Measuring inequalities in income is a crucial aspect of economic analysis. Several statistical tools and ratios have been developed for this purpose, each offering a different perspective on how income is distributed within a population. The question asks to identify which of the listed options are used to measure income inequalities.

Understanding Measures of Income Inequality

Let's examine each option provided:

1. Lorenz Ratio: This is closely related to the Gini coefficient. The Lorenz curve is a graphical representation of income distribution, plotting the cumulative percentage of total income against the cumulative percentage of households or individuals. The Lorenz Ratio, or Gini coefficient, is calculated from the area between the line of perfect equality (a 45-degree line where each percentage of the population earns the same percentage of income) and the actual Lorenz curve. A higher Gini coefficient (or Lorenz Ratio) indicates greater income inequality. The formula for the Gini coefficient (G) based on the area is often given conceptually as $G = \frac{A}{A+B}$, where A is the area between the line of perfect equality and the Lorenz curve, and B is the area under the Lorenz curve.
2. Theil's Index: This is another measure of income inequality. It belongs to the family of entropy indices and can be used to measure overall inequality as well as decompose inequality into different subgroups (e.g., inequality within regions vs. inequality between regions). Theil's Index is more sensitive to income differences at the extremes of the distribution compared to the Gini coefficient. There are two forms, one based on the logarithm of the ratio of individual income to mean income, and another based on the logarithm of the ratio of mean income to individual income. A common form is $T = \frac{1}{N} \sum_{i=1}^{N} \left(\frac{y_i}{\bar{y}}\right) \ln\left(\frac{y_i}{\bar{y}}\right)$, where $y_i$ is the income of individual $i$, $\bar{y}$ is the mean income, and $N$ is the population size.
3. Palma Ratio: This ratio specifically focuses on the income share of the richest 10% compared to the poorest 40% of the population. It is defined as the ratio of the income share of the top 10% to the income share of the bottom 40%. It highlights the inequality between the top earners and a significant portion of the lower-income group, often considered to be a more politically relevant measure of inequality than summary statistics like the Gini coefficient which average inequality across the entire distribution.
4. Deprivation Ratio: This measure is typically used in the context of poverty and well-being, not direct income inequality. A deprivation index or ratio measures the extent to which individuals or households lack essential goods, services, or living conditions necessary for a basic standard of living. While income level is a significant factor contributing to deprivation, the deprivation ratio itself measures the outcome (lack of essentials) rather than the distribution of income across the entire population. It's more about absolute or relative poverty and lack of resources than the overall shape of the income distribution curve.

Analyzing the Options for Measuring Income Inequality

Based on the descriptions:

• Lorenz Ratio (or Gini coefficient) is a standard measure of overall income inequality.
• Theil's Index is a standard measure of income inequality, sensitive to different parts of the distribution.
• Palma Ratio is a specific measure focusing on the gap between the top rich and the bottom poor/middle class based on income shares.
• Deprivation Ratio measures lack of essentials and is related to poverty and well-being, rather than the overall income distribution inequality itself.

Therefore, the measures used to quantify income inequalities among the given options are Lorenz Ratio, Theil's Index, and Palma Ratio.

Considering the options provided: A (Lorenz Ratio), B (Theil's Index), and C (Palma Ratio) are recognized measures for assessing income inequalities. Deprivation Ratio (D) is not typically classified as a measure of income inequality across the entire distribution.

Revision Table: Key Income Inequality Measures

Additional Information on Inequality and Deprivation

While income inequality measures focus on how income is distributed across a population, deprivation measures look at whether people have access to basic necessities like housing, food, healthcare, and education. Income can be a determinant of deprivation, but deprivation is a broader concept related to living standards and access to resources, often measured by looking at multiple indicators beyond just income level. Understanding both income distribution and deprivation is important for a complete picture of social welfare and inequality in a society.