Page 3 of 22 ZOOM + Mathematics of Income Inequality Income distribution amongst families is an important economic and social issue that has been brought to the public's attention over the past few years. The U.S Bureau of the Census has collected and analyzed data for income distribution amongst families. To quantify income distribution economists use Lorenz curves. Economists use the Gini coefficient to compare Lorenz curves over different years or different nations. Lorenz Curve The IPage 4 of 22 ZOOM + Lorenz Curve The Lorenz curve is the graph of the Lorenz function that shows the proportion of national income earned by a given percentage of the population. National Income (cumulative) 55% 40% 20% 10%% 20% 40% 100% Population (cumulatively ranked) poorest richest For example according to the above Lorenz curve 20% of the population receives 5% of the total income for all families in this year.Page 5 of 22 ZOOM + 100 Line of Equality (48 Degree) Cumulative share of income eamed Curve Lore 10014 Cumulative share of people from lowest to highest incomes Absolute equality would be if 10% of the population received 10% of the total Income. This is represented by the line y= x as shown above. The greater the area between the Lorenz curve and the line y= x the greater the income inequality. Gini Index The Gini Index allows more precise comparison of Lorenz curves. It is the proportion of the area taken up by the Lorenz curve (A) in relation to the overall area under the line of equality. In this project you will use Calculus to determine the Gini index for given Lorenz functions over a period of years. Subsequently you will discuss and synthesize the Gini indices and what this means for income distribution. Gini index = 2/[x-f(x)] ex where f(x) is the Lorenz functionPage 6 of 22 ZOOM Project 2 - Form C2 1. USA Data Collection [10 points] The following data, table 693 is from the US Census Bureau. It can be used to model a Lorenz function for a given year. As an example, let's consider income distribution in 1970. According to the data here, the poorest 206 (lowest 5#) of the population receives 4.19% of total income for families. Table (93. Share of Aggregate Income Received by Each Fifth and Top 5 Percent of Households 1970 10 3003 Third 174 108 562 11 Souron U.S. Census Bureau, Income Poverty and Health Insurance Coverage in the United States 2043. Current Population Report, P60-236RV, and Historical Tables-Tables Hi and H2, September 2009. (2015, Oct 31) retrieved from Census data In order to model the data, we need to consider cumulative population and cumulative percentage of national Income. Using the data from table 693 for 1970. the cumulative population and cumulative percentage of national income is calculated: Table 1 Cumulative Population (4) Cuntalative Income (19) Data Points (x7] X V 4.1% (0.2.0.041) 401 (20%% + 20%%) 14.9%% (4.186 - 10.814) (0.4,0.149) 60%% (40% - 20:) $2.5%% [14.986 - 17-495] (0.6.0.3231 56.34 (32.5% - 24506) (0.8.0.568) 100% (80% + 20%] 100% [56.35 - 43.3%] A power function is used to model the Lorenz function. Using regression and the data points above (in the last column in the table above). you should get the following equation:Page 7 of 22 ZOOM + As described above. select a year between 1970 and 1990 (do not select 1970 or 1990). this year is referred to as Year A. Use table 693 for the data for Year A. calculate the cumulative population and cumulative percentage of national Income, and create a table (like Table 1 for 1970). As described above. select a year between 1990 and 2008, this year is referred to as Year B. Use table 693 for the data for Year B, calculate the cumulative population and cumulative percentage of national income, and create a table (like Table 1 for 1970). All work for 1. USA Data Collection, 2. Data Analysis and 3. Global Data Collection should be included at the end of your report as an appendix. 2. Data Analysis Using Calculus [20 points] Gini Index for 1970 to 2014 Table 2 Year Lorenz Function Gini Index y = 0.92195 1970 0.378 Year A (Year between 1979 and 1990) = 0.88x159 1990 0.411 Year B [Year between 1990 and 2008] y -0.84x20 2012 0.456 y -0.84x- - 2014 0.479 Complete Table 2. . Enter the year you selected for Year A into table 2. Calculate the Lorenz function for Year A. using power regression as described for 1970. Use Calculus to determine the Gini Index for Year A. Show all your work. The Calculus has to be done by hand, include the hand written work as part of the appendix. Please type the work in the document using appropriate software thatPage of 22 ZOOM + . Enter the year you selected for Year B into table 2. Calculate the Lorenz function for Year B, using power regression as described for 1970. Use Calculus to determine the Gini Index for Year B. Show all your work. The Calculus has to be done by hand, please type the work in the document using appropriate software that has the capability to type mathematics. (3 decimal places) [10 points] 3. Global Data Collection [10 points] The following chart shows income inequality for selected countries: Inequality of incomes, 1901 to 2014 our Work This chart presents time-wries which are consistent over time for each country Before making cross-country comparisons please check the definitions on the "Sources" tab SUITS Bury Leavell Milwhen + Add country Go to https://ourworldindata.org/income-inequality! find the Inequality of Incomes interactive chart. You have to scroll down to see it. Create a chart with three countries, USA and two other countries selected from the following: Canada, France, Finland, Germany, Netherlands, Norway, Sweden . Countries can be added by selecting + Add countryPage B of 22 ZOOM + Inequality of incomes, 1901 to 2014 Our World in Dicta This chart presents time-series which are consistent over time for each country. Before making cross-country comparisons please check the definitions on the "Sources" tab. SOURCES Raly United Season + Add country 1080 2014 Go to https://ourworldindata.org/income-inequality/ find the Inequality of Incomes interactive chart. You have to scroll down to see it. Create a chart with three countries, USA and two other countries selected from the following: Canada, France, Finland, Germany, Netherlands, Norway, Sweden . Countries can be added by selecting + Add country Click on DATA, download the excel file, inequality-of-incomes-chartbook.csy. Open the excel file and find the GINI index for the two countries you selected other than the USA, make a note of the GINI index for 1970 (or closest year available), Year B (as above) and 2014. Put this information in a table. One table can be used for both countries, or a table for each country. [10 points]