GiniDecompLY

Gini Correlation Index

Gini correlation was first introduced in Schechtman and Yitzhaki (1987). It is a non-symmetric measure and can take the following two forms, depending on which variable is given in its actual values and which one is ranked:

If \(X\) and \(Y\) are random variables from continuous distribution functions \(F_X\) and \(G_Y\) respectively

\[ \Gamma(X,Y) = \frac{Cov(X,G_Y(Y))}{Cov(X,F_X(X))} \]

and similarly, \[ \Gamma(Y,X) = \frac{Cov(Y,F_X(X))}{Cov(Y,G_Y(Y))} \] The range of the gini correlation coefficient is \([−1, 1]\)

gini_corr function takes as first argument the variable in its actual values, and as second argument the variable of the distribution with the rank information. An optional argument for sampling weights is also available.

If we want to calculate the gini correlation coefficient of salaries distribution ranked with the total income distribution, then we can call the function as follow:

# Calculate the gini correlation between the salary and total income distributions

Salary_distribution = sample_income_data$wage
Total_income_distribution = rowSums(sample_income_data[3:6])

gini_corr(Salary_distribution, Total_income_distribution)
#> [1] 0.6307609

Gini decomposition by income sources

The Lerman and Yitzhaki decomposition method aims to understand the contribution of different income sources to overall income inequality.

Suppose the total income of households comes from 4 income sources as described in sample_income_data dataset. The decomposition of Lerman and Yitzhaki provide estimation of the contribution of each income source to the income inequality as measured by Gini coefficient.

\[ Gini(income) = 0.421 \]

The final output of the decomposition is represented by the Absolute Contibution for each income source which sum up to the overall \(Gini(income)\), and the Relative Contribution which sum up to

The steps of estimation include :

1- Calculation of the Share of each income source to the total income; 2- Calculation of Gini coefficients for each income source; 3- Calculation of Gini_corr (gini correlation coefficient) between the distribution each income source and the total income.

Absolute Contribution is calculated as follow:

Absolute Contribution = Share * Gini * Gini_corr

and then

Relative Contribution = Absolute Contribution / Gini(income)

gini_decomp_source provide a tibble containing all the components described above. It takes as a first argument the .data containing the income sources variables. And then we pass the columns names (or positions), separated by commas, indicating income sources.

sample_income_data |> 
  gini_decomp_source(wage, self_employment_rev, farming_rev, other_rev)

The function provides two optional arguments:

-.wgt for sampling weights; - and .by for results disaggregation, in that case the output will be a grouped tibble.

We can also pass the income sources variables by their positions in the data frame:

gini_decomp_source(sample_income_data, 3:6, .by = region, .wgt = sample_wgt)
#> # A tibble: 8 × 7
#> # Groups:   region [2]
#>   region income_source         Share  Gini Gini_corr Absolute_Contribution Relative_Contribution
#>   <chr>  <chr>                 <dbl> <dbl>     <dbl>                 <dbl>                 <dbl>
#> 1 Rural  farming_rev         0.412   0.815     0.854               0.287                 0.654  
#> 2 Rural  other_rev           0.301   0.402     0.395               0.0478                0.109  
#> 3 Rural  self_employment_rev 0.0674  0.907     0.471               0.0288                0.0655 
#> 4 Rural  wage                0.219   0.706     0.488               0.0755                0.172  
#> 5 Urban  farming_rev         0.00189 0.989     0.737               0.00138               0.00354
#> 6 Urban  other_rev           0.373   0.618     0.634               0.146                 0.374  
#> 7 Urban  self_employment_rev 0.165   0.827     0.327               0.0446                0.114  
#> 8 Urban  wage                0.460   0.648     0.665               0.198                 0.508

Gini income elasticity

The decomposition method described above can be represented in the following equation:

\[ Gini(Y) = \sum_k {S_k \Gamma(Y_k,Y) Gini(Y_k)} \]

Where \(Y_k\) and \(S_k\) are the distribution and the Share of income source \(k\), respectively. And \(Y\) denotes the distribution of the total income.

Thus, Absolute Contribution for the \(k^{th}\) income source is the product \(S_k \Gamma(Y_k,Y) Gini(Y_k)\)

Stark and al. (1986) showed that this decomposition equation offers a simple method to assess the marginal effect on total income inequality following a marginal percentage variation in income from a given source, equal for all households. Thus, the impact of an increase in income from source \(k\), for all households, in such a way that \(Y_k\) is multiplied by \((1 + e_k)\), where \(e_k\) is sufficiently small can be represented as follow :

\[ \frac{\partial Gini(Y)}{\partial e_k} = S_k (\Gamma(Y_k,Y) Gini(Y_k)- Gini(Y)) \]

This expression show a measure for the marginal contribution of source \(k\) to the overall inequality. It can be expressed differently to reveal the relative marginal contribution of inequality due to a marginal variation in the income from source \(k\):

\[ \frac{\partial Gini(Y) / \partial e_k}{Gini(Y)} = S_k (\frac{\Gamma(Y_k,Y) Gini(Y_k)}{Gini(Y)}- 1) = S_k(\eta_k-1) \]

Where

\[ \eta_k = \frac{\Gamma(Y_k,Y) Gini(Y_k)}{Gini(Y)} \] \(\eta_k\) will denote the elasticity of Gini index associated with a percentage change in the mean income (for each income source)

This definition clearly demonstrates that the marginal impact of a source of income depends on its income elasticity of the Gini index. Thus, a percentage increase in the income from a source with a lower \(\eta_k\) (higher) than 1, will decrease (increase), the overall income inequality. When the \(\eta_k\) is close to 1, it means that the variation in this source does not affect overall inequality.

gini_income_elasticity function calculate the Elsaticity along with the Marginal_Impact of a change in the mean of each income source on the overall Gini index.

