Question

Consider 3 individuals with incomes equal to 10,000, 20,000 and 30,000 euros.

a) They face an income tax rate equal to 10%. Compute the Gini coefficients for the income distribution before taxes and the income distribution after taxes.

b) Assume that a deduction from the tax base (taxable income) of 4,000 euros is introduced. Compute after tax incomes and the corresponding Gini coefficient.

c) Suppose that instead a proportional tax rate we have an income tax with increasing marginal rate such that:

Base Tax Rate

[0 , 10,000) 10%

[10,000 , 20,000)   20%

> 20,000 30%

Find the after tax Gini coefficient.

Answer 1

Gini Coefficient is a statistical measure of inequality in a given income distribution. It varies between 0 to 1. A higher value implies a higher inequality in income distribution and vice versa.

The Gini Coefficient is calculated in the following manner: The sum of scores in each income distribution is added and subtracted from 1. The score for each income distribution is arrived at by using the formula:

Score = Fraction of Income * (Fraction of Population + 2 * Fraction of Population that is richer)

(a) Here, total income of all earners is 60,000. Because there are only 3 individuals, fraction of population can be taken as 1/3 in all cases.

Fraction of Income (A)	Fraction of Population (B)	Fraction of Population that earns more (C)	Score = A*[B+(2*C)]
10000/60000 = 1/6	1/3	2/3	5/18
20000/60000 = 1/3	1/3	1/3	1/3
30000/60000 = 1/2	1/3	0	1/6

Gini Coefficient = 0.22

When all incomes are taxed at 10%, then total income becomes 54,000. Thus, new values in the table are as follows:

Gini Coefficient = 0.22

(b) When tax of 10% is applied by removing 4000 euros from every income, then total income is 55200.

Gini Coefficient = 0.22

(c) When taxes are not progressive, then total income is 46,000.

Gini Coefficient = 0.174

Thus we can verify that progressive taxes reduces inequality in the population.

Thanks!