## Sunday, 14 August 2016

### A torturous average vs. marginal tax rates example

In ECON110 last week, we ran out of time to go fully through an example on the difference between average and marginal tax rates, and the difference between progressive and regressive taxes. Fortunately, in the last week Jodi Beggs at Economists Do It With Models has just written a couple of posts on taxes (see here and here). So, I'm going to borrow from her second post to illustrate.

Jodi outlines an income tax regime that works as follows:
• 10% for income \$0 to \$9,275
• 15% for income \$9,275 to \$37,650
• 25% for income \$37,650 to \$91,150
• 28% for income \$91,150 to \$190,150
• 33% for income \$190,150 to \$413,350
• 35% for income \$413,350 to \$415,050
• 39.6% for income \$415,050+
Now, let's think about three taxpayers. Taxpayer 1 has income of \$30,000; Taxpayer 2 has income of \$90,000; and Taxpayer 3 has income of \$450,000. What are their average and marginal tax rates, and why does it matter?

The average tax rate is just the proportion of income that is paid in tax. That can be calculated as [Tax Paid]/[Income]. When the tax system is not straightforward then calculating the tax paid can take multiple steps (see below). The marginal tax rate is just the proportion of the next dollar earned that would be paid in tax. The difference between the two rates is important, and people don't always appreciate why. It's the marginal tax rate that matters for decision-making, since we make decisions at the margin. If we are thinking about whether to work another hour or not, the tax we pay on that next hour of wages is the relevant tax rate. However, as Jodi notes:
...while you mainly want to keep your marginal tax rate in mind for decision-making purposes, I guess it could make you feel better to calculate your average tax rate and be reminded that the federal government isn’t taking all of your money in income taxes.
Back to our example. The tax paid by Taxpayer 1 (on their income of \$30,000) is \$4,036.25. They pay \$927.50 on their first \$9,275 of income (\$9,275 * 10%), and then \$3,108.75 on the remaining \$20,725 of income between \$9,275 and \$30,000 (\$20,725 * 15%). The average tax rate for Taxpayer 1 is 13.45% (\$4,036.25 / \$30,000). Their marginal tax rate is 15% (if they earned one more dollar, that's the rate of tax they would pay on that dollar).

The tax paid by Taxpayer 2 (on their income of \$90,000) is \$18,271.25. They pay \$927.50 on their first \$9,275 of income (\$9,275 * 10%), then \$4,256.25 on the next \$28,375 of income up to \$37,650 (\$28,375 * 15%), and then \$13,087.50 on the remaining \$52,350 of income between \$37,650 and \$90,000 (\$52,350 * 25%). The average tax rate for Taxpayer 2 is 20.30% (\$18,271.25 / \$90,000). Their marginal tax rate is 25%.

Finally, the tax paid by Taxpayer 3 (on their income of \$450,000) is \$134,370. They pay \$927.50 on their first \$9,275 of income (\$9,275 * 10%), then \$4,256.25 on the next \$28,375 of income up to \$37,650 (\$28,375 * 15%), then \$13,375 on the next \$53,500 of income up to \$91,150 (\$53,500 * 25%), then \$27,720 on the next \$99,000 of income up to \$190,150 (\$99,000 * 28%), then \$73,656 on the next \$223,200 of income up to \$413,350 (\$223,200 * 33%), then \$595 on the next \$1,700 of income up to \$415,050 (\$1,700 * 35%), and then \$13,840.20 on the remaining \$34,950 of income between \$415,050  and \$450,000 (\$34,950 * 39.6%). The average tax rate for Taxpayer 3 is 29.86% (\$134,370 / \$450,000). Their marginal tax rate is 39.6%.

Phew! You might think that the above example was difficult. That's why we have tax software to do the work. But this example is relatively straightforward when you compare with the complex system of rebates and tax deductions that most tax systems include (in New Zealand we have low income rebates, Working for Families credits, as well as social security payments and accommodation supplements, all of which decrease as more income is earned). Once we factor in decreases in government transfers, rebates, and entitlements, we are calculating what we call the effective marginal tax rate.

Notice that in the case of all three taxpayers the marginal tax rate is greater than the average tax rate. That characterises an income tax system that is progressive. This is a tax system where, as incomes increase, the proportion of the income paid in tax increases. In contrast, for a regressive tax system the marginal tax rate is less than the average tax rate (and higher incomes are associated with a lower proportion of income paid in tax), while for a proportional tax system the marginal and average tax rates are the same (and the proportion of income paid in tax is the same at all levels of income).