the pgini with other measures. Section 5 provides the conclusion. 2. Definition of pgini
n n 1 1 pgini ≡ − . 2n2 µ i=1 j=1 yi yj 1
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(1)
H.K. Ryu / Economics Letters 118 (2013) 370–374
371
observations, BI =
n −1 Pi − Qi
1
n − 1 i=1
where Pi =
Pi
i n
and Qi =
i xj nγ j =1
(4)
where γ is the mean income and xj is the observed income of the jth individual. Wolfson (1994) introduced a scalar polarization index to show an insufficient explanation of the Gini for certain income changes. The lowered Gini coefficient is not necessarily a desirable consequence if the middle class is eliminated from society. The scalar polarization index is 4γ
|0.5 − L(0.5) − 0.5Gini| (5) m where m is the median income and γ is the mean income. The Wolfson polarization index is also used to compare and check its sensitivity with the income share changes of the poor group. P =
Fig. 1. Inconvenience measure.
3. Maximum entropy estimation of income shares from the pgini We show that the knowledge of pgini is equivalent to the knowledge of the first moment of inconvenience shares. The Lorenz curve for the inconvenience measure is defined as z
r (z ′ )dz ′
PL ≡ 0
where r (z ) is the inconvenience share function and the coordinate z is the population coordinate with z = 0 for the richest person with the least inconvenience and z = 1 for the poorest person with the largest inconvenience. Consider the partial integration of 1
zdPL = zPL(z )10 − 0
PL(z )dz = 1 − g 0
where g ≡ Since
Fig. 2. Inconvenience Lorenz curve.
1
1 0
PL(z )dz =
1−pgini . 2
dPL(z ) = r (z )dz .
A society has a total amount of inconveniences, i=1 (1/yi ) and n the mean inconvenience is µ = (1/n) i=1 (1/yi ). Each individual has inconvenience share ri and pgini can be rewritten as
n
The mean of the inconvenience share function is
µ1 =
1
zr (z )dz = 1 − g =
1 + pgini 2
0
ri =
1 1
(2)
nµ yi
pgini ≡
n n 1
2n i=1 j=1
ri − rj .
(3)
It can be shown that the pgini satisfies the necessary conditions required for an inequality measure stated in Fields and Fei (1978), the scale of irrelevance, symmetry, and rank-preserving equalization. The inconvenience Lorenz curve is defined with the sum of inconvenience shares. The richest person with least inconvenience is located at z = 0 and the poorest person with the most inconvenience is located at z = 1 (see Fig. 1). The pgini is equal to the ellipse area of Fig. 2. Other inequality measures assign different weights to the poor and rich individuals. The Bonferroni index is defined as 1
BI = 1 − 0
L(z ) z
dz
where L(z ) is the original Lorenz curve. Bonferroni (1930) and Ryu (2008) put heavier weights to the poor income groups. For discrete
.
(6)
Knowledge of the pgini is equivalent to the knowledge of the first moment of the true inconvenience share function. A similar result is also reported in Lerman and Yitzhaki (1984) for the original Gini coefficient. The inconvenience share distribution can be derived from the given first moment. Solving an entropy maximization problem as stated in Ryu (1993)
Maxr W ≡ −
r (z ) log r (z )dz
(7)
satisfying
zr (z )dz = µ1 ,
(8)
the Lagrangian method produces r (z ) = exp [a + bz] =
b eb − 1
exp[bz ]
(9)
where the normalization condition of the share function is used to remove a. Now the first moment condition (8) produces,
µ1 =
b eb
−1
1
z exp[bz ]dz = 0
1 + pgini 2
.
(10)
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H.K. Ryu / Economics Letters 118 (2013) 370–374
.560
.456
.556
.454 .452
.548
GINI
PGINI
.552
.544
.450 .448
.540 .536
.446
.532 .0034 .0036 .0038 .0040 Income shares of the poorest 5% group
.444 .0034 .0036 .0038 .0040 Income shares of the poorest 5% group Fig. 4. Gini and poorest 5% group.