The function has the same set of arguments as the previous function

sample_income_data |> 
  gini_income_elasticity(wage, self_employment_rev, farming_rev, other_rev,
                         .by = region,
                         .wgt = sample_wgt)
#> # A tibble: 8 × 7
#> # Groups:   region [2]
#>   region income_source         Share  Gini Gini_corr Elasticity Marginal_Impact
#>   <chr>  <chr>                 <dbl> <dbl>     <dbl>      <dbl>           <dbl>
#> 1 Rural  farming_rev         0.412   0.815     0.854      1.59         0.241   
#> 2 Rural  other_rev           0.301   0.402     0.395      0.362       -0.192   
#> 3 Rural  self_employment_rev 0.0674  0.907     0.471      0.973       -0.00182 
#> 4 Rural  wage                0.219   0.706     0.488      0.784       -0.0474  
#> 5 Urban  farming_rev         0.00189 0.989     0.737      1.87         0.00164 
#> 6 Urban  other_rev           0.373   0.618     0.634      1.00         0.000976
#> 7 Urban  self_employment_rev 0.165   0.827     0.327      0.693       -0.0506  
#> 8 Urban  wage                0.460   0.648     0.665      1.10         0.0480

Growth-redistribution impacts on social welfare function

In terms of social welfare, if individuals or households assess their level of well-being, on one hand, in absolute terms (that is, the income they have), and, on the other hand, in relative terms (how much they have compared to others), the level of social welfare can be represented as a function combining the level and inequality of well-being (Wodon and Yitzhaki, 2002; Essama-Nssah, 2000 & Sen, 1997). In other words, such a function combines both the social preference for more income and that for more equality. It is an increasing function of the mean and decreasing function of inequality (Essama-Nssah, 2000).

Sen (1997) showed, under the assumption that individual well-being is approximated by individual income and that the well-being of any pair of individuals equals that of the poorest, then the following expression can be considered as a function of social welfare underlying the Gini coefficient.

\[ SW = \bar{Y} (1-G)) \] Where \(\bar{Y}\) denotes average income and \(G\) the Gini index.

Using the decomposition of \(G\), the impact of a variation in an income source on the Social Welfare function is estimated according to the following formulation:

\[ \frac{\Delta SW}{SW} \mid _{Y_k} = \frac{S_k}{1-Gini(Y)} - \frac{S_k\Gamma(Y_k,Y)Gini(Y_k)}{1-Gini(Y)} \]

social_welfare_impact function provides a Growth-redistribution decomposition of the impact of a variation in each income source on the Social Welfare function.

The output of the function is three columns (added to the decomposition components described before)

• Total_Variation = \(\frac{\Delta SW}{SW} \mid _{Y_k}\);

• Growth_Effect = \(\frac{S_k}{1-Gini(Y)}\);

• Redistribution_Effect = \(- \frac{S_k\Gamma(Y_k,Y)Gini(Y_k)}{1-Gini(Y)}\)

sample_income_data |> 
   social_welfare_impact(wage, self_employment_rev, farming_rev, other_rev,
   .wgt = sample_wgt)
#> # A tibble: 4 × 7
#>   income_source       Share  Gini Gini_corr Growth_Effect Redistribution_Effect Total_Variation
#>   <chr>               <dbl> <dbl>     <dbl>         <dbl>                 <dbl>           <dbl>
#> 1 farming_rev         0.125 0.926     0.620         0.215               -0.123           0.0917
#> 2 other_rev           0.352 0.574     0.600         0.607               -0.209           0.398 
#> 3 self_employment_rev 0.136 0.864     0.424         0.234               -0.0858          0.148 
#> 4 wage                0.388 0.697     0.659         0.669               -0.308           0.362

References

Essama-Nssah B. (2000), Inégalité, pauvreté et bien-être social, fondements analytiques et normatifs, De Boeck Université, Bruxelles

Handcock, M. (2016), Relative Distribution Methods in the Social Sciences, Springer-Verlag, Inc., New York, 1999 ISBN 0387987789

Lerman, R. I., & Yitzhaki, S. (1985). Income Inequality Effects by Income Source: A New Approach and Applications to the United States. The Review of Economics and Statistics, 67(1), 151–156. https://doi.org/10.2307/1928447

Rawls J. (1971), A Theory of Justice, The Belknap Press of Harvard Univers

Schechtman, E., and Yitzhaki, S. (1987). A Measure of Association Based on Gini’s Mean Difference. Communications in Statistics: Theory and Methods A16:207–31.

Schechtman, E., and Yitzhaki, S. (1999) On the proper bounds of the Gini correlation, Economics Letters,Volume 63, Issue 2, p. 133-138, ISSN 0165-1765

Sen A. (1970) Collective Choice and Social Welfare, Holden-Day, ISBN:978-0-444-85127-7

Soudi K. (20) Stark, O., J. Taylor, and S. Yitzhaki. (1986). Remittances and Inequality. Economic Journal 96(383):722–40.

Wodon, Quentin and Yitzhaki, Shlomo (2002): Inequality and Social Welfare. Published in: A Sourcebook for Poverty Reduction Strategies , Vol. 1, (April 2002): pp. 75-104.