that the pre-fiscal policy income ranking should be preserved. In other words, if individual A was poorer than individual B before fiscal interventions, individual A should continue to be poorer than individual B after the interventions.

From theory, we know that the total _redistributive effect_ (RE) can be decomposed into two elements: the change in _vertical inequality_ (VE) minus _re-ranking_ (RR) (see Duclos and Araar, 2006, Chapter 8, p. 145; Lambert, 2002, Chapters 8-10; and Urban, 2009). The redistributive effect (RE) is equal to the difference between the Gini coefficient for incomes _before_ taxes and transfers, Gm, and the Gini coefficient for incomes _after_ taxes and transfers Gpost−fisc, or; Adding and subtracting Cpost−fisc the concentration coefficient for incomes _after_ taxes and transfers,<sup>26</sup> equation (1) can be decomposed into: Then the redistributive effect can be written as: where:

- VE is equal to the difference between the Gini coefficient for incomes _before_ taxes and transfers and the concentration coefficient for incomes _after_ taxes and transfers; if there is no re-ranking, RE = VE by definition because the concentration coefficient for incomes _after_ taxes and transfers will be identical to the Gini coefficient for incomes _after_ taxes and transfers;

- RR is equal to the difference between the Gini coefficient for incomes _after_ taxes and transfers and the concentration coefficient for incomes _after_ taxes and transfers.

The redistributive effect is diminished by re-ranking (RR) as clearly shown in equation (3). The VE measure is the Reynolds-Smolensky progressivity index (RS) and the RR measure is known as the Atkinson-Plotnick index of horizontal inequity.

It is very important to note that if taxes and transfers combined (i.e., the fiscal system) is globally progressive _but_ there is re-ranking, it is no longer true that the system is always necessarily equalizing.<sup>27</sup> In other words, to establish whether a fiscal system decreases inequality –and by how much—it is essential to carry out the empirical fiscal incidence analysis for the system as a whole.

# **3. Data and Assumptions**

To determine the size of each fiscal intervention at each step of income in Figure 1, we need to “map” the taxes and transfers from South Africa’s national accounts and administrative fiscal data to individual members in households. For this we use the 2010/11 Income and Expenditure Survey (IES) conducted by Statistics South Africa (Statistics SA, 2012a). This survey contains data on household income, expenditures, cash transfers, and utilization of educational services collected from 25,328 households across the country over a period of 12 months.<sup>28</sup> The allocation or “mapping” to individuals is obtained dividing the total tax paid or transfer received by each household by the total number of members in the household (excluding boarders and domestic workers).

> 26 Recall that a concentration coefficient (also known as quasi-Gini) differs from the Gini coefficient in that the households are still ranked by the income _before_ taxes and transfers.

> 27 See, Lambert (2002) and Duclos and Araar (2006).

> 28 The methodology used for data collection in the IES survey is an internationally accepted best practice and it is generally believed that the quality of the incomes and consumption data in the survey are good. However, there is some concern that the share of food consumption of the extreme poor in South Africa is much lower than one would expect, potentially pointing to some under-reporting at

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