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A progressive tax system is a method of taxation where the tax rate increases as the taxable income increases. This means that individuals with higher income levels pay a larger percentage of their income in taxes compared to those with lower income levels. This system is often embraced because it is deemed fairer; as income rises, the ability to pay taxes also increases, and thus the tax burden is distributed more equitably among taxpayers.

In the given exercise, the tax system is shown to be progressive because as income levels increase, the average tax rate also increases. For example, when income is £5000, the average tax rate is 18%. However, when the income rises to £50000, the average tax rate jumps to 28.8%. This progressive system ensures that those who earn more contribute a greater portion of their earnings to the tax pool.

The exercise also investigates what happens when the tax exemption threshold is raised from £2000 to £5000. Even with this adjustment, the system remains progressive as higher incomes continue to experience increasing average tax rates. However, the rate of increase between income levels becomes smaller, suggesting the system is less progressive, but still maintains its progressive nature.

The average tax rate is a critical concept in understanding how taxpayers are impacted by tax policies. It is calculated by taking the total amount of tax paid and dividing it by the total income. This ratio represents the overall tax burden as a percentage of income, providing insight into the impact of the tax system on different income groups.

For example, in the exercise, at a £5000 income level, the tax paid is £900, resulting in an average tax rate of 18%. At a £10000 income level, the tax paid is £2400, leading to an average tax rate of 24%. For a £50000 income, the tax paid is £14400, and the average tax rate is 28.8%. These calculations show how the average tax rate rises with income, aligning with the principles of a progressive tax system.

When the exemption threshold is increased to £5000, the average tax rates adjust accordingly. At a £5000 income, no tax is paid, hence the average tax rate drops to 0%. For a £10000 income, the rate lowers to 15%, and for £50000 income, it reduces to 27%. Despite these decreases, the average tax rate still increases with higher incomes, maintaining the system's progressive nature, albeit at a slower progression.

Taxable income refers to the portion of an individual's total income that is subject to taxation after deducting allowable exemptions, deductions, and other adjustments. Understanding taxable income is crucial for accurately determining tax liabilities.

In the exercise, the calculation of taxable income plays a pivotal role. Initially, for an income of £5000, only amounts exceeding the £2000 exemption are taxed, resulting in a taxable income of £3000. For a £10000 income, the taxable portion is £8000, and for £50000, it is £48000.

When the exemption threshold is increased to £5000, the taxable income decreases, altering the tax calculations. With this new exemption level, someone earning £5000 pays no tax as their entire income falls below the taxable limit. For a £10000 income, the taxable income shrinks to £5000, and for a £50000 income, it falls to £45000. These adjustments demonstrate how changes in exemptions can significantly impact the amount of taxable income and thus the overall tax liability, influencing the fairness and progressiveness of the tax system.