How does the application of definite integrals describe the inequality gap between Brazil and the United States and hence a decrease in global inequality?
The inequality of income is a significant challenge for developing countries “with little industrial and economic activity and where people generally have low incomes”. But also within developed countries “with a lot of industrial activity and where people generally have high incomes”1. The Global Inequality by Brank Milanovic2 gave me a fascinating insight into factors that influence inequality, and introduced me to the Kuznets Curve (Figure 1): an inverted U-shaped curve that shows the increase and reduction of inequality over income per capita. Economists claim that the world is currently undergoing its history’s second Kuznets curve due to globalization. Some economists argue that the global decrease in inequality will occur when new global catastrophes arise, while Milanovic thinks that convergence will lead to the turning point. Convergence occurs when developing countries, with high inequality rates, start catching up with lower inequality rates of more developed countries.
Figure 13: Kuznets Curve
This intriguing concept motivated me to investigate whether developing countries are actually closing the inequality gap with developed nations. In order to do this, I will calculate the Gini coefficient, a global measure of income inequality within nations, to compare how the United States (developed) and Brazil (developing) were doing in terms of inequality in 1986 and 2013.
I will then graph all data points to find out whether an overturn in the Kuznets curve is to be expected in the near future.
I did not take data in between this time period. I did this because one nation’s Gini coefficient, usually, consists of no major fluctuations between negative/positive rate of increase/decrease within a time period where no influential events occurred in relation to inequality. The past thirty to forty years has been one of those periods.
Gini Coefficient and Lorenz Curve
Each nation has a Gini coefficient ranging from 0 to 1. The 45-degree line, also known as the
line of equality, shows what perfect equality with a Gini coefficient of 0 would look like, where everyone in a nation would receive the same amount of income. If the Gini coefficient would be 1, ultimately, all but one person would receive the income and wealth while everyone else receives nothing.
The Lorenz Curve is currently the most prevalent representation of income inequality, and the Gini Coefficient is determined by the ratio of the area between the line of perfect equality and perceived Lorenz curve (Area A) to the area between the line of perfect equality and the x-axis (line of perfect inequality) (area A+B):
Figure 2: Lorenz Curve Figure 3: Formula Gini Coefficient
The y-axis of the diagram (Figure 2) is represented by the cumulative percentage of national income and the x-axis is represented by the cumulative population percentage from poorest to richest. The values are cumulative, meaning that percentages add up to each other to eventually reach and represent 100% of the population, and 100% of the share in national income. This means that the cumulative income at 20% of the population represents the amount of income that is shared with the poorest 20% of the population.
By means of collecting data from the World Bank of the United States’ and Brazil’s country distribution data (percentage share of income) from 1986 and 2013, the following table was created. The data is organized in quintiles and shows the income distribution between the lowest to the highest 20% of the populations.
Table 1: Income Distribution for Brazil and The United States in 1986 and 2013
Using Desmos (An online graphing calculator) and the data from Table 1, I created the Lorenz Curves for both countries (Figure 3). To do this, as aforementioned, I have to plot the points cumulatively so that curves would each reach 100% and the numbers from the lowest to highest 20% would add up to 1 (see Table 2).
Table 2: Cumulative %Income Distribution for Brazil and The United States in 1986 and 2013
Figure 4: Lorenz Curve of Brazil and the United States in 1986 and 2013
I will calculate the Gini Coefficient’s using the formula given by Figure 3 and the Lorenz Curves in Figure 4. I will first find it for Brazil in 1986. I have to separate area B into four rectangles and five triangles as shown in figure 5.
To find area B the values of the rectangles and triangles have been added. In order to find area A, Area B can be subtracted from 0.5. This is because 0.5 represents the area
under the line of perfect equality since the x- and y- axis both have a length of 1 (which divided by 2 is equal to 0.5). Through this we can also conclude the Area A + Area B is 0.5.
Knowing area A and area B (and the formula from Figure 3), the Gini Coefficient can be calculated as follows for Brazil in 1986:
Total area of rectangles:
(0.026*0.2)+(0.086*0.2)+(0.189*0.02)+(0.371*0.2) = 0.100
Total area of triangles
0.5(0.026*0.02)+(0.06*0.02)+(0.103*0.02)+(0.182*0.02)+(0.629*0.2) = 0.060
Total Area of B: 0.100+0.060 = 0.160
Area of A: 0.5 – 0.160 = 0.333
Gini Coefficient = = 0.666
Using the same method, I got that the Gini Coefficient for Brazil in 2013 is 0.441, and in 1986 the coefficient for the United States is 0.353, while in 2013 the coefficient was 0.380. This is more clearly represented through the following table:
Table 3: The Gini Coefficients and Area A + B for both countries in 1986 and 2013
As aforementioned, the higher the Gini Coefficient, the higher the inequality. In Table 3 we can see that Area B is the smallest in Brazil, especially in 1986, this is also where the inequality is the highest. A smaller area B, means that the Lorenz curve is further away from the line of perfect equality, hence this explains why a smaller area B correlates to a higher Gini coefficient and more inequality.
