Question

Calculate the range of possible Gini coefficients under the following conditions:

1) Median income equals average income,

2) Median income equals maximal income,

3) Average income equals maximal income.

Answer 1

The Gini coefficient is a measure of inequality of a distribution. It is defined as a
ratio with values between 0 and 1: the numerator is the area between the Lorenz curve
of the distribution and the uniform distribution line; the denominator is the area under
the uniform distribution line. It was developed by the Italian statistician Corrado Gini
and published in his 1912 paper "Variabilità e mutabilità" ("Variability and
Mutability"). The Gini index is the Gini coefficient expressed as a percentage, and is
equal to the Gini coefficient multiplied by 100. (The Gini coefficient is equal to half
of the relative mean difference.)
The Gini coefficient is often used to measure income inequality. Here, 0
corresponds to perfect income equality (i.e. everyone has the same income) and 1
corresponds to perfect income inequality (i.e. one person has all the income, while
everyone else has zero income).
The Gini coefficient can also be used to measure wealth inequality. This use
requires that no one has a negative net wealth. It is also commonly used for the
measurement of discriminatory power of rating systems in the credit risk
management.
Calculation
The Gini coefficient is defined as a ratio of the areas on the Lorenz curve
diagram. If the area between the line of perfect equality and Lorenz curve is A, and
the area under the Lorenz curve is B, then the Gini coefficient is A/(A+B). Since A+B
= 0.5, the Gini coefficient, G = 2A = 1-2B. If the Lorenz curve is represented by the
function Y = L(X), the value of B can be found with integration and:
In some cases, this equation can be applied to calculate the Gini coefficient
without direct reference to the Lorenz curve. For example:
• For a population with values yi, i = 1 to n, that are indexed in non-decreasing
order ( yi ≤ yi+1):
• For a discrete probability function f(y), where yi, i = 1 to n, are the points with
nonzero probabilities and which are indexed in increasing order ( yi < yi+1):
where:
and
• For a cumulative distribution function F(y) that is piecewise differentiable, has
a mean μ, and is zero for all negative values of y:
Since the Gini coefficient is half the relative mean difference, it can also be
calculated using formulas for the relative mean difference.
For a random sample S consisting of values yi, i = 1 to n, that are indexed in
non-decreasing order ( yi ≤ yi+1), the statistic: