4 Income Tax Revenue
The model of the previous section described, in terms of the changing parameters of lognormal distributions, the age-income profiles of succeeding cohorts of males and females in New Zealand. The parameter estimates applied to incomes from employment and self-employment. The aim of this section is to make use of the income model to obtain total income tax revenue in a range of time periods, aggregated over all existing cohorts, where each cohort's age-earnings profile can differ as a result of the ‘overtaking' phenomenon. In addition, population ageing means that, over time, the relative size of each cohort in the total population varies.
The basic framework and relationships are described in subsection 4.1. The calculation of total income tax revenue from a multi-step function, given the mean and variance of the relevant lognormal distribution, is then described in subsection 4.2. The accounting framework of analysis and the model of income profiles are applied separately to males and females, but for notational convenience the exposition here again does not make this distinction. Income tax projections for New Zealand are reported in subsection 4.3.
4.1 The Basic Framework
Suppose the income tax paid on an income of y is denoted T(y). This function depends on a number of parameters, typically income thresholds and marginal rates, and these may change over time. But they need not be specified at the present stage.
Let Tt,c denote the tax paid per employed person (for whom y > 0) in cohort c at time t. This is expressed as:
where, as above, 
is the two-parameter lognormal distribution function. Let Pt,c denote the number of individuals from cohort c who are alive at time t. Information on Pt,c can be obtained from data on demographic projections. Furthermore, wt,c denotes the participation rate (the proportion of the cohort who work) of cohort c at time t. [7]
Let Rt denote total income tax revenue at time t. This is expressed as:
The summation is indicated to begin at t - 100 to cover all cohorts alive at any time, assuming no one lives beyond 100 years.[8] Total income at time t, denoted Y t, can be written as:
where yt,c is given by (8) above. The overall average tax rate, ξt, is thus:
The following subsection explains how the Tt,c values can be obtained for a multi-step income tax function.
Notes
- [7] As explained in the previous section, no distinction is made here regarding different hours of work. An allowance was made when constructing the income data on which the profiles were estimated, rather than introducing separate treatments for workers in different hours categories.
- [8] However, in the application reported here the 75 and over age group was in fact aggregated into a single group.