Fig. 3. Pgini and poorest 5% group.
.598
Since the integration is a function of b, eb
b
eb − 1
=
1 + pgini 2
.
.596
(11)
Then b approaches zero if the pgini = 0 and b approaches infinity if the pgini = 1. Since the LHS of (11) is a monotonic increasing function of b, a given pgini coefficient uniquely determines b and the inconvenience share function r (z ). The reciprocal of r (z ) becomes an income function and an income share function is derived with normalization. Ryu (2012) showed similar calculations for the original Gini to derive an income share function.
.594 Bonferroni
1
µ1 ≡ − +
.590 .588 .586 .0034 .0036 .0038 .0040 Income shares of the poorest 5% group
4. Empirical results
Fig. 5. Bonferroni and poorest 5% group.
.496 .492
WOLFSON
The quintile shares of U.S. families for 2000–2009 are taken from the Tax Policy Center (2012). The pgini can be directly estimated using the quintile shares; however, the original kinked Lorenz curve is smoothed with a quadratic polynomial series to improve the pgini estimation accuracy. The Lorenz curve for z = [0, 0.4] is approximated with L1 (z ) = a + bz + cz 2 and the parameters are determined by the boundary conditions L(0), L(0, 2), and L(0, 4). For z = [0.2, 0.6], the Lorenz curve is approximated with L2 (z ) = d + ez + fz 2 and the parameters are determined by the boundary conditions L(0, 2), L(0, 4), and L(0, 6). Use L1 (z ) for z = [0, 0.2] and use [L1 (z ) + L2 (z )]/2 for z = [0.2, 0.4]. Ryu (2012) showed the kinked Lorenz curve produced Gini = 0.3657, the above smoothed Lorenz curve produced Gini = 0.3823, and the observed Gini = 0.3896 for 1983 U.S. family income shares. From the approximated Lorenz curves, income shares and inconvenience shares (normalized reciprocal shares) are derived for a hundred income groups. The scatterplot of the pgini coefficients and the poorest 5% group shares of U.S. families for 2000–2009 is plotted in Fig. 3. They are highly correlated with the correlation coefficient 0.994. The changes in pgini match exactly with the changes of poorest group income shares. In comparison, the Gini, Bonferroni, and Wolfson polarization index do not respond linearly with the poorest 5% group share movements. Fig. 4 shows the poorest 5% group share may decrease when the Gini indicates income inequality improvement. Figs. 5 and 6 show similar results with the Bonferroni index and Wolfson polarization index. Figs. 7–10 show the scatterplots of pgini (and other measures) and the 10% poorest group shares. The pgini is highly correlated with the 10% poorest group shares; however, other inequality measures show less linear movement with the poorest 10% group shares.
.592
.488 .484 .480 .476 .472 .0034 .0036 .0038 .0040 Income shares of the poorest 5% group Fig. 6. Wolfson and poorest 5% group.
The unknown income distribution can be estimated from the pgini using the maximum entropy method. Ryu (1993) derived the probability density function by maximizing the entropy subject to given conditions. The approximated results are shown in Fig. 11 for the 1983 U.S. family income data of the CPS. The original data has 66,227 observations; however, they are regrouped into a hundred income groups. The pgini accurately approximates the income shares for the very poor and very rich; however, its approximation is poor for the middle class. Ryu (2012) derived the share function from the Gini. The Gini based function showed a good approximation for the middle ranges, but showed poor performance for the very poor and rich ends.
H.K. Ryu / Economics Letters 118 (2013) 370–374
.560
.496
.556
.492
WOLFSON
PGINI
.552 .548 .544 .540
.488 .484 .480
.536
.476
.532 .0100 .0104 .0108 .0112 .0116 Incomes shares of the poorest 10% group
.472 .0100 .0104 .0108 .0112 .0116 Income shares of the poorest 10% group
Fig. 7. Pgini and poorest 10% group.
