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Average Marginal Income Tax Rates for New Zealand, 1907-2009 WP 12/04

4  Calculating AMTRs – Methodology

4.1  Applying the Barro-Sahasakul Approach

As noted in the introduction, the AMTR of interest here is the Barro and Sahasakul (1983) income-weighted average of individual effective marginal personal income tax rates. That is, we want to estimate the aggregate:

Equation 6 - income-weighted average of individual effective marginal personal income tax rates  .

where Yj is the personal income of taxpayer j, and Y is aggregate personal income across all j taxpayers. The EMTRjs are obtained from the tax schedule or suitably adjusted ‘effective' rates where those differ from statutory rates. The relevant tax rates and thresholds are then matched with information on (Yj/Y) from our income distribution data, inclusive of the income share of non-filing taxpayers. To avoid confusion, in the remainder of the paper we refer to marginal tax rates (MTR, EMTR) levied at the individual level using the subscript j; hence: MTRj, EMTRj.

Applying equation (6) to our data requires a number of simplifying assumptions. Firstly, from 1981-2009 the use of taxpayer unit record data ensures that the relevant MTRj or EMTRj of income tax is identified for each taxpayer. However, like the pre-1981 data, this dataset does not include the impact on EMTRjs of abatement of social welfare payments.

For data prior to 1981, we seek to match data from NZOYB and other sources on the distribution of gross assessable income with the relevant tax schedule. Since tax brackets are typically described with respect to net-of-exemptions assessable income, it is important to subtract those exemptions to identify net income and thereby the appropriate EMTRj to apply at each gross income level. For many taxpayers, the deduction of exemptions from their gross income will not alter their MTRj or EMTRj (eg, deduction of $5000 from gross income of $50,000 will not affect the MTRj where this MTRj applies over a net income band of $40,000-50,000. However, another taxpayer with the same $50,000 of gross income but $12,000 exemptions would face a different MTRj - that applicable to net income below $40,000.

We therefore need to deduct exemptions from gross (assessable) income to derive net (assessable) income in order to identify the relevant MTRj or EMTRj for each taxpayer. However, with aggregate-level, rather than individual-level, gross assessable income and exemptions data by gross income band, we do not know how many taxpayers (and associated fraction of gross income) would face a lower marginal tax rate than would be inferred from their gross income.

Treating our aggregate-level data as if they represented an individual within each income band would mean that either all or no income would shift MTRj bands as a result of adjusting for exemptions. Instead we (i) assume that the impact of exemptions is to move individuals by no more than one MTRj band; and (ii) use the ratio of total exemptions to gross assessable income in each band to weight the MTRjs for each band, m. This yields an EMTRj estimate reflecting the exemptions adjustment:

EMTRj,m = (em/ym) MTRj,m-1 + (1 - em/ym) MTRj,m         (7)

Where (em/ym) is the exemptions/income ratio in band m, and MTRj,m (MTRj,m-1) is the MTRj in band m (m-1), (m > 0) and MTRj,0 = 0, captures the general personal exemption. To examine the sensitivity of our AMTR calculations to this assumption, we also report AMTRs where no shifts in MTR brackets, based on exemptions data, has been assumed.

4.2  Examples of AMTR calculations

Table 2 shows an example of the AMTR calculations - for the 1980 income year - when there were relatively few (six) income tax brackets and MTRjs. Since the income tax schedule defines taxable income as income net of exemptions (deductions), the income brackets in row 1 are defined with respect to net income. The MTRj for each income bracket is shown in row 2. Row 3 provides an estimate of the EMTRj faced by individuals in each tax bracket, adjusted for the impact of exemptions.[14] This adjustment weights the MTRjs in each bracket by the ratios of exempt income (row 5) to gross income (row 4). For example, approximately 7% of income in the <4,500 bracket in 1980 was exempt from tax; we therefore assume that this fraction of income faces the MTRj of the bracket immediately below; in this case, 0%. In view of the small amount of exemptions (averaging 6% of gross income), the resulting impacts on the 1979/80 EMTRs in row 3 are small.

Table 2 - The AMTR calculations, 1980
Taxable income bracket ($) <4,500 4,500-10,000 10,000-11,000 11,000-16,000 16,000-22,000 >22,000 Total
Statutory MTR 14.5% 36.5% 41.5% 48.0% 55.0% 60.0%
EMTR 13.5% 35.2% 41.2% 47.6% 54.6% 59.8%
Gross assess-able income 1,225,465 4,565,695 991,170 3,550,580 1,583,880 1,304,220 13,221,010
Exemptions 85,775 275,315 66,130 235,960 90,860 45,870 799,910
Income share (%) 9% 35% 7% 27% 12% 10% 100%
Income-weighted AMTR = 41.70%

The relevant gross income shares are calculated in row 6. In principle, non-filer income is also added before estimating the gross income shares in row 6 though, as noted above, this is not relevant after 1958. Applying the row 6 weights to the EMTRjs in row 3 yields the AMTR (=41.70%) for 1980. It can be seen that this is dominated by the large shares (nearly 65%) of income in the $4.5-10k and $11-16k income brackets facing EMTRjs of 35.2% and 47.6% respectively.

The case in Table 2 illustrates a relatively straightforward year. Most years, however, involve multiple marginal tax rates across income levels and a variety of additional complications including:

  1. earned and unearned income distinctions (1921-50) with each facing different MTRjs[15]
  2. estimation of EMTRjs where statutory rates do not measure effective rates; eg, separate non-filer incomes and abatement of thresholds
  3. The multi-slope tax function where EMTRjs rise with every pound of income (1914-39); in that case we calculate AMTRs based on EMTRjs at the mid-points in income classes across the income distribution
  4. income classes from income distribution data that approximate the income bands in the tax structure, requiring some re-grouping of data on incomes, exemptions etc
  5. simultaneous application of several different taxes at various rates to a given income including social security taxes, and special ‘war taxes’
Figure 4 - Income distribution and tax structure, 1950
Figure 4 - Income distribution and tax structure, 1950   .

Figure 4 illustrates the more complex 1950 income distribution and tax structure. This overlays income distribution data with the individual EMTRjs. The rates rise in multiple steps from 4% at an income of £200 to 76% at incomes over £4000. This yields the AMTR of 29% shown. Exemptions data are used to adjust the EMTRjs to approximate the effect of people moving to lower income brackets. This adjustment has a particularly large impact at the bottom of the income distribution; for the £200-300 bracket the effective tax rate drops from 21%, before allowing for exemptions, to 4%. That is, the availability of exemptions shifts a large fraction of taxpayers into the 0% tax rate applied to net incomes up to £200.


  • [14]Note that this is not an EMTR as conventionally defined since no individual faces this rate. Rather it reflects the weighted average of rates faced by taxpayers in that, and the adjacent, income brackets.
  • [15]Data on earned incomes was collected until 1956 but earned and unearned income faced the same tax rate from 1951-1956.
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