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Policy Analysis Using the Decomposition of the Gini by Non-marginal Analysis

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The Gini Methodology

Part of the book series: Springer Series in Statistics ((SSS,volume 272))

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Abstract

The objective of this chapter is to demonstrate the usefulness of several decompositions of the Gini (and the EG) in order to analyze the strengths and the weaknesses of various policies. We concentrate on distributional issues. The other component of the problem of tax reform—the estimation of the marginal cost of taxation—is identical to the description given in Chap. 14 hence it will not be repeated here.

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Notes

  1. 1.

    In a less technical language, exchangeability up to a linear transformation means that the joint distributions of the two variables are symmetric with respect to each other.

  2. 2.

    117% from 1995 to 2006, adding it up to 25% from 1990 to 1995, according to Tables 4 at http://www.cso.ie/releasespublications/documents/economy/2006/nie2006tables1995-2006excel.xls and http://www.cso.ie/releasespublications/documents/economy/HistoricalNIETables1970-1995excludingFISIM1.xls.

  3. 3.

    Source: QNHS (Ireland) and HES (Israel).

  4. 4.

    Source: Statistical Yearbook of each country.

  5. 5.

    Some researchers tend to use equalized income but to weight unequalized persons. This causes internal contradiction leading to ambiguous results (see Ebert 1997, 1999). Throughout this chapter we weight households by their number of equalized persons only.

  6. 6.

    The Irish accounting period is a year, while the Israeli one is 3 months. Finkel, Artsev, and Yitzhaki (2006) find that the Gini coefficient calculated from a 3-month accounting period was by nearly 4% higher than the index based on a 12-month period. Other estimates are provided by Creedy (1979, 1991), Burkhauser and Poupore (1997), and Gibson, Huang, and Rozelle (2).

  7. 7.

    Ireland, Central Statistics Office, EU Survey on Income and Living Conditions (EU-SILC 2006), Table 1. (http://www.cso.ie/releasespublications/documents/eu_silc/current/eusilc.pdf).

  8. 8.

    Israel, Central Bureau of Statistics, Press Release 13 August 2007. (http://www.cbs.gov.il/hodaot2007n/15_07_150e.pdf). As mentioned above, this figure is based on the Income Survey, which we do not use here. The same calculation based on the Household Expenditure Survey would yield the Gini coefficient = 0.380.

  9. 9.

    We are indebted to Joel Slemrod for pointing out this issue.

  10. 10.

    See Chap. 13.

  11. 11.

    This type of problems may also occur when incomes are registered on a cash flow base rather than on an accrual basis. Different sources of income such as capital gains, farm income, and other types of capital income, which are registered according to realization, may have different accumulation and distribution patterns over time. Relying on snap-shots of the distribution may exaggerate the impact of those incomes on inequality in the long run.

  12. 12.

    For an alternative and interesting view, see Fields and Ok (1996, 1999) who present an axiomatic characterization to absolute changes in incomes.

  13. 13.

    Fields and Ok (1999) refer to this type of mobility measures as correlation-based mobility index. An example of such an index is Hart’s index discussed by Shorrocks (1993).

  14. 14.

    An important property of the Gini correlation is that the bounds are identical for all marginal distributions. This property does not hold for the Pearson correlation coefficient (Schechtman & Yitzhaki, 1999). This means that one minus Pearson’s correlation coefficient cannot serve as an index of mobility because a change in the shape of one of the marginal distributions, that does not affect the transition process of the ranks, may change the value of the correlation.

  15. 15.

    See Feldstein (1976). See also the measures of progression in the income tax (Lambert 2, Chap. 6).

  16. 16.

    One could divide the symmetric and asymmetric Gini indices of mobility by two in order to keep the indices between zero and one.

  17. 17.

    A turnover matrix is a matrix the elements of which add up to one. A transition matrix is a matrix of which the elements in the rows add up to one. Usually transition matrices represent the conditional probabilities, while the elements of a turnover matrix represent the joint probability distribution of the two variables.

  18. 18.

    The mobility index that is the closest to the one suggested in this chapter is Bartholomew’s (1982) index of mobility which is based on the expected value of the absolute difference in the values attached to categories in the initial and final distributions. However, Bartholomew’s index is sensitive to the initial and final marginal distributions, and therefore may give a misleading picture of the transition process. For example, assume that everyone in the society is promoted by one category. Bartholomew’s index would indicate transition although there is no change in the ranking of the members. On the other hand, the Gini mobility index is not affected by linear transformations of the marginal distributions. See Boudon (1973, pp. 51–54) for a discussion of the properties of Bartholomew’s index.

  19. 19.

    Note, however, the important contribution by Geweke, Marshall, and Zarkin (1986) who analyze mobility indices in a continuous time framework.

  20. 20.

