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# 3  Aggregate revenue elasticities

This section examines aggregate tax revenue elasticities, which are the most relevant from the point of view of tax forecasting and planning, and possible automatic stabilisation properties of the tax structure. Subsection 1 presents the basic formulae required and subsection 2 reports results for New Zealand in the case of equiproportional income changes. The implications of non-equiproportional income changes are examined in subsection 3.

## 3.1  Elasticity formulae

Aggregate revenue elasticities can readily be expressed as a tax-weighted sum of the individual elasticities. Letting and denote respectively total income tax revenue and total income, the aggregate income tax revenue elasticity is:

(15)

Evaluation of the aggregate elasticity therefore requires, in addition to information about the income distribution (for computation of the income tax shares ), knowledge of the extent to which individuals’ incomes change when aggregate income changes, reflected in the term where the condition must hold. A typical simplifying assumption is that all incomes increase by the same proportion, so that and the aggregate income rax revenue elasticity is a simple tax-share weighted average of individual values.

The aggregate consumption tax elasticity can be calculated, following (15), using:

(16)

where is aggregate consumption tax revenue. Furthermore, since total revenue is the elasticity of total revenue with respect to aggregate income can be found as a tax-share weighted average of the income and consumption tax revenue elasticities.

## 3.2  Estimates of aggregate revenue elasticities

This subsection uses the analytical expressions in subsection 1, along with the assumption of equiproportional income changes, to examine how aggregate revenue elasticities vary with aggregate income levels in New Zealand.

One approach would be to use detailed information, in the form of a large data set containing data on individual taxable incomes.[6] However, the method used here (since such individual data are available only to a highly restricted group of users in New Zealand) is to parameterise the distribution, based on grouped income distribution data, and then to produce a simulated distribution of incomes by taking random draws from the fitted distribution.

Figure 5 in the appendix shows the New Zealand grouped income distribution in 2001, and discusses the application of a lognormal distribution to summarise the data.[7] It was found that a mean and variance of the logarithms of incomes of and provide a reasonable approximation to parameterise a lognormal income distribution. These values imply an arithmetic mean income of \$26,903.[8] Each aggregate revenue elasticity is obtained using a simulated population of 20,000 individuals, drawn at random from the distribution. As the results reported here assume that all incomes increase by the same proportion, the relative dispersion of incomes remains constant as incomes change over time.

Figure 3 shows aggregate elasticity profiles for income and all consumption taxes, and for total tax revenues, as incomes increase over a range of average income levels.[9] The consumption tax profile shown is for the non-proportional consumption case. These profiles are approximately linear with the income and consumption tax elasticities at slightly below 1.3 and 0.9 respectively. However, the income tax elasticity reveals a tendency to rise slightly and then decline at higher average income levels. This decline occurs when a large proportion of the income distribution faces the highest marginal income tax rate.[10]

Figure 4 shows how consumption tax elasticities depend on assumed saving behaviour. Revenue elasticity estimates are noticeably higher for the proportional consumption and ‘no savings’ cases (profiles A and B), but decline more rapidly as income rises, compared to the non-proportional case in profile C. For example, at mean income levels of around \$30,000, elasticities of about 1.0 and slightly below 0.9 are obtained from profiles A and C respectively. Figure 4 also suggests that the effect on the revenue elasticity of ignoring savings is not substantial provided, when income increases, the proportion of income consumed remains approximately constant.

#### Notes

• [6]Some studies use an entirely numerical approach, by imposing small income increases on each individual in the data set and examining the resulting tax changes, rather than using explicit formulae such as those given above.
• [7]In Giorno et al. (1995), aggregate income tax revenue elasticities were obtained by fitting lognormal distributions, using information for each country about only the ratios of the first and ninth deciles to the median income. Values of individual elasticities were computed at 16 points on the income distribution. An aggregate elasticity was obtained as the ratio of average (income weighted) marginal rates to that of average (income weighted) average tax rates. (The weights were obtained from the first moment distribution of the associated lognormal distribution). Unfortunately the ratio of averages is not equivalent to the average of ratios, which is the required measure.
• [8]In the lognormal case, the arithmetic mean income is derived as
• [9]The 20k values were selected randomly from an initial distribution with a lower mean of logarithms than the 2001 distribution. The average income increase, with a fixed variance of logarithms of income, was achieved simply by increasing all incomes by a fixed proportion each year. The non-equiproportional case involves a more complex process of income change, as shown below.
• [10]For the highest average income shown, about one quarter of taxpayers are above the top threshold.
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