Income tax from monopoly
Income tax is not always easy to calculate

Slopes of tax brackets

Income tax plays a big role in all of our lives, once we get out into the working world.

In this step we will have a look at how many modern western countries organize their income tax bracket levels. It might be quite interesting to find out how different countries do this.

Australian bank notes in wallet© “Australian bank notes in wallet” by Martin Kingsley/Wikimedia Commons CC BY 2.0

The bracketed nature of income tax

Income Tax is money that you pay the government, and generally the amount you pay is roughly corresponding to how much you make — with some important qualifications.

In most countries Income Tax only kicks in once a person’s income gets to a certain point, and at first has a fixed percentage rate for additional monies earned after that initial point. But then at some higher income, you move into a higher tax bracket, where the slope of the tax/income line increases.

This is a prime example of how important it is for ordinary people to have a basic understanding of linear relationships, and how the geometrical notion of a slope has a tangible, direct and crucial relevance to all of us in our daily lives.

The current story for Australia:

The following rates for 2014-15 apply from 1 July 2014 in Australia. Here is a graph showing yearly gross income on the horizontal axis, and income tax on the vertical axis. Let’s mark the brackets , , etc., with corresponding lines , , etc.

Graph of income tax versus gross income

Use the graph to estimate the following questions about the line segments involved.

Q1 (M): What are your guesses for the slopes of the line segments , and ? What are the units that these slopes are measured in?

Q2 (M): What would be your guess for the equation of the line (assuming we extended it)?

Q3 (E): What would be the -intercept of the line segment if we extended it to a line?

Taxable income Tax on this income
0 – $18200 Nil
$18201 – $37000 19c for each $1 over $18200
$37001 – $80000 $3572 plus 32.5c for each $1 over $37000
$80001 – $180000 $17547 plus 37c for each $1 over $80000
$180001 and over $54547 plus 45c for each $1 over $180000

Interpreting this table

Please have a good look at the above table with a view of understanding precisely what it says. The table in fact has some slopes hidden inside it. Can you see what they are?

For an income less than $18K (here K is a convenient abbreviation for one thousand), there is no tax, so the line has slope 0. Between an income of $18.2K and $37K, the table is telling us that the base tax payable is $0.19 for every dollar in Income. This is exactly a slope, since it is telling us that with every increase by $1 in the horizontal direction, there is an increase of $0.19 in the vertical direction. So in this interval, the line has slope

For the maximum income in this bracket, namely $37K, the tax payable is thus . So that number $3572 is not arbitrarily made up–it is a direct consequence of the tax rate in this income bracket.

Q4 (M): What would be the difference between earning $80K and $100K in Australia, after tax?

Tax brackets and rates around the world?

It is a very interesting question to ask about income brackets and corresponding tax rates around the world. So here perhaps participants from different countries around the world might be able to share with us: what are the lowest and highest tax rates in your country for individual income taxes? We would expect socialist inclined countries like Sweden to have quite steep tax rates for higher income earners. Other countries with more capitalist orientations will have a flatter tax structure. Where does your country fit in?

Answers

A1. Here are the exact values, which we can determine from the table. The slope of is , that of is , that of is , and that of is . Did you get something that was close to these values? In each case, the slopes are unit-less, since they are obtained as ratios of dollars to dollars, so that the dollars cancel out.

A2. The line passes through the points and and we know it has slope . So its precise equation is

or

or

Well done if you were able to get something close to this, just from the graph.

A3. The -intercept for line is .

A4. The difference in the after tax incomes is . The reason is that the extra attracts a tax rate of which amounts to .

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Maths for Humans: Linear and Quadratic Relations

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