    The transition matrix is a special case of a doubly stochastic matrix, where each column and each row add up to one, as discussed by Marshall and Olkin (1979, Chap. 2), although each element should be multiplied by a constant. A similar situation arises when the variable is a binary variable: although the probability is a continuous variable, the realization of the variable in the sample is either one or zero. Traditionally transition matrices have been applied to discrete distributions, due to grouping. The fact that we are not dealing with groups is not due to an inability to handle groups. Rather, we define the transition matrix without grouping in order to avoid the loss of intra-group differences in ranks and thereby inequality, which may be relevant for calculating the inequality index. Note that since the mobility index is a sufficient statistic for the informational content of the transition matrix for our purpose, there is no need to construct the transition matrix and therefore its size is irrelevant in practice.

  21. 21.

    Using the framework proposed in this section, Beenstock (2002a, 2002b) analyzes intergenerational mobility in Israel; Fisher and Johnson (2006) apply the methodology using consumption data from the USA, Wodon (2) applies the methodology to mobility and risk during the business cycle in Argentina and Mexico, and Wodon and Yitzhaki (2003a) look at wage inequality over time in Mexico.

  22. 22.

    Silber (1995) developed an index of the intensity of change in ranking, which is equal to twice the Plotnick index. Using (15.9) it means that it is also equal to the Gini mobility index.

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Authors and Affiliations

  1. Department of Economics, The Hebrew University, Mount Scopus, Jerusalem, Israel

    Shlomo Yitzhaki

  2. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Edna Schechtman

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  1. Shlomo Yitzhaki
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Appendix 15.1

Appendix 15.1

1.1 Proof of (15.8)

The proof consists of finding upper and lower bound for GY(α). The upper bound is

$$ \begin{array}{*{20}{c}} {{{\rm{G}}_{{{\rm{Y}}(}}}{{_{\alpha }}_{)}} = {\rm{2COV}}[\alpha {{\rm{Y}}_{{1}}} + ({1} - {\rm{\alpha }}){{\rm{Y}}_{{2}}},{\rm{F}}({\rm{Y}}({\rm{\alpha }}))]} \hfill \\ { = {\rm{2\alpha COV}}[{{\rm{Y}}_{{1}}},{\rm{F}}({\rm{Y}}({\rm{\alpha }}))] + {2}({1} - {\rm{\alpha }}){\rm{COV}}[{{\rm{Y}}_{{2}}},{\rm{ F}}({\rm{Y}}(\alpha ))]} \hfill \\ { \leqslant {2}\alpha {\rm{COV}}\left[ {{{\rm{Y}}_{{1}}},{\rm{F}}\left( {{{\rm{Y}}_{{1}}}} \right)} \right] + {2}({1} - \alpha ){\rm{COV}}\left[ {{{\rm{Y}}_{{2}}},{\rm{F}}\left( {{{\rm{Y}}_{{2}}}} \right)} \right] = \alpha {{\rm{G}}_{{1}}} + ({1} - \alpha ){{\rm{G}}_{{2}}}.} \hfill \\ \end{array} $$

Recall that Y1 and Y2 are normalized with μ1 = μ2 = 1. The derivation of the upper bound is based on Cauchy-Shwartz inequality, which can be utilized to show that for all Yj and Yk, COV[Yj, F(Yk)] ≤ COV[Yj, F(Yj)].

The lower bound is obtained from

$$\openup 5pt \begin{array}{rcl} {{{\rm{G}}_{{{\rm{Y}}}}}_{{(\alpha) }} = {\rm{2COV}}[\alpha {{\rm{Y}}_{{1}}} + ({1} - \alpha ){{\rm{Y}}_{{2}}},{\rm{F}}({\rm{Y}}(\alpha ))]} \hfill \\{ = {2}\alpha {\rm{COV}}[{{\rm{Y}}_{{1}}},{\rm{F}}({\rm{Y}}(\alpha ))] + {2}({1} - \alpha ){\rm{COV}}[{{\rm{Y}}_{{2}}},{\rm{F}}({\rm{Y}}(\alpha ))]} \hfill \\{ \ge {\rm{Max}}[0,{2}\alpha {\rm{COV}}[\left( {{{\rm{Y}}_{{1}}},{\rm{F}}\left( {{{\rm{Y}}_{{2}}}} \right)} \right] + {2}({1} - \alpha ){\rm{COV}}\left[ {{{\rm{Y}}_{{2}}},{\rm{F}}\left( {{{\rm{Y}}_{{1}}}} \right)} \right]} \hfill \\{ = {\rm{Max}}[0,\alpha {{\rm{G}}_{{1}}}{\Gamma_{{{12}}}} + ({1} - \alpha ){{\rm{G}}_{{2}}}{\Gamma_{{{21}}}}].} \hfill \\\end{array} $$

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Yitzhaki, S., Schechtman, E. (2013). Policy Analysis Using the Decomposition of the Gini by Non-marginal Analysis. In: The Gini Methodology. Springer Series in Statistics, vol 272. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4720-7_15

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