Deriving a General Formula for the Gini Coefficient
The equation in figure 3 can be used to find the Gini Coefficient. It is, however, difficult to determine an accurate value of A and B independently by just analyzing a given diagram.
In order to find the area of B we previously summed the areas of the four rectangles and five triangles (see Figure 5).
This addition can also be expressed using sigma notation:
where f(xi) represents the height of the rectangles and ?xi the width of each rectangle.
The calculation would become more accurate if there was an infinite number of rectangles adding up to the area under the curve.
Figure 5 represents an equation where the approximations will be more accurate through using an infinite number of rectangles. This is fundamental in finding a general formula that could be used to find the exact area under the Lorenz curve (Area B). This limit is known as a definite integral which can be used to calculate the area under the Lorenz Curve. In calculus, definite integrals’ are expressed by the change between the values of an integral by specific lower and upper limits which are needed to identify where the area under the curve starts and ends.4
In the case of measuring the area under the Lorenz curve, the area is bounded by 0 < x < 1 . Where x = 1 represents the 100% of the cumulative population distribution and x = 0 represents the decimal value for 0% of the cumulative population distribution. Therefore, I would replace a and b (Figure 6) with 1 and 0 which results in the area under the Lorenz curve (area B) being: Now that we know definite integral for the area under the Lorenz Curve, the General Formula can be algebraically derived since: Proving the General Formula for the Gini Coefficient To test the accuracy of the general formula, I decided to use the world bank's5 statistics on Switzerland's Gini Coefficient in 2012. Given that I know the cumulative population's income share per quintile of the population, I can create a line of best fit that represents the Lorenz Curve through Microsoft Excel using Table 4. Table 4: Data used to construct the Lorenz curve for Switzerland in 2012 When the line of best fit is created with the cumulative percentage against the percentage of the population in quintiles, an equation for the Lorenz curve is generated.
This polynomial function can then be substituted into the general formula (where f(x) represents the Lorenz curve) to find the Gini Coefficient. The integral of the Lorenz curve equation can be found by the power rule:
I would then apply the definite boundaries for 0 < x < 1 (because these are the x values where the curve starts and ends) to find the value of B or : The value of can then be subsisted into the general formula, to find the Gini Coefficient: Using data from the worldbank, and the derived general formula, our estimate of Switzerland's Gini Coefficient in 2012 is 0.303. Worldbank themselves found the Gini Coefficient to be 0.314. Since there is a difference of about 1% or 0.01 in Gini Coefficient, it can be concluded that the derived general formula is not fully accurate because there is only one permanent function expressed as f(x) and there only five data points for each of the four curves of which this is based, this creating a best fit line that is not fully accurate though definitely close enough to approximate a reliable representation of a nation's Gini coefficient. Applying the General Formula to find the Gini Coefficient for Brazil and the United States Using the derived general formula and the data from Table 1 together with the aforementioned steps, I found that Brazil's Lorenz Curve equation in 1986 was y = 1.5616x2 - 0.6855x + 0.0486.
I substituted this equation into the general formula, , as f(x):
The resultant Gini coefficient for Brazil in 1986 ending up being 0.546.
Using the same method, I got that the Gini Coefficient for Brazil in 2013 is 0.494, and in 1986 the coefficient for the United States is 0.365, while in 2013 the coefficient was 0.394. This is
more clearly represented through the following table:
United States 1986
United States 2013
Table 5: Polynomial Functions and Gini Coefficients used and calculated using general formula
To analyze the inequality gap and the according Gini coefficients that have been calculated for Brazil and the United States, Table 6 has been created to compare the values of the Gini Coefficient using the Economics Formula (Figure 3) and the values attained from the definite integral method using the derived general formula. These two values have also been compared with the World Bank’s officially recognized Gini coefficients.
Table 6: Comparing the two calculated values and the worldwide recognized Gini Coefficients
By means of analyzing Table 6, it becomes clear that the Gini coefficients obtained from the economics formula do not completely resemble the values from the general formula and the World Bank. The average percent difference (error) is 12.922% compared to the worldwide recognized value from the World Bank. Through this, we can conclude that the method doesn’t completely misrepresent the official Gini Coefficient and national income distributions, though there are clear method and calculation errors driving the limitations.
