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## UNSW Sydney

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” 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 $${\normalsize x}$$ axis, and income tax on the vertical $${\normalsize y}$$ axis. Let’s mark the brackets $$\normalsize{B1}$$, $$\normalsize{B2}$$, $$\normalsize{B3}$$ etc., with corresponding lines $$\normalsize{l1}$$, $$\normalsize{l2}$$, $$\normalsize{l3}$$ etc.

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 $$\normalsize{l_2}$$, $$\normalsize{l_3}$$ and $$\normalsize{l_4}$$? What are the units that these slopes are measured in?

Q2 (M): What would be your guess for the equation of the line $$\normalsize{l_2}$$ (assuming we extended it)?

Q3 (E): What would be the $$\normalsize{y}$$-intercept of the line segment $$\normalsize{l_2}$$ 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

## 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?

A1. Here are the exact values, which we can determine from the table. The slope of $$\normalsize{l_2}$$ is $$\normalsize{0.19}$$, that of $$\normalsize{l_3}$$ is $$\normalsize{0.325}$$, that of $$\normalsize{l_4}$$ is $$\normalsize{0.37}$$, and that of $$\normalsize{l_5}$$ is $$\normalsize{0.45}$$. 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 $$\normalsize{l_2}$$ passes through the points $$\normalsize{[18200,0]}$$ and $$\normalsize{[37000, 3572]}$$ and we know it has slope $$\normalsize{s=0.19}$$. So its precise equation is

$\Large{(y-0)/(x-18200)=0.19}$

or

$\Large{y=0.19(x-18200)}$

or

$\Large{y=0.19x-3458}.$

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

A3. The $${\normalsize y}$$-intercept for line $$\normalsize{l_2}$$ is $$-0.19 \times 18200 = -3458$$.

A4. The difference in the after tax incomes is $$\normalsize{\12600}$$. The reason is that the extra $$\normalsize{\20000}$$ attracts a tax rate of $$\normalsize{37\%},$$ which amounts to $$\normalsize{\7400}$$.