GINI DECOMPOSITION AND GINI INCOME ELASTICITY UNDER INCOME VARIABILITY

I. INTRODUCTION

The Gini Income Elasticity (GIE hereafter) is a parameter that measures the impact on the (extended) Gini index of income inequality of a marginal proportional change in an income source (e.g., Lerman and Yitzhaki, 1985; Stark et al., 1986). Estimates of the GIE have been used, among others, for assessing the impact on inequality of increases in outlays for social programs (e.g., Lerman and Yitzhaki, 1994; Wodon and Yitzhaki, 2002). However, while an increase in funding for programs such as unemployment benefits may reduce inequality in mean income, it may also reduce income variability or risk, and this is not taken into account in the traditional decomposition of the Gini index. In this note, we show that under income variability (assuming that the observations represent a draw from states of nature), an income source has three GIEs corresponding to the three ways in which the income source affects the certainty equivalent income: (i) the mean income from the source over time, (ii) the variability in income from the source and (iii) the correlation over time between income from the source and income from other sources. The total impact on inequality of a marginal change in income from the source is then a function of these three GIEs.

II. METHOD

Consider a situation where total income is the sum of income from J sources. The standard decomposition of the Gini index is

(1)

where y_j is income from source j, F is the rank of the individual (or household) in the distribution of total income, F_j is the rank in the distribution of income from source j, m_j is the source's mean income, and m is mean total income. In equation (1), R_j is the Gini correlation between total income and income from source j, G_j is the Gini of source j, and S_j is the source's share of total income. It has been shown by Stark et al. (1986) that the impact on inequality of a marginal proportional change in income from source j is such that

(2)

where the GIE, denoted by η_j, is defined as

(3)

A GIE higher (lower) than one denotes an income source that is inequality increasing (decreasing) at the margin.

In this note, we extend this decomposition and the concept of the GIE to a context in which individuals are subject to income variability over time. Our strategy consists in considering the income variable y in (1) as the certainty equivalent income of the individual, and to apply the decomposition to that certainty equivalent.¹ If the actual income for individual i is denoted by x_i– this is a random variable with density function g_i(·), the certainty equivalent income y_i is defined through

(4)

where does not represent the Bernoulli utility function of the individual, but rather a social judgement on the impact of income variability on welfare. Denote by μ_i the individual's mean income over time. The cost of variability, , is defined by

(5)

Using a Taylor expansion, an approximation of this cost is

(6)

where is the Arrow–Pratt absolute risk aversion index measured at μ_i, and is the variance of the individual's income. In this note, for simplicity, we assume that the social planner has a normative evaluation of risk aversion. This normative evaluation is represented by a constant absolute risk aversion (denoted by ρ), so that

(7)

Return now to the fact that there are J sources of income with means over time μ_ij for individual i so that

(8)

and

(9)

where is the covariance between a and b. We have

(10)

where . Using (8)–(10), (7) may be rewritten as

(11)

where is the cost of the income variability of source j, and is the cost (or benefit) due to the covariance between source j and other income sources. If we denote by and the mean values over the population as a whole of μ_ij, φ_ij and ξ_ij, and as before by m the mean value of y_i, applying (1)–(11) yields for each income source three components in the decomposition of the Gini

Using a notation similar to (1), this yields

(12)

Consider now a small proportional change in an income source by a factor of e, such that y_ik(e) = (1 + e) μ_ik+ (1 + e)²φ_ik+ (1 + e) ξ_ik. It is shown in the appendix that

(13)

Dividing by G, we get

(14)

where

With cross-section data, since we do not observe income variability, only the first term on the right-hand side of (14) appears, so that whether an income source increases or decreases inequality at the margin is solely determined by whether the GIE is above or below one. In a panel setting, we have instead three sources of impact at the margin, and thereby three GIEs related to (i) the mean income from the source over time, (ii) the variability of the source, and (iii) the correlation between the source and other sources over time. For example, a source such as unemployment benefits may reduce inequality in certainty equivalent income at the margin not only through targeting individuals with comparatively low mean income, but also through reducing income variability for these individuals, with this beneficial effect appearing through the third GIE in (14).

One last point: to introduce flexibility in the measurement of inequality, we may use the extended Gini coefficient (Yitzhaki, 1983), in which case the weights placed on various parts of the distribution of the certainty equivalent income will depend on a parameter υ. A value of 2 yields the standard Gini. A higher (lower) value places more (less) weight on lower parts of the distribution. All the results will remain valid. We will have

(15)

where

(16)

and

(17)

and are defined analogously. Following a small proportional change in income from a source, we will have

where

III. CONCLUSION

Most of the work for assessing the impact of public transfers on inequality is based on cross-sectional data. Yet many transfers are designed not only to reach the poor, but also to offset the impact of income variability on welfare and on vulnerability. In this note, we have extended the source decomposition of the Gini index of inequality to show how to evaluate the impact at the margin on inequality of a proportional increase in program outlays under income variability. When considering inequality in certainty equivalent income, each income source can be said to have three Gini Income Elasticities corresponding to the three terms appearing in the Taylor approximation of the individual's certainty equivalent income, namely the mean income from the source, the variability of the source over time, and the covariance between the source and other income sources over time. The importance of such sources of variability remains an empirical matter that goes beyond the scope of this note.

The Gini coefficient has a unique underlying social welfare function that is based on the rank of individuals. In future research it would be interesting to propose a more general framework to account for income variability. This will be in line with the work of Makdissi and Wodon (2002), Duclos et al. (2005, 2008) and Makdissi and Mussard (2008a,b) who propose more general frameworks to analyse tax and transfer reforms for all social welfare function that obey Fishburn and Willig (1984) generalized transfer principles.