## To calculate the marginal tax rate, we need the specific tax rates for each income bracket in the tax system. Since you have not provided those details, I will explain how to calculate the marginal tax rate for different tax systems.

1. Proportional Tax System:

In a proportional tax system, also known as a flat tax system, the tax rate remains the same regardless of the level of income.

To calculate the marginal tax rate for the proportional tax system as income rises from $50,000 to $100,000, we need to know the tax rate. Let's assume the tax rate is 25% for this example.

The increase in income is $100,000 - $50,000 = $50,000.

The extra taxes paid on the additional income would be 25% of $50,000 = $12,500.

Therefore, the marginal tax rate is $12,500 / $50,000 = 0.25 or 25%.

Similarly, for the income rise from $100,000 to $200,000, we can use the same tax rate of 25%.

The increase in income is $200,000 - $100,000 = $100,000.

The extra taxes paid on the additional income would be 25% of $100,000 = $25,000.

So, the marginal tax rate is $25,000 / $100,000 = 0.25 or 25%.

2. Regressive Tax System:

In a regressive tax system, the tax rate decreases as income increases. This means that individuals with lower incomes pay a higher percentage of their income in taxes compared to those with higher incomes.

To calculate the marginal tax rate for a regressive tax system, we need to know the specific tax rates for each income bracket. Let's assume the tax rates are as follows:

- For income up to $50,000: 30%

- For income between $50,001 and $100,000: 20%

- For income above $100,000: 10%

Using these tax rates, we can calculate the marginal tax rate for income rises in the following manner:

- For the income rise from $50,000 to $100,000: We assume the tax rate remains constant at 30% throughout this income range, so the marginal tax rate would be 30%.

- For the income rise from $100,000 to $200,000: In this scenario, the individual's income falls under different tax brackets. The first $50,000 would be taxed at 30%, and the remaining $100,000 would be taxed at 20%. Therefore, we can calculate the marginal tax rate by dividing the extra taxes paid on the additional income by the increase in income.

Let's assume the extra taxes paid for the $100,000 increase in income is $30,000 (30% of $50,000) + $20,000 (20% of $100,000) = $50,000.

The increase in income is $200,000 - $100,000 = $100,000.

Thus, the marginal tax rate is $50,000 / $100,000 = 0.5 or 50%.

3. Progressive Tax System:

In a progressive tax system, the tax rate increases as income rises. This means that individuals with higher incomes pay a higher percentage of their income in taxes compared to those with lower incomes.

To calculate the marginal tax rate for a progressive tax system, we need to know the specific tax rates for each income bracket. Let's assume the tax rates are as follows:

- For income up to $50,000: 10%

- For income between $50,001 and $100,000: 20%

- For income above $100,000: 30%

Using these tax rates, we can calculate the marginal tax rate for income rises in a similar method as above.

For the income rises from $50,000 to $100,000 and from $100,000 to $200,000, we assume that the tax rates remain constant at 20% and 30% respectively for the respective income ranges.

Therefore, the marginal tax rates for both scenarios would be the same as the tax rates themselves: 20% and 30% respectively.

Please note that the actual marginal tax rates for each tax system will depend on the specific tax rates and income brackets defined in the tax system.