AnaBel Dawson

Discovering if a particular resource is distributed evenly amongst members of a population is a very important political and economic question. For instance, a country may want to know if its distribution of wealth is becoming more or less equitable over time. Additionally, economists may want to measure which country has the most equitable income distribution. How would we answer these questions? Well… we can use our knowledge of integrals and the Fundamental Theorem of Calculus!

For starters, let represent the fraction of the resource owned by the poorest fraction  of the population. If the resources were distributed evenly throughout the population, then any fraction of the population would have the same fraction of the total resource. This means that (for all values of between 0 and 1). For example, when the resource is distributed evenly, then any 20% of the population will have 20% of the resource. Similarly, any 30% will have 30% of the resource, etc. 

Since represents the fraction of the resource owned by the poorest fraction of the population, then . In plain English, the poorest 0% of the population owns nothing, which is 0% of the resource. With this knowledge, it makes sense that F(1) = 1. The poorest 100% of the population would own all of the resource (100% of the resource). Values over 1 are not practically possible because we cannot account for more than 100% of the population or more than 100% of available resources. is an increasing function, since it represents an accumulating value. As increases, the fraction of the population that is included in the calculation of the function also increases. As a result, the fraction of the resources owned by that population must also increase. In other words,  any increase in wealth must be greater or equal to what was gained from a similarly sized population (since the concern here is only on the poorest fraction of the population). Because this occurs for every increment, is concave up when put on a graph. 

However, resources are not always distributed evenly amongst a population. When a resource is not dispensed evenly, individuals may want to discover how evenly the resource is distributed. Gini’s Index of Inequality, , is one way to measure how evenly the resource is distributed. In other words, it is a summary measure of income inequality that measures the dispersion of income across the entire income distribution.  It is defined by:

In this equation, is a measure of inequality. When graphed, is equal to the area below the line of perfect equality minus the area below the Lorenz curve, divided by the area below the line of perfect equality. In other words, it is double the area between the Lorenz curve and the line of perfect equality. Explanation for this graph is shown below

The portion of the graph under the Lorenz curve is called the fairest distribution. Smaller areas mean a fairer distribution of resources across the population. For example, here is a graphical representation of Gini’s Index of Inequality. 

As shown above, the ideal distribution, , depicts perfect equality. The smaller the area Gini’s Index encompasses, the closer the distribution gets to the ideal distribution (or fairest) distribution which represents total equality.  Two graphs of countries are shown below. Using Gini’s Index, we can conclude that country A has a more equitable distribution of wealth than country B.

Gini’s Index of Inequality can also be viewed as the measure of deviation from perfect equality. The minimum possible value of Gini’s Index of Inequality, G, is 0. When G is equal to zero, the resource is distributed equally among members of the population. In this instance, the graph of a straight line with a slope of 1. Perfect equality has been achieved, with each person owning an equal share of the resources. The further a Lorenz curve deviates from this straight line (when G1 = 0), the higher the Gini coefficient becomes and less equal the society. Therefore, the maximum possible value of Gini’s Index of Inequality is 1.0.  This occurs when all of the resource is owned by one person or group. In this case, the graph of represents the percentage of the maximum area between the Lorenz curve and the line of absolute equality. Here, the distribution of resources across the population has reached total inequality.

It is important to not mistake Gini’s Index for an absolute measurement of income. For instance, a high-income country and a low-income country can have the same Gini coefficient if the incomes are similarly distributed in each country. Below are graphs of perfect equality and perfect inequality which are mathematically based on the Lorenz curve (which plots the proportion of the total income of the population (x-axis) that is cumulatively earned by the bottom x% of the population. 

Congratulations! Now you have the knowledge to understand some very important questions surrounding the distribution of resources. Understanding this concept will allow you to compare countries and understand which populations are most affected by unequal distribution of resources. Unequal distribution of natural resources is one of the biggest perpetrators in economic and geopolitical power relations that can influence major conflict. In the long run, this can help raise awareness for populations and countries struggling with inequality, and help those struggling countries achieve perfect equality of resource distribution. 

Sources

• Catalano , M. T., Leise, T. L., & Pfaff, T. J. (2009). Measuring resource inequality: The Gini coefficient. https://digitalcommons.usf.edu/cgi/viewcontent.cgi?article=1032 context=numeracy
• Gini Index . Databank. (2023). https://databank.worldbank.org/metadataglossary/world-development-indicators/series/SI.POV.GINI
• Hughes-Hallett, D., Gleason , A. M., Connally, E., Kalaycıoğlu, S., Lahme, B., Lomen, D. O., Lock, P. F., & Lovelock , D. (2014). In Applied Calculus (5th ed., pp. 328–329). essay, Wiley.
• Introduction to Inequality. International Monetrary Fund . (2020, July 5). https://www.imf.org/en/Topics/Inequality/introduction-to-inequality
• Ramzai, J. (2020, April 27). Clearly explained: Gini coefficient and Lorenz curve. Medium. https://towardsdatascience.com/clearly-explained-gini-coefficient-and-lorenz-curve-fe6f5dcdc07
• Your guide to the Lorenz curve and Gini coefficient | indeed.com. Indeed.com. (2022, October 11). https://www.indeed.com/career-advice/career-development/lorenz-curve