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# 4  Comparisons over Time

The previous sections of this paper have discussed alternative income distribution comparisons for a single time period. The fact that the redistributive effect of any tax system cannot be evaluated independently of the population (the pre-tax income distribution) raises the question of how comparisons can be made over time, where typically both the population and the tax structure are different. In fiscal incidence studies the question is thus: has the income tax and transfer system become more or less redistributive? The difficulty is therefore to isolate the marginal effect of the tax policy change from that of the population change.

Suppose that two cross-sectional household surveys are available. Let Ti denote the tax structure for i = 0,1 (an initial period and subsequent period respectively). Similarly let Pi denote the population in period i. There are therefore four possible Gini inequality measures of both gross market income and disposable income; denote these by Gm (Pi,Tj) and Gd (Pi,Tj) for i,j = 0,1. Indeed, these four Gini measures could be obtained using each of the combinations of income concept and unit of analysis discussed above. It is assumed here that each survey contains enough information about the characteristics of households so that the disposable incomes of each population can be computed for each of the tax structures.

To simplify the discussion using the hypothetical households introduced above (and minimise the data to be presented), suppose the only difference between the two populations is that in the second period there is a fifth household consisting of two children and two adults with market incomes of 80 and 20 (so again the households all have the same total income). Suppose the second-period disposable incomes are as show in Table 7: these may be compared, for the first four households, with those in Table 1. The total household disposable incomes are thus

[78, 78, 100, 87, 102]

and the disposable incomes per adult equivalent person are

[78, 45, 48.1, 36.1, 42.3].
Gross market income Disposable Income HH A1 A2 C1 C2 A1 100 - - - 78 - - - 60 40 - - 43 35 - - 75 25 0 - 68 32 0 - 100 0 0 0 87 0 0 0 80 20 0 0 72 30 0 0

Consider comparisons using distributions 5 and 6 in Table 4, that is, the distributions of income per adult equivalent person, using the individual as unit of analysis. It may be tempting to compare the Gini measures of disposable income in each period, giving Gd (P0,T0) = 0.1171 and Gd (P1,T1) = 0.1016. This comparison would conclude that the policy reform has reduced inequality. But this would be a spurious comparison. Alternatively, it may be tempting to compare, for the two periods, the percentage reduction in the Gini when moving from market to disposable income: in this case they are both 17%, suggesting no change in the redistributive effect of taxes as a result of the policy change.[23] However, the separate effects of tax and population changes can be obtained as follows.

In order to identify the appropriate marginal effects of tax policy and population changes, it is first useful to consider the following decomposition:[24]

The first term in square brackets on the right hand side of (4) is the population effect given tax structure 1, and the second term in square brackets is the tax policy effect given initial population 0. Appropriate computation for the hypothetical data gives:

The reduction in inequality of disposable income per adult equivalent person is the term on the left hand side of (5), that is 0.1016 - 0.1171 = -0.0155. The population effect is negative, since 0.1016 - 0.1220 = -0.0204). The policy effect is actually positive, since 0.1220 - 0.1171 = 0.0049. Thus the effect of the tax policy change, measured using the population of the initial year, is to increase inequality.

However, there is another possible decomposition of the change in inequality, since:

The first term in square brackets on the right hand side of (6) is the population effect given tax structure 0, while the second term is the tax policy effect given population structure 1. Computation gives:

In this case the population effect is again negative (-0.0187) and the policy effect is again positive (0.0032). Both effects are smaller in absolute terms but of course give the same overall reduction in the Gini measure.[25] Faced with two values for each of the marginal effects, one approach is to obtain the unweighted arithmetic mean, giving a tax policy effect of 0.00405 and a marginal population effect of -0.01955.[26] The overall reduction in inequality of disposable income per adult equivalent person in the present example (comparing the second cross-sectional dataset with the first) arises because the inequality-reducing marginal effect of the population change outweighs the inequality-increasing marginal effect of the tax policy change.

### Notes

• [23]The percentages are rounded to the nearest integer here as in Table 5, but in this case they are equal when given to two decimal places, both being 16.65%.
• [24]This kind of decomposition can be extended to allow for labour supply responses to the tax policy change, but rapidly becomes more complex as the number of decompositions increases. See Bargain (2010) and, for a range of extensions including the use of a money metric welfare measure, see Creedy and Hérault (2011).
• [25]Computations based on the use of the equivalent person as the unit of analysis (distributions 7 and 8 in Table 4), give a percentage reduction in the Gini when moving from market to disposable income in the second period of 20% (compared with 18%, as shown in Table 5). This could not be used to suggest that the policy change has produced a more redistribute tax structure. Use of this alternative unit of analysis produces positive marginal policy effects on the change in inequality of disposable income, in agreement with the use of the individual as unit (though absolute values are smaller).
• [26]This average is recommended by Shorrocks (2011), who links it to the Shapley Value, familiar from game theory.
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