# 3.3  Non-equiproportional income changes

This subsection relaxes the assumption of equiproportionate income changes, used in the previous subsection and in the vast majority of studies. In line with the present approach of using parametric specifications at a fairly high level of aggregation, subsection 1 presents a function to describe the systematic variation in with Subsection 2 presents revised aggregate elasticities based on estimates of the dynamic specification.

## 3.3.1  A specification

It is convenient to specify a functional form for the variation in with . A suitable form, involving just one parameter, is:

(17)

where is the mean of logarithms of income (the logarithm of geometric mean income).[11] This means that if and is less than geometric mean income, the elasticity, is greater than unity, and vice versa, so that (17) involves equalising changes. If income changes are disequalising. This specification can thus be used to examine the sensitivity of aggregate revenue elasticity measures to variations in the standard assumption of .[12]

In examining non-equiproportionate changes, according to equation (17), it is also useful, when increasing the 20,000 simulated incomes from one year to the next, to impose random proportionate income changes, in addition to the systematic equalising or disequalising tendency reflected in . Without such changes, annual income inequality changes too rapidly. The specification in (17) is consistent with the following dynamic process. Let denote individual ’s income in period , and let denote the mean of logarithms in period , with as the geometric mean. The generating process can be written as:

(18)

where is . Equation (18) can be rewritten as:

(19)

Hence the variance of logarithms of income in period 2, , is given by

(20)

The variance of logarithms is therefore constant when .[13]

## 3.3.2  Aggregate elasticities

Estimation of equation (19) was carried out for a range of pairs of consecutive years during the 1990s, using information from large samples of IR5 and IR3 filers. The results suggest a relatively stable value of of around . This reflects a substantial degree of regression towards the (geometric) mean; indeed, with no random component of income change this would have the effect of halving income inequality in as little as three years. If is combined with the variance of logariths of income remains constant over time.[14] These values produce an aggregate income tax revenue elasticity, at 2001 mean income, of about . For the proportional and non-proportional consumption functions respectively, the aggregate consumption tax revenue elasticities are and giving corresponding total tax revenue elasticities of and respectively.

Regression towards the geometric mean therefore reduces the aggregate revenue elasticities. This arises because, for those above the geometric mean income, the value of is reduced, and vice versa for those below the geometric mean. The aggregate elasticity, from (15), is a tax-share weighted average of these terms, and in view of the fact that increases as increases, the lower values of at the upper income levels dominate.

To give some idea of the sensitivity of results to the variation in , consider a value of which requires for a stable degree of income inequality. The aggregate income tax elasticity, again at 2001 mean income, is nowwhile the consumption tax elasticities are and for proportional and non-proportional consumption functions (giving total revenue elasticities of and ).

### Notes

• [11]In the case of lognormal income distributions, this is also the median income.
• [12]For non-equiproportional changes, the aggregate revenue elasticity can now be less than 1; for example it is zero if the only incomes which increase are below a tax-free threshold and none crosses the threshold.
• [13]In general, the variance of logarithms of income increases if the regression coefficient, , exceeds the correlation between log-incomes in the two periods. If the variance of logarithms of incomes eventually becomes stable at . On dynamic income specifications, see Creedy (1985).
• [14]This value is in fact similar to that estimated for IR3 filers, though the values obtained for IR5 filers were lower, at about 0.1. Given that these samples do not constitute all taxpayers, and in practice inequality is relatively stable, it is appropriate here to model a stable variance of logarithms.
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