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Social Indicators Research

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01-12-2012

Income Inequality Indices Interpreted as Measures of Relative Deprivation/Satisfaction

Authors: Luis José Imedio-Olmedo, Encarnación M. Parrado-Gallardo, Elena Bárcena-Martín

Published in: Social Indicators Research | Issue 3/2012

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Abstract

This paper considers different ways of making comparisons between individuals in terms of deprivation and/or satisfaction. This allows the Gini index, the Bonferroni index and the De Vergottini index to be interpreted as social deprivation measures as well as social satisfaction measures. The inequality measures that belong to the β family, or linear combinations of them, are obtained when using different weighting schemes to average the deprivation and satisfaction associated with each income level. Particularly, the generalised Gini indices (Yitzhaki, Int Econ Rev 24:617–628 in 1983), the indices proposed by Aaberge (J Econ Inequal 5(3):305–322, 2007) or those proposed by Imedio-Olmedo et al. (J Public Econ Theory 13(1):97–124, 2011) can be used to evaluate social deprivation or social satisfaction in an income distribution.

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This concept initially appeared in studies in the field of sociology to justify certain aspects of collective behaviour: Stouffer et al. (1949), Davis (1959), Runciman (1966), Gurr (1968) and Crosby (1976, 1979). From an economic standpoint, Runciman’s (1966) approach has had the greatest impact. He defines relative deprivation as follows: a person is relatively deprived of Z when (1) he does not have Ζ, (2) he sees some other person or persons, which may include himself at some previous or expected time, as having Z, (3) he wants Ζ, and (4) he sees it as feasible that he should have Z.

Other references related to deprivation in terms of income are: Yitzhaki (1982), Chakravarty and Chakraborty (1984), Berrebi and Silber (1985), Paul (1991), Chakravarty et al. (1995), Podder (1996), Chakravarty (1997, 2007), Chakravarty and Mukherjee (1998), Imedio-Olmedo et al. (1999), Ebert and Moyes (2000), Bárcena (2003), Imedio-Olmedo and Bárcena-Martín (2003), and Magdalou and Moyes (2009).

This definition is motivated by Runciman (1996): “…relative deprivation is the extent of the difference between the desired situation and that of the person desiring it”.

Some proposals, such as that of Berrebi and Silber (1985), avoid the first step. We think that it is essential to explicitly state how to make inter-comparisons in such formulations.

Bárcena and Imedio (2008) make some considerations about these indices in this regard.

The De Vergottini index is the only one that does not belong to β.

To avoid being repetitive, on occasion we only use the term deprivation when referring to both deprivation and satisfaction.

As pointed by Easterlin (1995), Alesina et al. (2004) and Berg and Veenhoven (2010) in the individual’s deprivation the absolute level of income is not as important as the relative income position of the individuals, i.e. the comparison with the income position of the relevant others.

Proof of all the results of this paper are available upon request from the authors.

F is sometimes assumed to be continuous in order to obtain theoretical results in a simpler manner. In such a case, \( {\text{f}}({\text{x}}) = {\text{F}}^{\prime } ({\text{x}}) \) is the density function of the distribution.

In the following equality, if the minimum income is x₀ > 0, then \( {\text{B}}(0) = \mathop {\lim }\limits_{{{\text{x}} \to 0^{ + } }} {{({\text{L}}({\text{p}})} \mathord{\left/ {\vphantom {{({\text{L}}({\text{p}})} {\text{p}}}} \right. \kern-\nulldelimiterspace} {\text{p}}}) = {\text{L}}^{\prime } (0^{ + } ) = {{{\text{x}}_{ 0} } \mathord{\left/ {\vphantom {{{\text{x}}_{ 0} } {{\upmu}}}} \right. \kern-\nulldelimiterspace} {{\upmu}}} \).

Their distribution functions are: \( {\text{F}}_{{[0,{\text{x}}]}} ({\text{z}}) = {{{\text{F}}({\text{z}})} \mathord{\left/ {\vphantom {{{\text{F}}({\text{z}})} {{\text{F}}({\text{x}})}}} \right. \kern-\nulldelimiterspace} {{\text{F}}({\text{x}})}}, \) \( 0 < {\text{z}} \le {\text{x,}} \) \( {\text{F}}_{{\left[ {{\text{x}},{\text{x}}_{\text{M}} } \right]}} ({\text{z}}) = {{\left( {{\text{F}}({\text{z}}) - {\text{F}}({\text{x}})} \right)} \mathord{\left/ {\vphantom {{\left( {{\text{F}}({\text{z}}) - {\text{F}}({\text{x}})} \right)} {\left( {1 - {\text{F}}({\text{x}})} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - {\text{F}}({\text{x}})} \right)}}, \) \( {\text{z}} \ge {\text{x}} . \)

