1. Measures of Inequality and Their Applications in Indonesia
Takahiro Akita Master of Public Management and Administration (MPMA), Rikkyo University December 5, 2017

2. Popular Measures of Inequality: Why do we use the following inequality measures?
Gini Coefficient
Coefficient of Variation (CV)
Theil Indices: Theil Index L and Theil Index T
Theil Indices are among the Generalized Entropy Class of Inequality Measures

3. Four Criteria for Inequality Measurement
Inequality measures should meet the following four criteria
1. Anonymity
2. Income homogeneity
3. Population homogeneity
4. Pigue-Dalton Transfer Principle
Gini, CV and Theil Indices satisfy these criteria.　

4. 1. Anonymity
If an income distribution Y is obtained from another income distribution X by a permutation of X, then these two income distributions, X and Y, are equally unequal
Example:
Income distribution in 2000
X = (Yamada 2, Kotani 3, Kataoka 5, Kakinaka 6, Akita 9)
Income distribution in 2010
Y = (Akita 2, Kotani 3, Kakinaka 5, Yamada 6, Kataoka 9)
These two income distributions are equally unequal. Measures of inequality should have the same value for these two distributions.

5. 2. Income Homogeneity
If an income distribution Y is obtained from another income distribution X by multiplying everyone’s income by the same positive scalar multiple , then these two income distributions, X and Y, are equally unequal
Example 1
Income distribution in Income distribution in 2010
X = (2, 3, 5, 6, 9) Y = (4, 6, 10, 12, 18) = 2X
Measures of inequality should have the same value for these two distributions.
Example 2
Income distribution in 2010 in US dollars for Village A
X = (2, 3, 5, 6, 9)
Income distribution in 2010 in Japanese Yen for Village A (exchange rate = 100 yen/US dollar)
Y = (200, 300, 500, 600, 900) = 100X

6. 3. Population Homogeneity
If an income distribution Y is obtained from another income distribution X by replicating each income an integer number of times, then these two income distributions, X and Y, are equally unequal
Example
Income distribution for Village A: X = (2, 3, 5, 6, 9)
Income distribution for Village B: X = (2, 3, 5, 6, 9)
Income distribution for (Village A + Village B):
Y = (X, X) = (2, 2, 3, 3, 5, 5, 6, 6, 9, 9)
Measures of inequality should have the same value for income distributions X and Y.

7. 4. Pigue-Dalton Principle of Transfers
If a distribution Y is obtained from another distribution X by transferring a positive amount of income from a relatively rich person to a relatively poor person while holding all other incomes the same as before where even after the transfer, the rich person is still richer than the poor person, then distribution Y is more equal than distribution X.
Example
Income distribution X = (2, 3, 5, 6, 9)
Income distribution Y = (2, 3+1, 5, 6, 9-1) = (2, 4, 5, 6, 8)
Measures of inequality should have a smaller value for income distribution Y than distribution X.

8. Lorenz Curve and Gini Coefficient
= 2 × (Area between Lorenz curve and 45 degree line)
0 ≤ Gini ≤1
Gini = 0  perfect equality
Gini = 1  perfect inequality
Income distribution
Ranking 1 2 3 4 5
Total
Income 6 9 25
Cumul. Pop Share
0.20
0.40
0.60
0.80
1.00
Cumul. Income Share
0.08
0.64

9. Gini Coefficient satisfies four properties
Formula for Gini  2 𝑛𝜇 cov(ranking( 𝑥 𝑖 ), 𝑥 𝑖 )
X = (2, 3, 5, 6, 9)  Gini = 2 (5)(5) 3.4 = 0.272
2. Income homogeneity
Y = (4, 6, 10, 12, 18)  Gini = 0.272
3. Population homogeneity
Y = (2, 2, 3, 3, 5, 5, 6, 6, 9. 9)  Gini = 0.272
4. Pigue-Dalton transfer principle
Y = (2, 3+1, 5, 6, 9-1) = (2, 4, 5, 6, 8)
 Gini = < 0.272
Ranking (xi)
Income (xi) 1 2 3 5 4 6 9
No of persons (n)
Mean income ()
Covariance
3.4
Gini
0.272

10. Coefficient of Variation (CV) satisfies four properties
Formula for CV  Sample STD 𝜇
where Sample STD = 1 𝑛−1 𝑥 𝑖 −𝜇 2
X = (2, 3, 5, 6, 9)  CV = 0.548
2. Income homogeneity
Y = (4, 6, 10, 12, 18)  CV = 0.548
3. Population homogeneity
Y = (2, 2, 3, 3, 5, 5, 6, 6, 9. 9)  CV = 0.548
4. Pigue-Dalton transfer principle
Y = (2, 3+1, 5, 6, 9-1) = (2, 4, 5, 6, 8)
 CV = < 0.548
Ranking (xi)
Income (xi) 1 2 3 5 4 6 9
No of persons (n)
Mean income ()
(Sample) standard deviation (STD)
2.739 CV
0.548

