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# Appendix 1: The NZ ‘Multi-Slope' Income Tax System 1914-1939

This Appendix describes the so-called ‘multi-slope' marginal tax rate schedule. This is in contrast to the more usual ‘multi-step' tax schedule in which marginal tax rates rise in steps at specified income thresholds and are constant between thresholds, and where each MTR applies to additional income above each threshold. The multi-slope tax schedule of 1914-39, on the other hand, typically applied to incomes in excess of an initial threshold income level but the marginal tax rate specified in the schedule increased with every additional pound that an individual earned and each higher rate applied to all income (above an initial tax-exempt level of income where applicable), not just the increment.

Below we describe this system using two years to illustrate: 1914 and 1917. The 1917 case involved two additional features: an additional ‘special war tax' (1917- 1920) and an initial income exemption abated beyond a specified higher level of income.

## A1.1 The 1914 tax structure

Let y be taxable income before the exemption, A. An exemption of 300 pounds applies for all taxpayers. Let t* be the marginal tax rate specified in the tax schedule as levied on assessable income, so that the total tax paid is:

T = τ*(Y - a) where a = 300         (A1)

The marginal tax rate, from (A1), is:

τ = dT/dy = τ* + (y - a)(dτ*/dy)         (A2)

For 300 ≤ y ≤ 400: t = τ* = 0.025 (6 pence per pound; there are 240 pennies in a pound)

For incomes above 400, the value of τ* increases by 3/400ths of a penny for each pound increase in income. Hence:

For 400 < y ≤ 1400:

τ* = 0.025 + (3/400) (1/240) (y-400)

⇒ τ* = 0.0125 + 0.00003125y         (A3)

Note that τ* = 0.025 at y=400 and τ* = 0.05625 at y=1400. Using (2) and (3) it can be shown that:

τ = 0.003125 + 0.0000625y         (A4)

giving τ = 0.028125 at y=400 and τ = 0.090625 at y=1400. Values of τ for 400 < y < 1400 are on a straight line between these two points; see Figure 1.

For 1400 < y ≤ 2400, the lower rate of increase in τ* (1/(400*240)) yields:

τ* = 0.04168 + 0.000010417y         (A5)

such that τ*= 0.05625 at y=1400 and τ*= 0.0667 at y=2400. For this income range, using (2) and (5) it can be shown that:

τ = 0.038552 + 0.000020834y         (A6)

Equation (6) yields: τ = 0.06772 at y=1400 and τ = 0.08855 at y=2400. Values of τ for y between 1400 and 2400 are on a straight line between these two points. Beyond 2400, the marginal tax rate is specified as a constant 14 pennies per pound, or 5.83%.

This reveals that the EMTR can be higher, and sometimes considerably higher, than τ* during the 1914-39 period due to the impact of the multi-slope aspect of the schedule. The maximum rate does not generally apply at the highest income levels, where the ‘slope aspect' is absent. The slope is further amplified when there is abatement of the tax-free threshold, a1, as occurred during 1917-35.

Figure A1 also reveals that the EMTR varied between about 2.5% and 9% in 1914 for those who were liable to pay tax and file tax returns. Most income earners did not earn sufficient income to exceed the tax-free threshold in this period - by our estimates (see below) only around 10% of employees were tax filers and not all of those were assessed as tax-liable (for example, if their assessable income fell below £300). Hence, when weighted by taxpayer incomes (see section 5), the average ‘effective' marginal tax rate across tax filers was around 5% in 1914, but for all income earners combined it was only around 0.5%.

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