There is no valid upper bound for any distribution. For a given distribution with maximum income x_M, \( {\text{V}} \in [0,({{{\text{x}}_{\text{M}} } \mathord{\left/ {\vphantom {{{\text{x}}_{\text{M}} } {{\upmu}}}} \right. \kern-\nulldelimiterspace} {{\upmu}}}) - 1]. \)

A relative inequality index (does not change when the variable is multiplied by a constant), I, is a compromise index if μI is an absolute index (does not change when a constant is added to the variable). Analogously, an absolute index, J, is a compromise index if J/μ is a relative index (Blackorby and Donaldson 1978).

An index shows inequality aversion if the Pigou–Dalton Principle of Transfers is satisfied. This principle states that an income transfer from a richer to a poorer individual that leaves their ranks in the income distribution unchanged (progressive transfer) reduces income inequality. The B index shows greater inequality aversion than the G index. The use of B is appropriate if the focus is placed on the left-hand side of the distribution (Nygard and Sandström 1981). In contrast, V attaches more weight to the incomes on the right-hand side of the distribution and its aversion to inequality is smaller than that of G.

The Rawlsian leximin focuses on the poorest individual of the population. Between two distributions, the distribution with the greater minimum income is preferred or, in the event of equality, the distribution in which the minimum income is less frequent. This approach is derived from the theory of social justice defined by Rawls (1971).

Imedio-Olmedo and Bárcena-Martín (2007) describe this aspect in great detail, comparing the behaviours of α and γ families in this context.

If individuals focus on status instead of income, the rank in the income distribution is the relevant issue. Imedio and Bárcena (2003) discuss this point of view.

In Definition 1 the individual identifies his situation with his income. Now, an individual with income x (x not being the minimum income) feels a kind of altruism in the sense that the individual identifies (through the mean income) his situation with that of those who are worse off. This stance increases his deprivation with respect to the one in Definition 1.

For example, if we work with a uniform distribution defined over \( [0,{\text{x}}_{\text{M}} ] \), \( {\text{M}}({\text{x}}) - {{{\text{x}} = ({\text{x}}_{\text{M}} - {\text{x}})} \mathord{\left/ {\vphantom {{{\text{x}} = ({\text{x}}_{\text{M}} - {\text{x}})} 2}} \right. \kern-\nulldelimiterspace} 2} \) is strictly decreasing, while \( {\text{x}} - {\text{m}}({\text{x}}) = {{\text{x}} \mathord{\left/ {\vphantom {{\text{x}} 2}} \right. \kern-\nulldelimiterspace} 2} \) is strictly increasing. Nonetheless, for the distribution \( {\text{F}}({\text{x}}) = \left( {{{\text{x}} \mathord{\left/ {\vphantom {{\text{x}} {{\text{x}}_{\text{M}} }}} \right. \kern-\nulldelimiterspace} {{\text{x}}_{\text{M}} }}} \right)^{1/2} , \) \( {\text{M}}({\text{x}}) - {\text{x}} \) shows a relative maximum in p = 1/4, p = F = (x) and it is decreasing for \( {\text{p}} > 1/4. \) The function \( {\text{x}} - {\text{m}}({\text{x}}) \) is increasing.

A SWF, W, is consistent with the inequality index I if \( {\text{I}}({\text{F}}) \le {\text{I}}({\text{G}}) \Leftrightarrow {\text{W}}({\text{F}}) \ge {\text{W}}({\text{G}}) \) is verified for any two distributions F and G with the same mean income.

Bárcena-Martín et al. (2003) use another approach to achieve the same result.

Specifically, for \( {\text{t}} \ge 1, \) the index I(1,t) shows greater aversion to inequality than I(2,t).

Both indices show less inequality aversion than the Gini index. The direct comparison between them would be in terms of the degree of concavity of their respective social preference distribution.

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Title: Income Inequality Indices Interpreted as Measures of Relative Deprivation/Satisfaction
Authors: Luis José Imedio-Olmedo
Encarnación M. Parrado-Gallardo
Elena Bárcena-Martín
Publication date: 01-12-2012
Publisher: Springer Netherlands
Published in: Social Indicators Research / Issue 3/2012
Print ISSN: 0303-8300
Electronic ISSN: 1573-0921
DOI: https://doi.org/10.1007/s11205-011-9912-8

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