Lorenz curve

The Lorenz curve is a graphical representation of the inequality of a distribution, usually income or wealth. It is a curved line that plots the cumulative percentage of total income or wealth held against the cumulative number of recipients, starting with the poorest individual or household. The greater the inequality, the further the Lorenz curve will deviate from the line of perfect equality, which is a diagonal line running from the lower left to the upper right of the graph. The area between the Lorenz curve and the line of perfect equality is an indicator of the degree of inequality in the distribution.

Example of lorenz curve

• The most common example of a Lorenz curve is a distribution of income. Suppose that a country has 100 people and each one has an income ranging from $0 to $100,000. The Lorenz curve would plot the cumulative percentage of total income (in this case, 100%) against the cumulative number of people, starting with the poorest person. The curve would show that the poorest person has 0% of the total income, while the wealthiest person has 100% of the total income. The area between the Lorenz curve and the line of perfect equality (the diagonal line running from the lower left to the upper right of the graph) is an indication of the degree of inequality in the distribution.
• Another example of a Lorenz curve is a distribution of wealth. Suppose that a country has 100 people and each one has a net worth ranging from $0 to $10 million. The Lorenz curve would plot the cumulative percentage of total wealth (in this case, 100%) against the cumulative number of people, starting with the poorest person. The curve would show that the poorest person has 0% of the total wealth, while the wealthiest person has 100% of the total wealth. Again, the area between the Lorenz curve and the line of perfect equality is an indication of the degree of inequality in the distribution.
• A third example of a Lorenz curve is a distribution of resources in a population. Suppose that a population of 100 people has access to varying amounts of food, water, and other resources. The Lorenz curve would plot the cumulative percentage of total resources (in this case, 100%) against the cumulative number of people, starting with the least privileged person. The curve would show that the least privileged person has 0% of the total resources, while the most privileged person has 100% of the total resources. As before, the area between the Lorenz curve and the line of perfect equality is an indication of the degree of inequality in the distribution.

Formula of lorenz curve

The Lorenz curve is used to measure the inequality of a distribution, usually income or wealth, using the following formula:

$$L(x) = \frac{\sum_{i=1}^x F_i}{\sum_{i=1}^n F_i}$$

Where L(x) is the Lorenz curve, $$F_i$$ is the ith value of the cumulative frequency distribution, and n is the total number of observations.

The Lorenz curve plots the cumulative percentage of total income or wealth held against the cumulative number of recipients, starting with the poorest individual or household. For example, if the cumulative frequency distribution was [10, 30, 50, 80], then the Lorenz curve would be as follows:

$$L(1) = \frac{10}{140} = 0.0714$$

$$L(2) = \frac{40}{140} = 0.2857$$

$$L(3) = \frac{90}{140} = 0.6429$$

$$L(4) = \frac{140}{140} = 1.0000$$

The area between the Lorenz curve and the line of perfect equality is an indicator of the degree of inequality in the distribution. The greater the inequality, the further the Lorenz curve will deviate from the line of perfect equality, which is a diagonal line running from the lower left to the upper right of the graph.

Types of lorenz curve

The Lorenz curve is a graphical representation of the inequality of a distribution, usually income or wealth, and there are several types of Lorenz curves. These include:

• The Standard Lorenz Curve: This is the most commonly used type of Lorenz curve, which plots the cumulative percentage of total income or wealth held against the cumulative number of recipients, starting with the poorest individual or household.
• The Equality Line: This is a diagonal line that runs from the lower left to the upper right of the graph and shows the perfect equality of a distribution.
• The Gini Curve: This type of Lorenz curve uses the Gini coefficient to measure the inequality of a distribution.
• The Concentration Curve: This type of Lorenz curve plots the cumulative percentage of total income or wealth held against the cumulative number of recipients, but it also considers the concentration of wealth among the different income or wealth groups.
• The Relative Lorenz Curve: This type of Lorenz curve plots the cumulative percentage of total income or wealth held against the cumulative number of recipients, but it takes into account the relative wealth or income of each group.

Advantages of lorenz curve

The Lorenz curve is a useful tool for understanding the degree of inequality in the distribution of wealth or income. It is a visual representation of the gap between the richest and poorest members of society, providing an insight into the level of inequality in the distribution. Below are some of the advantages of the Lorenz curve:

• It is a simple and easy to understand way of visualizing the inequality in a distribution.
• It can be used to compare distributions over time, making it useful for observing changes in the level of inequality in a given population.
• It can be used to compare distributions across different countries or regions, making it an effective way of measuring global inequality.
• It can be used to identify policies and strategies that could be used to reduce inequality.
• It can be used to measure the impact of economic policies and public programs on inequality.

Limitations of lorenz curve

The Lorenz curve is a useful tool for visualizing the degree of inequality in a given distribution, but it has some limitations. These include:

• The Lorenz curve does not take into account the number of individuals or households in the distribution. It only looks at the total income or wealth, not the individual or household amounts. This means that the Lorenz curve may not accurately represent the true extent of inequality in a given distribution.
• The Lorenz curve does not account for the varying levels of need of individuals or households. Because the Lorenz curve is based purely on the total income or wealth of the population, it does not take into account the needs of different individuals or households. This means that the Lorenz curve may not accurately reflect the relative degree of inequality in a given distribution.
• The Lorenz curve does not account for the different levels of taxation or government welfare benefits. As the Lorenz curve does not account for government policies such as taxation and welfare benefits, it does not accurately reflect the true extent of inequality in a given distribution.
• The Lorenz curve does not take into account the time period of the distribution. Since the Lorenz curve does not take into account the time period of the distribution, it may not accurately represent the changing levels of inequality over time.

Other approaches related to lorenz curve

In addition to the Lorenz curve, there are several other approaches for measuring inequality in a distribution. These include:

• The Gini coefficient: This is a measure of inequality that is derived from the Lorenz curve, and is calculated by dividing the area between the Lorenz curve and the line of perfect equality by the total area of the graph.
• The Atkinson index: This measure of inequality is based on the idea that the utility of an additional unit of income is higher for the poorer members of society than for the richer members.
• The Theil index: This measure of inequality is based on the idea that the gains from economic growth are distributed unequally, with a higher proportion of the gains going to the richer members of society.
• The Hoover index: This measure of inequality is based on the idea that the inequality increases when the income of the richer members of society increases more than the income of the poorer members.

In summary, there are several approaches for measuring inequality in a distribution, including the Gini coefficient, the Atkinson index, the Theil index, and the Hoover index. All of these measures are based on different assumptions regarding the distribution of economic gains and losses, and are useful for measuring the degree of inequality in a given distribution.

Lorenz curve — recommended articles

Poverty gap — Harmonic mean — Percentage point — Total factor productivity — Index of diversity — Corruption perception index — GDP deflator — Confidence level — Average annual growth rate

References

• Gastwirth, J. L. (1972). The estimation of the Lorenz curve and Gini index. The review of economics and statistics, 306-316.
• Kakwani, N. C. (1977). Applications of Lorenz curves in economic analysis. Econometrica: Journal of the Econometric Society, 719-727.