Distributive Analysis and Negative Incomes

By:

Abdelkrim Araar &

Jean-Yves Duclos

2008-06-21

      In much of distributive analysis, it is supposed that wellbeing indicators are positive. In the presence of negative indicators, some of the usual distributive indices and curves cannot be used. This short note suggests how one can choose indices and curves to do distributive distributive analysis in these contexts.

 Negative incomes and  inequality

1.1- Negative incomes and  inequality indices

      Two main classes of  inequality indices exist: relative and absolute ones. Relative indices use the dispersion of incomes around the mean and normalised it by the mean. One  virtue of this normalisation is making relative inequality indices scale invariant. This poses a problem however in the presence of negative incomes when average income is zero or negative. 

      When relative indices fall to assess inequality,  absolute inequality indices can still be used since they continue to preserve their useful properties. Moreover, when wellbeing indicators are expressed in real terms, these indices may be found acceptable and easily interpretable as capturing absolute differences in well-being.

Recommended references:

-- Absolute and relative inequality indices and some useful properties for inequality.

1. Kolm, S.-C. (1976a): Unequal inequalities I. Journal of Economic Theory, 12, 416-442.
2. Kolm, S.-C. (1976b): Unequal inequalities II. Journal of Economic Theory, 13, 82-111.

1. Chen, C.-N.; Tsaur, T.-W.; Rhai, T.-S. (1982): The Gini coe cient and negative income. Oxford Economic Papers, 34, 473-478.
2. Berreby, Z.M.; Silber, J. (1985): The Gini coe cient and negative income: A comment. Oxford Ecomomic Papers, 37, 525-526.
3. Chen, C.-N.; Tsaur, T.-W.; Rhai, T.-S. (1985): The Gini coe cient and negative income: A reply. Oxford Economic Papers, 37, 527-528.
4. Araar Abdelkrim (2006), Computating  the CTR Index: Stata module, CIRPEE, Université Laval.

1.2 - The Absolute Gini Index (AGI)

Among absolute inequality indices, there is the absolute Gini Index of inequality. The following references recall the definition of the basic inequality axioms and decompose the AGI index across groups and income sources.

Recommended references:

1. Araar Abdelkrim (2006), The Absolute Gini Coefficient of Inequality: decomposability and Stochastic Dominance:CIRPEE & PEP, Université Laval, mimeo

2. Araar Abdelkrim (2006), On the Decomposition of the Gini Coefficient: an Exact Approach, with an Illustration Using Cameroonian Data, CIRPÉE - Working Paper: 06-02, Université Laval.

1.3- Decomposing the AGI index across groups or income sources.

Recommended references:

1. Araar Abdelkrim (2006), The Absolute Gini Coefficient of Inequality: decomposability and Stochastic Dominance:CIRPEE & PEP, Université Laval, mimeo

2. Araar Abdelkrim (2006), Decomposing the AGI index by groups Stata module, CIRPEE, Université Laval.

3. Araar Abdelkrim (2006), Decomposing the AGI index by income sources Stata module, CIRPEE, Université Laval.

1.4- Negative incomes and  inequality dominance

1. Araar Abdelkrim (2006), The Absolute Gini Coefficient of Inequality: decomposability and Stochastic Dominance:CIRPEE & PEP, Université Laval, mimeo

2. Araar Abdelkrim (2006), Lorenz Curves Stata module, CIRPEE, Université Laval. (with this module, one can draw the Absolute Lorenz Curve: see the help for this module)

3. Araar Abdelkrim (2006), Deprivation Curves Stata module, CIRPEE, Université Laval. (with this module, one can draw the Absolute Deprivation Curves: see the help for this module)

Negative incomes and  Poverty

2.1- Negative incomes and  poverty indices

2.2- Negative incomes and  poverty dominance

2.3- Negative incomes, FGT index and Shapley decomposition by income sources

For 2.1, 2.2 and 2.3, see the following mimeo.

Araar Abdelkrim & Duclos Jean-Yves  (2006), Poverty and Negative Incomes, CIRPÉE, Université Laval, mimeo.