Tax Progressivity Notes

Forms of Progressivity

Tax Progressivity

Tax progressivity relates the tax to the tax base, $T = f(B)$, where:

• $T$ is the tax.

• $B$ is the tax base.

The relationship can be progressive, proportional, or regressive.

• Average tax rate: The ratio between the tax and the tax base.

• Marginal tax rate: The ratio between the change in tax and the change in the tax base.

Tax System Classifications

• Progressive: Marginal rate > Average rate

• Proportional: Marginal rate = Average rate

• Regressive: Marginal rate < Average rate

Types of Progressivity

The forms of progressivity are:

• Constant marginal rate

• Increasing marginal rate

Constant Marginal Rate Progressivity

Achieved through:

• Deduction from the tax base.

• Tax credit (detraction from the tax).

Progressivity by Deduction

Involves a fixed deduction from the tax base with a single legal tax rate.

Example: $Ty = 0.3(Y - 100)$

• Marginal tax rate = 0.3

• Average tax rate = $0.3 - \frac{30}{Y}$

The minimum taxable income is 100.

Progressivity by Detraction

A fixed amount is subtracted from the total tax amount after applying the legal tax rate to the entire tax base. Example: $Ty = 0.3Y - 30$

Increasing Marginal Rate Progressivity

Two main forms:

• Continuous progressivity

• Progressivity by brackets

Continuous Progressivity

The tax function's form is directly defined. If $f( . )$ is strictly convex, the marginal rate increases from left to right along the curve.

Progressivity by Brackets

m levels of taxable income determine m+1 brackets, each with corresponding increasing rates.

Example:

• Up to 1000: 10%

• Over 1000 up to 3000: 20%

• Over 3000: 30%

For an income of 4000:

• $Ty = 0.1 * 1000 + 0.2 * 2000 + 0.3 * 1000 = 800$

• Average tax rate = $\frac{800}{4000} = 0.2$

• Marginal tax rate = 0.3

Local Measures of Progressivity

• Tax liability elasticity: The ratio between the % change in tax and the % change in taxable income.

• Net income elasticity: The ratio between the % change in net income and the % change in taxable income.

Global Measures of Progressivity

Evaluate tax progressivity across the entire income distribution using summary indicators:

• Gini Index (G): Measures inequality (0 = complete equality, 1 = complete inequality).

• Overall Redistribution Index (R): $R = G<em>{pre} - G</em>{post}$. Higher R indicates greater redistribution.

• Reynolds-Smolensky Index: Measures the overall redistributive effect of a tax.

• Kakwani Index: Measures the global progressivity of a tax. K = 0 if the tax is proportional. Higher values indicate greater progressivity.