On the Decomposition of the Esteban and Ray Index by Income Sources
Abstract
:1. Introduction
2. Matrix Representation of the Esteban and Ray (1994) ER Polarization Index
3. Decomposing the ER Index by Income Sources
- -
- The higher , the higher the degree of polarization of the distribution of total income.
- -
- The higher , the higher the degree of polarization of the distribution of total income.
- -
- If is positive, the higher this correlation measure, the higher the degree of polarization of the distribution of total income. However, if it is negative, it will have a negative impact on the overall Esteban and Ray index ER.
- -
- Similarly, if is positive, the higher this correlation measure, the higher the degree of polarization of the distribution of total income. However, if it is negative, it will have a negative impact on the overall Esteban and Ray index ER3.
4. A Short Empirical Illustration
- Benefits (benefits) that include: old-age and survivor’ benefits, unemployment benefits, sickness benefits, disability benefits, education-related allowances, family/children related allowances, social exclusion not classified elsewhere, housing allowances
- Income from rental of a property or land, interest, dividends, profit from capital investments in unincorporated business (property and interest)
- Income available before including sources 1 and 2 (income before)
5. Concluding Comments
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. The Similarity between the Decomposition by Income Sources of the Gini Index and of the ER Index
Country | Mean Total Income | Income Before % | Benefits % | Property and Interest % | Total Population |
---|---|---|---|---|---|
AT | 26,662.48 | 70.0% | 27.3% | 2.7% | 7,963,391 |
BE | 24,520.2 | 74.0% | 24.6% | 1.3% | 9,319,177 |
BG | 4164.49 | 76.4% | 21.9% | 1.7% | 6,235,715 |
EE | 11,043.97 | 82.3% | 16.6% | 1.1% | 1,113,681 |
EL | 9161.76 | 75.7% | 19.8% | 4.5% | 8,092,137 |
ES | 16,370.34 | 71.9% | 24.4% | 3.6% | 42,446,793 |
FR | 25,730.84 | 65.1% | 24.4% | 10.5% | 55,793,599 |
HR | 6663.01 | 80.3% | 18.3% | 1.4% | 3,225,726 |
HU | 5474.74 | 75.6% | 23.2% | 1.2% | 8,332,493 |
LT | 7742.34 | 81.6% | 16.6% | 1.8% | 2,417,930 |
LV | 8135.06 | 80.3% | 18.6% | 1.2% | 1,708,676 |
PL | 6912.37 | 80.9% | 18.2% | 0.9% | 32,623,207 |
PT | 10,892.61 | 79.1% | 17.7% | 3.2% | 8,183,986 |
RO | 2850.82 | 82.4% | 17.5% | 0.1% | 15,991,057 |
RS | 3214.21 | 74.2% | 25.0% | 0.8% | 5,432,579 |
SE | 29,761.2 | 77.6% | 17.6% | 4.8% | 7,647,944 |
SI | 13,678.07 | 73.8% | 23.5% | 2.7% | 1,794,388 |
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1 | Esteban and Ray (1994) refer to the natural logarithm of income rather than to income. We will make a somehow similar assumption by stating that the mean income of a given group refers in fact to its mean income relative to the mean income in the whole population. To simplify the notations, we do not introduce the population mean income in the formulations. |
2 | |
3 | Expression (13) reminds us of the decomposition of the Gini index by income sources (see, Lerman and Yitzhaki 1985) where the contribution of an income source to the overall Gini index is a function of the share of this source in total income, of the Gini index of this source and of the Gini-correlation between this source and total income. In (13) the contribution of an income source to the overall ER index is a function of the two components of the ER index for this source, and of two correlation measures. However the share of the source does not appear. In Appendix A, we provide a more detailed decomposition where the parallel with the traditional decomposition of the Gini index by income sources becomes evident. |
4 | For a survey of equivalence scales and related income distribution issues, and some comparisons of scale relativities, see Coulter et al. (1992). |
5 | When, in expression (1), we divide the income data by the average income and assume that β = 1, ER will equal to twice the traditional Gini index. What is called the absolute Gini index, is actually the product of the Gini index by the mean, so that when β = 1 and we use absolute incomes and not relative incomes in (1) ER will be equal to twice the absolute Gini index. |
Measure Computed | Value for Total Income | Relative Contribution of Income Before | Relative Contribution of Benefits | Relative Contribution of Property Income and Interest |
---|---|---|---|---|
Average income with absolute contribution of income sources | 15,634 | 11,060 | 3626 | 948 |
Average income with relative contribution of income sources | 100% | 70.70% | 23.20% | 6.10% |
ER with parameter β equal to 2.5 computed on basis of relative incomes (relative contributions of income sources) | 0.038 | 62.4% | 25.4% | 12.2% |
ER with parameter β equal to 1 computed on basis of relative incomes (relative contributions of income sources) | 0.645 | 65.6% | 24.9% | 9.5% |
ER with parameter β equal to 2.5 and logarithms of incomes | 0.045 | |||
ER with parameter β equal to 1 (like Gini) and logarithms of incomes | 0.577 |
Measure Computed | Value for Total Income | Relative Contribution of Income Before | Relative Contribution of Benefits | Relative Contribution of Property Income and Interest |
---|---|---|---|---|
Average income with absolute contribution of income sources | 17,048 | 12,233 | 3892 | 924 |
Average income with relative contribution of income sources | 100% | 71.7% | 22.8% | 5.4% |
ER with parameter β equal to 2.5 computed on basis of relative incomes (relative contributions of income sources) | 0.024 | 54.4% | 28.6% | 17.0% |
ER with parameter β equal to 1 computed on basis of relative incomes (relative contributions of income sources) | 0.413 | 59.6% | 27.7% | 12.8% |
ER with parameter β equal to 2.5 and logarithms of incomes | 0.026 | |||
ER with parameter β equal to 1 (like Gini) and logarithms of incomes | 0.478 |
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Bárcena-Martín, E.; Silber, J. On the Decomposition of the Esteban and Ray Index by Income Sources. Econometrics 2018, 6, 17. https://doi.org/10.3390/econometrics6020017
Bárcena-Martín E, Silber J. On the Decomposition of the Esteban and Ray Index by Income Sources. Econometrics. 2018; 6(2):17. https://doi.org/10.3390/econometrics6020017
Chicago/Turabian StyleBárcena-Martín, Elena, and Jacques Silber. 2018. "On the Decomposition of the Esteban and Ray Index by Income Sources" Econometrics 6, no. 2: 17. https://doi.org/10.3390/econometrics6020017
APA StyleBárcena-Martín, E., & Silber, J. (2018). On the Decomposition of the Esteban and Ray Index by Income Sources. Econometrics, 6(2), 17. https://doi.org/10.3390/econometrics6020017