Tax cuts, and misinterpreted average and marginal tax rates

Many, many people don't understand the difference between average and marginal tax rates. Articles like this one by Leicester Gouwland don't help:

The bulk of the default tax collected is from people who have moved from the 17.5 per cent bracket to the 30 per cent bracket, that is a 71 per cent increase in tax.

The people who have moved from the 10.5 per cent bracket to the 17.5 per cent have suffered a 66 per cent increase in tax, however, if they now earn more than $24,000, they qualify for the independent earner rebate.

Neither of those two statements is correct. To see why, let's take a step back first. The average (income) tax rate for a taxpayer is the tax that they pay divided by their income. So, if a taxpayer have income of $50,000 and pay $8,020 in tax, their average tax rate is 16.04% (8,020 / 50,000). The marginal tax rate is the proportion of the next dollar they earn that would be paid in tax.

Most income tax systems are described in terms of marginal tax rates, as Gouwland does early in his article for New Zealand:

The current tax brackets on personal income are; 10.5 per cent for income up to $14,000; 17.5 per cent for income up to $48,000; 30 per cent for income up to $70,000; and then 33 per cent for any income more than $70,000.

So, for our taxpayer that has income of $50,000, their marginal tax rate (the tax rate they would pay on the next dollar they earn) is 30%, even though their average tax rate is only 16.04%.

Now, this is where most people go wrong. A person who moves tax brackets doesn't face a huge increase in their average tax rate, only their marginal tax rate. It's not correct to say that, for someone who moves "from the 17.5 per cent bracket to the 30 per cent, that is a 71 per cent increase in tax". To illustrate, let's take a taxpayer who was previously earning $45,000 (in the 17.5 per cent tax bracket) and give them a pay rise to $50,000 (in the 30 per cent tax bracket). Does their tax payment go up by 71% (as Gouwland suggests)? Hell no. It only goes up by 16.3%, from $6,895 to $8,020. [*] That's still a big increase, but it's certainly not 71%! And remember that their before-tax income has gone up by 11.1% as well.

Similarly, has someone who has moved "from the 10.5 per cent bracket to the 17.5 per cent ...suffered a 66 per cent increase in tax"? Let's take a taxpayer who was previous earning $12,000 (in the 10.5 per cent tax bracket), and increase their income to $16,000 (in the 17.5 per cent tax bracket). Their tax payment goes from $1,260 to $1,820, an increase of 44.4% (but in the context of a 33.3% increase in pre-tax income).

So, more care is needed in interpreting average and marginal tax rates, and articles like Gouwland's certainly don't help.

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[*] The tax paid by a taxpayer with an income of $45,000 is calculated as [$14,000 x 0.105] + [($45,000 - $14,000) x 0.175] = $6,895. The tax paid by a taxpayer with an income of $50,000 is calculated as [$14,000 x 0.105] + [($48,000 - $14,000) x 0.175] + [($50,000 - $48,000) x 0.3] = $8,020.

[**] The tax paid by a taxpayer with an income of $12,000 is calculated as [$12,000 x 0.105] = $1,260. The tax paid by a taxpayer with an income of $16,000 is calculated as [$14,000 x 0.105] + [($16,000 - $14,000) x 0.175] = $1,820.