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1 According to Wikipedia, Occupy Wall Street was a protest movement that began in September of 2011 in New York City s Wall Street financial district. The movement was intended to raise awareness of social and economic inequality, greed, corruption, and the perceived undue influence of corporations on government particularly from the financial services sector. Income inequality is not a new idea. Economists have developed various methods to measure income inequality and have been tracking it over time. The most widely used measure of income inequality appears to be the Gini coefficient. This is computed from the Lorenz curve, which was created by Max Lorenz in This handout focuses on how this curve and the Gini coefficient are computed. Let s begin with the Lorenz curve. Source: There are several resources available on the internet that describe how to graph a Lorenz Curve. Once such resource is Charting Income Inequality: The Lorenz Curve. 1

2 Calculations for the Lorenz Curve The calculations behind the creation of a Lorenz curve will be discussed in the context of the following example. This example was taken from Charting Income Inequality: The Lorenz Curve (link given above). Example #1: Suppose the income of five students is given by following quantities. $7,800; $12,957; $2,417; $8,489; and $10,072 Step #1: Order the incomes from smallest to largest and compute the total income. Person Income people at or income income at or Total 2

3 Step #2: Compute the following quantities. The proportion of people at or below each The proportion of income at or below each Person Income 1 $2,417 people at or = 1 / 5 = 0.20 income = 2,417 / 41,735 = income at or $7,800 ❶ $8, ❸ 4 $10, ❷ $12, Total $41,735 Give details for how each of the specified values would be computed in the above table. 1. Explain how the value in Box ❶ would be computed. o o What is this value? Provide a written description of the meaning/interpretation of this value. 2. Explain how the value in Box ❷ would be computed. o o What is this value? Provide a written description of the meaning/interpretation of this value. 3

4 3. Explain how the value in Box ❸ would be computed. o o What is this value? Provide a written description of the meaning/interpretation of this value. Plotting the Lorenz Curve The Lorenz Curve plots the proportion of people at or below each (x-axis) against the proportion of income at or below each (y-axis). So, we will focus on the values in the highlighted columns in the table below. Person Income people at or income income at or 1 $2, $7, $8, $10, $12, Total $41, Use the outcomes computed above to plot the Lorenz Curve for the incomes provided in Example #1. Sketch this curve onto the graph on the next page. 4

5 Using Excel to do Calculations Next, we will use Excel to calculate the quantities computed above. Open the Lorenz Curve excel file, which can be found on our course website. Enter all of the necessary formulas in Excel to complete the table as was done above. Answer the following questions. 5. What is the appropriate Excel formula for cell B7? 5

6 6. What is the appropriate Excel formula for cell D3? 7. What is the appropriate Excel formula for cell E3? 8. Explain why the value in cell E6 should be 1. Example #2: Complete income equality may be unrealistic; however, it is important that we visualize what the Lorenz Curve should look like when there is complete income equality. In your Excel sheet, enter five new incomes in cells B2 through B6 that are all the same, indicating complete income equality. An example is shown below. Once you change the values in column B, you Excel sheet should update automatically in columns D and E if you have entered the formulas correctly. Plot the Lorenz Curve for the situation of complete income equality. 6

7 9. Consider the Lorenz curve from above. What proportion of income is attained when the proportion of people is at 40%? How about the proportion of people at 60%? Do these values make sense? Discuss. 10. Come up with five new incomes that would result in an extreme Lorenz Curve for the situation of significant income inequality. Plot the resulting Lorenz Curve on the above plot, as well. How is the Lorenz Curve for significant income inequality different from the curve for income equality? Discuss. Example #3: In some under-developed countries, livestock (i.e., farm animals) are used for monetary purposes. Consider the following Lorenz Curve based on livestock for a particular area in Africa. Answer the following questions. 11. What proportion of cattle is owned by the top 20% of households? 12. Does this area in Africa appear to have equality in its livestock holdings? Discuss. 7

8 Example #4: In this example we will consider income inequality in the United States. Many people have studied this issue over time. For example, Emmanuel Saez, Director of the Center of Equitable Growth at UC - Berkeley, has created substantial resources on this topic. Source: "Income Inequality in the United States, " with Thomas Piketty, Quarterly Journal of Economics, 118(1), 2003, 1-39 (Longer updated version published in A.B. Atkinson and T. Piketty eds., Oxford University Press, 2007) (Tables and Figures Updated to 2011 in Excel format, January 2013) The following table was provided in an Excel file from the above source. 8

9 Next, we will use these values from this table to construct a Lorenz Curve so that we can investigate income inequality in the U.S. Consider the income shares in Fill in the values in the table below so that you can construct the Lorenz Curve for Person Income people Income income at or ❶ ❷ After obtaining the necessary values, we can create a plot of the Lorenz curve. This plot is shown next. 9

10 Lorenz curve for US in 1990 Next, data from the same table was used to also find the Lorenz curve for Consider the following graph that compares the Lorenz curves from 1990 to

11 13. What has happened to the Lorenz Curve over time in the United States? Do you think the Lorenz Curve has moved substantially from 1990 to 2011? 14. What do these Lorenz Curves imply about income equality in the U.S. in 2011 relative to 1990? Explain. 11

12 The Gini Coefficient The Gini coefficient is a commonly used measure to quantify the level of income inequality. Once we have created the Lorenz curve, we can easily calculate the Gini coefficient. Source: 12

13 The Gini coeffient is a proportion of area and thus is restricted to be between 0 and 1. A value near 0 implies income equality A value near 1 implies substantial income inequality. Consider a graph of the Gini Coefficient of countries around the world. Source: Wiki. 15. Discuss the above graph. What countries appear to have a high Gini Coefficient? Which countries appear to have income equality? Discuss. 16. Does income equality (or lack thereof) in the United States appear to differ from that of other developed countries in the world? Discuss. 13

14 Computing the Gini Coefficient The Wiki page indicates that the Gini Coefficient is typically calculated using calculus methods. We will approximate the Gini Coefficient by counting boxes on the plots of our Lorenz curves. These calculations are shown below. Lorenz Curve for US 2011 The Gini Coefficient is the proportion of area between the Line of Equality and the Lorenz Curve to the area under the Line of Equality For example, let s compute the Gini Coefficient for the United States in AAAAAAAA BBBBBBBBBBBBBB LLLLLLLL oooo EEEEEEEEEEEEEEEE aaaaaa LLLLLLLLLLLL CCCCCCCCCC GGGGGGGG CCCCCCCCCCCCCCCCCCCCCC = TTTTTTTTTT AAAAAAAA UUnndddddd LLLLLLLL oooo EEEEEEEEEEEEEEEE = =

15 Example #5 17. Complete the following table for the U.S. in Person Income people Income income at or Sketch the Lorenz curve for the U.S. in 2004 on the graph below. 19. Use this picture to estimate the Gini coefficient for the U.S. in