The calculations made through the definite integrals only had a 3.574% average percent difference (error) from the world bank’s official Gini Coefficient value. This would probably be more accurate if there were more data points available, instead of just five quintiles, I would personally assume the World Bank has more than five data points available to construct their Lorenz Curves.
As aforementioned, there is a remarkable difference between the official value and the one obtained from the economics formula where the areas of the four rectangles and five triangles were added to find the area under the Lorenz Curve. The shapes seem to (see Figure 4) cover the whole area, but in reality, the shapes do not cover the curved edges. The Lorenz Curve also was by no means successive since the Lorenz Curve was simply was made of five discontinuous lines (as opposed to a best-fit curve using the general formula and Microsoft Excel as opposed to Desmos).
The calculated Gini Coefficient for Brazil in 1986 was 0.546 and decreased in 2013 to 0.494. As aforementioned, when the Gini Coefficient is higher, the Lorenz curve is further away from the line of equality, hence the decrease in Gini Coefficient is a decrease in inequality. The Gini coefficient lowered by 0.052, in other words, inequality was reduced by about five percent (0.052 = 5.2%) over a period of 27 years. This is a big decrease since the average decrease in inequality within this timeframe, according to the world bank, is also about 5 percent (0.584 – 0.528 = 0.056). Meanwhile, the calculated Gini Coefficient for the United States in 1986 was 0.365, which increased to 0.394. While Brazil, as a developing country has decreased inequality by big margins, the United States has had inequality rates increased by almost 3 percent (0.394-0.365 = 0.029).
The aforementioned concept of convergence, where developing nations and developed nations could ultimately overturn and decrease global inequality by means of tightening the inequality gap6, is definitely existent. The question is if this proves that convergence is actually occurring, considering there are more developing and developed nations with other inequality rates.
It is hard to show through four Gini Coefficients if global convergence is actually happening and if the Kuznets Curve will reach its maximum point in the near future, which is the point at which global inequality would ultimately start decreasing instead of increasing (see Figure 1). Even though Brazil does still have higher inequality rates compared to the United States, the fact that they are closing into each other at a relatively fast rate, does give a very solid indication that global inequality might decrease in the near future.
The investigation was successful in the way that I was able to compare different methods of calculating the Gini Coefficient (using definite integrals being a more accurate method, and the addition of rectangles and triangles being a less-reliable method) and find reliable and accurate sets of data to find out if convergence is actually existent on a global scale. The rates of growth/decline in inequality for two very powerful nations were also successfully calculated to confirm this. It is, however, important to note, that these calculations do not represent a fully accurate Gini coefficient.
As aforementioned, the calculations made from the general formula are 3.574% off from the calculations made by the world bank. This can be because the world bank only released statistics for five groups of income shares (quintiles) per population, through which my Lorenz curve was created by only five data points, whereas the world bank themselves might have had many more data points (that are not publicly available) to construct a more accurate Lorenz curve and ultimately a more accurate Gini Coefficient.
Even though it’s very intriguing to have a focused investigation on only two particular countries, showing how their inequality rates are increasing/decreasing over a certain period of time, and being able to take in-depth conclusions from this, is not fully possible. The investigation would have needed to include more countries, both developing and developed for which data mostly wasn’t available.
In conclusion, however, when considering the rate at which the United States and Brazil are closing the inequality gaps, it is quite reasonable to predict that the Kuznets Curve’s maximum point, might be a point reached in the near future.
1 “Meaning of “developed country” in the English Dictionary.” Cambridge Dictionary, dictionary.cambridge.org/dictionary/english/income.
2 Milanovic, Branko. Global Inequality: A New Approach for the Age of Globalization. 3rd ed., Cambridge, Belknap Press of Harvard Univ. press, 2016.
3 “‘Introducing Kuznets Waves: How income Inequality Waxes and Wanes over the Very Long Run.” Economists View, 24 Feb. 2016, economistsview.typepad.com/ economistsview/2016/02/introducing-kuznets-waves-how-income-inequality-waxes-and-wanes-over- the-very-long-run-.html.
4 “Definition of definite integral in English:.” Oxford Dictionary, en.oxforddictionaries.com/definition/definite_integral.
5 “GINI index (World Bank estimate).” The World Bank, data.worldbank.org/indicator/SI.POV.GINI.
6 Milanovic, Branko. Global Inequality: A New Approach for the Age of Globalization. 3rd ed., Cambridge, Belknap Press of Harvard Univ. press, 2016.