5.3 An Instrument Based on Expected Tax Rate
Given a distribution of income for each individual, conditional on income in the two years preceding the tax change, it is possible to calculate an expected tax rate. This will not generally correspond to a statutory marginal rate in the multi-rate structure. As before, let yj,i denote individual i's income at time j, and let μj denote arithmetic mean log income at time, j. The process of relative income change is the same as described in (7) above, which allows for serial correlation and regression towards the mean in the process of relative income change. Rearranging this equation gives:
Assuming that E(uj,i ) = 0 and V (uj,i ) = σu2 are the constant mean and variance of uj,i for all j, taking expectations gives:
and the variance of logarithms of conditional income is:
As explained earlier, in the context of the tax change in New Zealand, it is necessary to obtain values relating to 2002, given incomes in 1999 and 1998. Hence, moving forward one year gives:
with a variance of logarithms of:
Finally, moving a further year forward gives:
with a conditional variance of logarithms of:
These last two expressions can be used to give the mean and variance of log-income in 2002 conditional on income in 1999 and 1998. The variance is of course the same for each individual.
The expected tax rate for the individual in period j + 2, given a set of tax thresholds and rates, is obtained as follows. Suppose the income tax function has rates tk for k = 1,...,K applying between income thresholds ak and ak+1 where a1 = 1 and aK+1 = ∞. First let E (log yj+2,i | yj-1,i yj-2,i = μj+2,i and V(log yj+2,i|yj-1,i,yj-2,i) = σj+22, with:
On the assumption that the u are normally distributed, log-income is normally distributed and the probability that the individual falls into the kth bracket is:
where N (h|0,1) is the area to the left of h of a standard normal distribution. Here N(ξj+2,K+1,i|0,1) = 1 and N(ξj+2,1,i|0,1) = 0.
The expected tax rate for the individual, E(τj+2,i) is thus:
This gives the expected tax rate instrument, τi*[E (τ2,i)] = E (τj+2,i), for each individual.