Taxes, Redistribution and Efficiency

A number of measures of income inequality and of the redistributional impact of income taxes and social transfers are available. Since one of our objectives is to highlight the role of value judgements in reaching inequality conclusions we report the familiar Gini and Atkinson inequality indices, using a variety of ‘inequality aversion’ assumptions for the Atkinson index, A.[5] In considering the redistributive impact of income taxes and social transfers (such as Working for Families), we consider how these measures change for different definitions of pre-tax, post-tax and post-tax-and-transfer income, in addition to Kakwani and Reynolds-Smolensky measures of tax progressivity. The next subsection summarises those measures.

Measures of Inequality and Redistribution

The Gini index of pre-tax income, y, G_y, is the familiar measure of the area contained between the Lorenz curve for y and the line of perfect equality (the 45^o line). The Lorenz curve describes the relationship between the cumulative proportion of individuals, N, (or other income unit) and the cumulative proportion of incomes, F(y), where individuals are ranked in ascending order of their incomes. Formally G_y can be defined as:

where F(y_i) is the proportion of people with incomes below y_i, and is mean income. Alternatively G_y can be expressed as:[6]

The Gini measure can in fact be derived from a special kind of social welfare, or evaluation, function in which welfare is viewed in terms of ‘unfairness’. If welfare is defined as the average of the minimum income from all pairwise income comparisons, it is possible to show that the welfare function is equivalent to:

This type of expression for the evaluation function involves two summary measures of the distribution, the arithmetic mean and the Gini coefficient. It is a convenient ‘compression’ of the form involving all individual values of income, and is referred to as an ‘abbreviated’ welfare function. In particular, it allows the implications for the trade-off between ‘equity and efficiency’, implied by the value judgements, to be transparent.[7]

The Atkinson measure of inequality is instead based on a ‘wastefulness’ judgement regarding inequality, whereby a more equal distribution of the same total income could produce an improvement, from the point of view of the judge, and hence a higher social welfare. The social welfare function, in terms of individual values, is:

where ε reflects the relative inequality aversion of the judge (reflecting the concavity of the welfare function). Increases in higher incomes are given a relatively lower weight than for lower incomes. The Atkinson index relies on the concept of an equally-distributed equivalent value of total income, y_e , which yields the same social welfare as the current distribution. This is given by:

The Atkinson index, A (ε), is the proportional difference between arithmetic mean income and the equally distributed equivalent:

Values of ε close to zero imply a mild aversion to inequality, and thus a low preference to redistribute towards the bottom of the income distribution. That is, only a small sacrifice in total incomes would be tolerated in order to achieve income equality. A value of e greater than 1 implies a stronger aversion to inequality (i.e. a willingness to sacrifice a large fraction of income in the redistributive process).

To illustrate, HES data used in section 2 show that average gross taxable income in New Zealand in 2006-07 was $33,503. Using the individual incomes within the HES, and a value of ε = 0.5 to calculate y_e yields y_e= $27,153. That is, ε = 0.5 represents a significant degree of inequality aversion - a willingness to sacrifice around 20% of total income in order to achieve full equality. A value of ε = 0.2 implies a much lower willingness to sacrifice income (8%), while ε = 1.0 implies a 40% sacrifice. In effect the value of A(ε) captures the proportionate reduction in mean income that the value of ε implies; hence A(ε) = 0.2 implies a 20% income sacrifice to achieve equality. This is approximately achieved here with e = 0.5.

The rearrangement of (8) shows that the social welfare function associated with the Atkinson measure can also be expressed in abbreviated form as:

Hence the abbreviated functions using the Gini and Atkinson measures are very similar, despite the fact that they are based on quite different value judgements. In each case they imply that, in terms of the trade-off between equity and efficiency, a 1 per cent increase in equality, 1 - I (where I = G(ν), A(ε)), would be traded for a 1 per cent reduction in arithmetic mean income. Alternatively, a 1 per cent increase in total income would be just sufficient to compensate for a 1 per cent increase in inequality (as measured by the relevant index).

The Gini and Atkinson indices can be compared for alternative definitions of pre-tax, post-tax, or post-tax-and-transfer, incomes where in the case of the Gini, individuals are ranked according the value of the income measure under consideration. However for some comparisons it is useful to rank individuals according to their pre-tax incomes even when considering a post-tax income definition. This gives rise to the Concentration index, C, such that if y and z are respectively pre- and post-tax incomes then G_z is the Gini measure of post-tax incomes (ranked by z) while C_z is the concentration measure, which takes essentially the same form as the Gini except that individuals are ranked by their pre-tax incomes.

Using these definitions, the Reynolds-Smolensky, L, and Kakwani, K, measures of the redistributive effect of a tax can be defined. This requires a Concentration index of income tax payments, C_t, which measures the extent of inequality in individuals' tax payments, ranked by their pre-tax incomes. Thus, tax progressivity is measured by:

The Reynolds-Smolensky index captures the difference in the Ginis of pre- and post-tax incomes, while the Kakwani index measures the disproportionality of tax payments relative to pre-tax incomes. With a proportional tax, the two K components, C_t and G_y, are the same (the Concentration and Lorenz curves coincide) so that K = 0. Thus, with a tax/transfer system that is progressive at all income levels, the concentration curve of tax payments lies outside (that is, further from the 45^o line than) the Lorenz curve of pre-tax incomes. Progressive taxes yield 0 ≤ K, L ≤ 1.

Kakwani (1977) shows that K and L differ depending on the extent of re-ranking of individuals between pre- and post-tax incomes and the aggregate ratio of tax payments to incomes, such that:

where g is the aggregate tax ratio and P is an index of re-ranking given by:

In the absence of any re-ranking K and L are proportional, with L deviating from K depending on the size of the aggregate tax take. As a result, tax reforms that are not revenue-neutral can yield changes in K and L that are not proportional.