Income Distribution: Gini Coefficient and Lorenz Curve
This section explores how income is distributed within countries at different levels of development. We will focus on two key measures: the Gini coefficient and the Lorenz curve. These tools provide insights into income inequality.
The Lorenz Curve
The Lorenz curve is a graphical representation of income distribution. It plots the cumulative percentage of total income earned against the cumulative percentage of the population.
Suggested diagram: A Lorenz curve showing a perfectly equal distribution (45-degree line) and various levels of inequality.
Key features of the Lorenz curve:
The curve starts at the bottom left, representing the poorest segment of the population.
A perfectly equal distribution would result in a straight diagonal line (the 45-degree line) where the percentage of the population equals the percentage of total income.
The further the Lorenz curve bows away from the 45-degree line, the greater the income inequality.
The Gini Coefficient
The Gini coefficient is a numerical measure of income inequality. It represents the area between the Lorenz curve and the line of perfect equality (the 45-degree line), expressed as a percentage.
Formula:
$$G = \frac{\int_0^P A(x) dx}{\int_0^P x dx} \times 100$$
Where:
G is the Gini coefficient.
A(x) is the cumulative distribution function of income.
P is the total income of the population.
Interpretation of Gini coefficients:
Gini Coefficient
Interpretation
0
Perfect equality (everyone has the same income)
1
Perfect inequality (one person has all the income)
0 - 0.2
Low inequality
0.2 - 0.4
Moderate inequality
0.4 - 0.6
High inequality
0.6 - 0.8
Very high inequality
0.8 - 1
Extremely high inequality
Calculating the Gini Coefficient (Simplified Example)
While the integral formula is the theoretical definition, we can use simplified methods for estimation, especially with data sets.
Example: Consider a population of 5 people with the following incomes:
Person 1: $10,000
Person 2: $15,000
Person 3: $20,000
Person 4: $30,000
Person 5: $100,000
Steps:
Calculate cumulative percentages of income:
Person
Income
Cumulative Income
Cumulative Percentage of Total Income
1
$10,000
$10,000
$10,000 / $125,000 = 8%
2
$15,000
$25,000
$25,000 / $125,000 = 20%
3
$20,000
$45,000
$45,000 / $125,000 = 36%
4
$30,000
$75,000
$75,000 / $125,000 = 60%
5
$100,000
$175,000
$175,000 / $125,000 = 140%
Calculate cumulative percentages of population:
Person
Population
Cumulative Population
Cumulative Percentage of Total Population
1
1
1
1 / 5 = 20%
2
1
2
2 / 5 = 40%
3
1
3
3 / 5 = 60%
4
1
4
4 / 5 = 80%
5
1
5
5 / 5 = 100%
Plot the Lorenz curve: Plot the cumulative percentage of income against the cumulative percentage of the population.
Calculate the area between the Lorenz curve and the line of perfect equality: This area represents the inequality.
Calculate the Gini coefficient: $$G = \frac{Area \ between \ Lorenz \ curve \ and \ line \ of \ equality}{Area \ under \ the \ line \ of \ equality} \times 100$$
Income Distribution and Development Levels
Income distribution patterns vary significantly across countries at different levels of development:
Low-income countries: Often exhibit high levels of income inequality, with a small percentage of the population controlling a large share of the wealth. This can be due to historical factors, unequal access to resources, and limited opportunities.
Middle-income countries: May show a gradual reduction in inequality as economies develop and social safety nets are strengthened. However, disparities can still be significant.
High-income countries: Generally have lower levels of income inequality compared to low- and middle-income countries, although inequality can be increasing in some developed nations due to factors like globalization and technological change.
Understanding income distribution is crucial for analyzing economic development, social welfare, and policy interventions aimed at reducing poverty and promoting greater equality.