May 16, 2014

# Averages, percentages, nets, and GST

Our only Prime Minister, on Radio NZ

“I utterly reject those propositions. Twelve percent of households pay 76 percent of all net tax in New Zealand,” he said.

I’ve written about “net tax” before, both on StatsChat and elsewhere. It has to be defined and analysed carefully and non-intuitively in order to get these sorts of results.

Suppose we had an imaginary population divided into three groups. The ‘Low’ group, of 10 people, each pay \$1000 in income tax, \$1000 in GST, and receive \$1500 in cash benefits. The “Middle” group, of 5 3 people, each pays \$4000 in income tax, \$3000 in GST, and receives no cash in benefits. The one person in the “High” group pays \$17000 in income tax, \$8000 in GST, and receives no cash in benefits.

According to Mr Key’s definition, the high-income group pays 71% of the “net tax”.  The middle-income group pays 50% of the “net tax”, and the low-income group pays -21% of the “net tax”.  That’s even though every person in this imaginary population pays more in tax than they receive in cash benefits.

There are three strange things going on here. The first is that GST is ignored. That’s obviously just wrong — GST is just as real as income tax.  The second is that cash benefits are treated differently from all other categories of government expenditure, even other categories such as subsidised medications that provide a direct, quantifiable individual benefit.  The third is that percentages behave strangely when you have a mixture of negative and positive numbers.  It’s quite possible, by choosing the subsets of the NZ population correctly, to find a group that pays well over 100% of the “net tax”.

Percentages become a lot less useful when they aren’t bounded by 100, and people who want to communicate accurately should avoid them in that situation.  And if you want to distinguish income tax revenue from GST revenue, you should clearly explain what you’re doing and why.

Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient See all posts by Thomas Lumley »

• Megan Pledger

The actual proportion of people paying 76% of the taxes is only 12% at this time but over a life time a goodly proportion of the population would have been in that group (and in the others) as they transition from school, to uni, to new hire, to middle management equivalent, to upper management equivalent, to retiree.

And putting it like that makes it seem less one-sided i.e. only a small proportion of people at a time pay a lot of tax but a large proportion of people have or will pay a lot of tax at some time in their life.

• Thomas Lumley

Indeed, but that would be true even for a reasonable definition of net tax.

There’s a real problem with a definition of net tax that can have two non-overlapping subsets of the population both paying more than 70% of the net tax.

• DAVID Harrison

But the other thing that has been overlooked is that the low paid people spend all their money. Therefore they pay a larger proportion of GST.

The highly paid usually invest or bank a good proportion of their income, therefore they get Interest on their savings, and pay tax on that. But they pay a far smaller proportion of GST than the low paid.

• Thomas Lumley

That hasn’t been overlooked, it’s the whole point. However, since rents, mortages, and houses are not taxed, and low-paid people in Auckland or ChCh often pay an unreasonable fraction of their income on housing, it’s not obvious exactly how the GST burden is spread.