373
Fig. 10. Wolfson and poorest 10% group.
.456 .454
GINI
.452 .450 .448 .446 .444 .0100 .0104 .0108 .0112 .0116 Income shares of the 10% poorest group Fig. 8. Gini and poorest 10% group.
.598 .596
Bonferroni
.594 .592
Fig. 11. Comparison of observed shares with the approximated shares.
.590
the most exiting inequality measures are more or less insensitive to the relative income changes of the poorest group. Yitzhaki (1983) and others extended the Gini coefficient to reflect a preference for inequality and Bonferroni (1930) provided a heavier weight to the poor groups in his inequality index. In this paper, an inconvenience level is defined with the reciprocal value of observed income. The inequality of inconvenience levels is measured with the Gini formula. This measure is highly correlated with the income shares of bottom 5% poorest group and is called the pgini (the bottom poor sensitive Gini coefficient) in this paper. The inconvenience Lorenz curve is plotted and the pgini is equal to the ellipse area above the Lorenz curve. The unknown true share function can be derived approximately from the given pgini value using the maximum entropy method. The derived function accurately approximated the shares of the very poor and very rich; however, the approximation for the middle ranges was inaccurate. In the future, the inconvenience inequality measure can be extended for the Theil index and Atkinson index (1970) to study the parallel discussions of results derived from the income shares
.588 .586 .0100 .0104 .0108 .0112 .0116 Income shares of the poorest 10% group Fig. 9. Bonferroni and poorest 10% group.
The pgini and the poorest 5% group shares move more or less linearly; however, the poorest 5% group share cannot be used as an inequality measure because it does not satisfy the required conditions of an inequality measure and it cannot be used to derive the underlying unknown income share functions. 5. Conclusion The purpose of evaluating the income inequality measure is to recognize the miserable conditions of the poorest group; however,
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H.K. Ryu / Economics Letters 118 (2013) 370–374
and the inconvenience shares (reciprocal income shares). The standard income inequality measures the relative happiness of the various income groups and the inconvenience inequality measures the relative pains of the various inconvenience groups. Acknowledgment This Research was supported by a Chung-Ang University Research Grant in 2012. References Atkinson, A., 1970. On the measurement of inequality. Journal of Economic Theory 2, 244–263. Bonferroni, C., 1930. Elemente di Statistica Generale. Libereria Seber. Firenze. Dagum, C., 1997. A new approach to the decomposition of the Gini income inequality ratio. Empirical Economics 22, 515–531.
Donaldson, D., Weymark, J., 1983. Ethically flexible Gini indices for income distributions in the continuum. Journal of Economic Theory 29, 353–358. Fields, G., Fei, J., 1978. On inequality comparisons. Econometrica 46 (2), 303–316. Kakwani, N., 1980. On a class of poverty measures. Econometrica 48 (March), 437–446. Lerman, R., Yitzhaki, S., 1984. A note on the calculation and interpretation of the Gini index. Economics Letters 15, 363–368. Ryu, H., 1993. Maximum entropy estimation of density and regression functions. Journal of Econometrics 56, 397–440. Ryu, H., 2008. Maximum entropy estimation of income distributions from bonferroni indices. In: Chotikapanich, Duangkamon (Ed.), Modeling Income Distributions and Lorenz Curves. Springer, pp. 193–211. Ryu, H., 2012. Looking for the standard shape of income distributions. Journal of Economics, Kumamoto Gakuen University 19, 71–94. The Tax Policy Center, 2012. A joint venture of the Urban Institute and Brookings Institution. http://www.taxpolicycenter.org/taxfacts/displayafact.cfm?Docid= 330. Wolfson, M., 1994. Conceptual issues in normative measurement: when inequalities diverge. American Economic Review 84 (2), 353–358. Yitzhaki, S., 1983. On an extension of the Gini inequality index. International Economic Review 617–628.