11. Theil Index L satisfies four properties
Formula for Theil L  1 𝑛 ln 𝜇 𝑥 𝑖 = 1 𝑛 ln⁡(𝜇 −ln( 𝑥 𝑖 ))
X = (2, 3, 5, 6, 9)  Theil L = 0.131
2. Income homogeneity
Y = (4, 6, 10, 12, 18)  Theil L = 0.131
3. Population homogeneity
Y = (2, 2, 3, 3, 5, 5, 6, 6, 9. 9)  Theil L = 0.131
4. Pigue-Dalton transfer principle
Y = (2, 3+1, 5, 6, 9-1) = (2, 4, 5, 6, 8)
 Theil L = < 0.131
Ranking (xi)
Income (xi) 1 2 3 5 4 6 9
No of persons (n)
Mean income ()
Theil Index L
0.131

12. Relative vs Absolute Inequality Measures
If inequality measures satisfy income homogeneity, then they are called relative inequality measures; if not, then they are called absolute inequality measures.
Relative inequality measures: Gini coefficient, Coefficient of Variation and Theil Indices,
Absolute inequality measures: Sample Variance and Sample Standard Deviation
Example:
X = (2, 3, 5, 6, 9)  Sample STD = 2.739
Y = (4, 6, 10, 12, 18) = 2X  Sample STD = 5.477

13. Useful Methods for Inequality Analysis using Gini Coefficient, Coefficient of Variation and Theil Indices
Decomposition of inequality by income sources (or expenditure components) using Gini coefficient and coefficient of variation (CV)
Decomposition of inequality by population subgroup (for example, by rural and urban sectors, educational groups, gender, age groups) using Theil indices

14. Decomposition of Inequality by Income Sources using Gini Coefficient and Coefficient of Variation (CV)
Objective: analyze which income sources are inequality increasing or decreasing sources
Total household income
= (1) wage income + (2) interest income + (3) income from agriculture + (4) income from own business + (5) remittance
𝑥 𝑖 = 𝑥 1𝑖 + 𝑥 2𝑖 + 𝑥 3𝑖 + 𝑥 4𝑖 + 𝑥 5𝑖
𝐺= 𝑤 1 𝑅 1 𝐺 1 + 𝑤 2 𝑅 2 𝐺 2 + 𝑤 3 𝑅 3 𝐺 3 + 𝑤 4 𝑅 4 𝐺 4 + 𝑤 5 𝑅 5 𝐺 5
𝐺 𝑘 = Gini for income source k
𝑤 𝑘 = income share of income source k
𝑅 𝑘 = rank correlation ratio of income source k

15. Decomposition of Inequality by Population Subgroup using Theil Indices (1)
Objective: analyze to what extent disparity between population groups contributes to overall income (or expenditure) inequality
Example:
Suppose that a country is divided into rural sector and urban sector.
Country
L = Theil L for the Country
Rural Sector
𝐿 𝑅 = Theil L for Rural
𝑝 𝑅 = Population share for Rural
𝜇 𝑅 = Mean income for Rural
Urban Sector
𝐿 𝑈 = Theil L for Urban
𝑝 𝑈 = Population share for Urban
𝜇 𝑈 = Mean income for Urban

16. Decomposition of Inequality by Population Subgroup using Theil Indices (2)
Then overall inequality as measure by Theil index L can be decomposed as follows: 𝐿=𝐿 𝑊 + 𝐿 𝐵 = within-group inequality + between-group inequality 𝐿 𝐵 = income inequality between rural and urban sectors in mean income () 𝐿 𝑊 = income inequality within rural and urban sectors = 𝑝 𝑅 𝐿 𝑅 + 𝑝 𝑈 𝐿 𝑈 𝑝 𝑘 = population share of sector k 𝐿 𝑘 = Theil index L for sector k

17. Decomposition of Inequality by Population Subgroup using Gini Coefficient
Overall inequality as measure by Gini coefficient can be decomposed as follows:
𝐺 =𝐺 𝑊 + 𝐺 𝐵 + 𝐺 𝑅
= within-group inequality + between-group inequality + residual
In the case of Gini, residual component is positive when the income distributions for the rural and urban sectors overlap, while it is zero when the income distributions do not overlap.

18. Applications My recent paper on inequality in Indonesia using inequality decomposition methods based on Susenas
Akita, T., 2017, ‘Educational expansion and the role of education in expenditure inequality in Indonesia since the 1997 financial crisis’, Social Indicators Research, Springer, 130:
 Decomposition of educational inequality by location (rural and urban sectors) using Gini coefficient
 Decomposition of expenditure inequality by location (rural and urban sectors) using Theil index L
 Decomposition of expenditure inequality by educational groups (primary, secondary and tertiary groups) in each sector using Theil index L

19. Educational Expansion and the Role of Education in Expenditure Inequality in Indonesia since the 1997 Financial Crisis
Background
Education is thought to be a major determinant of wage income and a positive relationship is likely to exist between educational inequality and the distribution of income.
Whether the expansion of education, which has occurred over the last decades, has narrowed or widened educational inequality is thus of policy relevance in Indonesia.

