AP Calculus Final Project 2023: Gini Index Part I: Income Distribution The distribution of income in our society is a concept of ongoing interest to economists, politicians, public policy analysts, and other concerned individuals. In a capitalistic society such as the US, perfect equity in income distribution is neither possible nor desired. There would be no incentive to develop new products if you weren't able to capitalize on your invention. However, there is also a limit to how much of the total income should be controlled by a small group. Some suggest that this inequity in income distribution is playing an important role in the unrest apparent in Tunisia, Egypt, Yemen, and Bahrain. In the US, are the "rich getting richer, and the poor getting poorer" and is the "middle class disappearing" as some politicians suggest? And if so, how could you tell? To quantify distribution of income in a country, economists consider the percent of the country's total income that is earned by certain groups of the population. To understand how this is done, we will consider a very small society consisting of the individuals with the following jobs and salaries: Administrative $28,369 Public Relations $39,913 Support Specialist President of the $400,000 Advertising $40,424 country Mail Carrier $36,619 CEO $100,271 Electrical Engineer $62,201 Congressman $150,000 Secretary $23,311 Teacher $33,123 Pediatrician $113,510 Governor $110,346 Head Nurse $48,000 Migrant fa rm worker $2,500 Drafter $37,500 Farm worker $7,500 Mechanic $29,521 College Basketball $260,000 Coach Firefighter $27,976 Microbiologist $55,411 Cashier $15,184 Forensic Science $32,864 Technician Travel Agent $27,373 Librarian $42,120 Aircraft Mechanic $42,370 1. Organize the data in order of increasing income. And find the total income for this society. 2. We will now divide the population up into fths. Since there are 25 people, there will be 2515=5 people in each of the fifths. We will refer to the 5 people with the lowest incomes as the "lowest fifth", the next set of 5 people as the "second fifth, and so on. The last set of 5 people we generally refer to as the "highest fifth." Find the total income of each fifth. 3. Obtain the percentage of income for each fifth by dividing the income of the fifth by the total income of the whole society. 4. In a perfectly equitable society, what would we get for these percentages? Economists have developed a very intuitive measure of the inequality of the distribution of incomes by considering the cumulative income. The cumulative data can be obtained by adding up the percentages for each fth. For example, suppose we had a society with the following distribution of income: Fifths of Households Percent of Income Lowest fifth 11.5 Second fifth 15 Third fifth 16.5 Fourth fifth 20 Highest fifth 37 Table 1: Percentage of Total Income The lowest onerfifth of families earned 11.5% of the total income. The lowest twoefths of families earned {11.5 +15}% = 26.5% of the total income. The lowest threefifths of families earned (11.5 + 15 + 16.5)% = 43% of the total income. Continuing in this way, we obtain: Fifths of Households Cumulative Percent of Income Lowest fth 11.5 Lowest two fifths 26.5 Lowest three fifths 43 Lowest four fifths 63 All fifths 100 Table 2: Cumulative Percentage of Total Income 5. Find the cumulative percentage distribution for our sample minicountry. A graph of the data in Table 2 can be obtained by plotting the cumulative proportional distribution of aggregate income versus the proportion of the population, as shown below. The percentages should be converted to decimal numbers, that is, the lowest twofths earning 26.5% of the aggregate income is represented by the point (0.4,0265). The points (0,0) and (1,1) are included because 0% of the households earn 0% of the income and 100% of the households earn 100% of the income. 1 09 08- 0.?- 0.6- 0.5 0.4 9/ U 3 02 / S" 0.1 0.2 0.3 0.4 0.5 0.6 0.? 0.3 0.9 1 A curve that models data of the type (proportion of households, cumulative proportion of aggregate income} is called a Lorenz curve. 6. Graph the data (proportion of households, cumulative proportion of aggregate income} for our sample society. Then sketch in the Lorenz curve that fits this data. 7. What would a graph of this data look like for a perfectly equitable society? What would be the equation of the Lorenz curve for a perfectly equitable society? Explain why