20. Objectives, Data and Methods of the Paper
Examine the role of education in expenditure inequality in Indonesia under educational expansion since 1997 financial crisis.
Data: monthly household expenditure data from Susenas from to 2011
Three Decomposition Methods:
1. Decomposition of educational inequality using Gini coefficient
2. Blinder Oaxaca decomposition
3. Hierarchical decompositin of Theil index

21. Analytical Framework
Analyses are conducted in an urban-rural dual framework
Reasons
1. urban-rural expenditure disparity is one of major determinants of expenditure inequality
2. difference in educational attainment levels is thought to play an important role in urban-rural expenditure disparity
3. notable differences exist between urban and rural sectors in structure of educational attainment levels and magnitude of expenditure inequality.

22. Method 1: Decomposition of Educational Gini by Urban and Rural Sectors
Examine relationship between level and inequality of educational attainment in urban-rural setting using educational Gini
Educational Gini = inequality in the distribution of years of education as measured by Gini index
Total education Gini (G) can be additively decomposed into within- sector Gini (GWS), between-sector Gini (GBS) and residual term (GR) as follows
G = GWS + GBS + GR
If educational distributions of urban and rural sectors do not overlap, then GR = 0.

23. Method 2: Blinder-Oaxaca Decomposition of Expenditure Difference by Urban Rural Sectors
Analyze to what extent urban-rural difference in educational endowments contributes to urban and rural difference in mean per capita expenditure
Urban-rural difference in mean per capita expenditure
= (1) urban-rural difference in per capita difference explained by urban-rural differences in explanatory variables such as education, age, age squared, household sixe, gender (endowments effect)
+ (2) unexplained part.

24. Method 3: Hierarchical Decomposition of Expenditure Inequality by Location & Education by Theil Index (1)
Investigate the roles of education in expenditure inequality in an urban-rural framework
Total expenditure inequality (T) can be decomposed hierarchically into
(1) between-sector inequality component (TBS),
(2) within-sector between-group inequality component (TWSBG),
(3) within-sector within-group inequality component (TWSWG)
T = TBS + TWSBG + TWSWG

25. Result 1: Educational Expansion and Educational Inequality
Level of education (measured by mean years of education) has increased steadily in both urban and rural areas since : 8.4 ⇒ 9.2 years in urban; 5.1 ⇒ 6.3 years in rural (educational expansion).
Speed of educational expansion has been faster in rural than urban areas.
This educational expansion has been associated with decrease in educational inequality

26. Trend in Average Level of Education in Rural and Urban Areas

27. Trend in Educational Inequality

28. Result 2: Decomposition of Educational Gini by Urban & Rural Areas
Expansion of basic education in rural areas lowered educational disparity between urban & rural sectors (0.12 ⇒ ) and educational inequality within rural (0.39 ⇒ 0.34).
Contrib. of residual term rose significantly (19 ⇒ 25%) ⇒ magnitude of overlap in distribution of educational attainment between urban & rural sectors increased
Unlike expenditure inequality, urban sector has much smaller educational inequality than rural sector.

29. Result 3: Blinder-Oaxaca Decomposition by Urban and Rural Sectors
Urban-rural difference in educational endowments is major factor of urban-rural expenditure disparity, as difference in mean years of education accounts for more than 30% of the urban-rural disparity.
Due in large part to declining urban-rural educational disparity, urban-rural expenditure disparity has narrowed since the mid-2000s.

30. Result 4: Hierarchical decomposition of Expenditure Inequality by Theil
Expend. disparity between educational groups in urban sector is major contributor to overall expend. inequality & its contribution has increased since 2000 (10 ⇒ 15%).
In urban sector, tertiary educ. group has largest w-group inequality & its contribution to overall expend. inequality has risen (11 ⇒ 16%).
Expansion of higher educ. in urban sector has played an important role in recent rise in overall expend. inequality by raising not only disparity between educational groups but also inequality within tertiary educational group.

31. Conclusion and Policy Implications 1
Basic educ. policies still serve as an effective means to mitigate expenditure inequality, as they could reduce not only educational gap between urban & rural sectors but also educ. inequality within rural sector by raising general educ.level.
Government should improve quality of primary & secondary education & thereby raise gross enrolment rate (GER) in tertiary education, as GER in 2010 at 23.1% is still very low as compared to neighboring Asian countries.

32. Conclusion and Policy Implications 2
Higher education policies is also crucial, since expansion of higher education in urban areas seems to be one of main factors of recent rise in overall expenditure inequality.
Government should introduce policies that could raise general quality level of higher education.
Government should also introduce policies that could promote linkages between industry and academe to remove the mismatch between the demand and supply.

33. Terima kasih banyak. Sampai ketemu lagi!