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Name:_________________________

Date:___________________________ Calculus | Packer Collegiate Institute Income Inequality1

Band:________

Prelude: Estimation
Imagine if you line up all the adults in the United States based on yearly income. Then divide up this line of people into 5 equal parts. Add up all the wealth for each of these groups. What percent of the total income of all the adults in the US do you think each of these groups will get? Estimation of True Distribution: Second fifth Third fifth Fourth fifth

Lowest fifth

Highest fifth

What do you think the distribution ought to be, in a healthy economy? Estimation of Ideal Distribution: Second fifth Third fifth Fourth fifth

Lowest fifth

Highest fifth

Now well look at your distributions graphically Put lines separating each fifth, and color each section a different color. Estimation of True Distribution:

Estimation of Ideal Distribution:

To find out the true distribution, go to: http://bit.ly/wealthdistribution


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This activity is largely taken from http://courses.ncssm.edu/math/apcalcprojects/econ/Gini_Index_Student_Handout.pdf

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 werent 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 countrys 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:

1. Organize the data in order of increasing income. And find the total income for this society.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Total income for each fifth:


Lowest Fifth Second Fifth Third Fifth Fourth Fifth Highest Fifth

Total income for all people:

2. Obtain the percentage of income for each fifth by dividing the income of the fifth by the total income of the whole society.
Lowest Fifth Second Fifth Third Fifth Fourth Fifth Highest Fifth

3. In a perfectly equitable society, what would we get for these percentages? (Note: I am not talking about the best society I am asking you to take the word equitable literally.)
Lowest Fifth Second Fifth Third Fifth Fourth Fifth Highest Fifth

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 fifth. For example, suppose we had a society with the following distribution of income:

The lowest one-fifth of families earned 11.5% of the total income. The lowest two-fifths of families earned (11.5 +15)% = 26.5% of the total income. The lowest three-fifths of families earned (11.5 + 15 + 16.5)% = 43% of the total income. Continuing in this way, we obtain:

4. Find the cumulative percentage distribution for our sample mini-country.


Lowest fifth Lowest two fifths Lowest third fifths Lowest four fifths All fifths

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 two-fifths earning 26.5% of the aggregate income is represented by the point . 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.

A curve that models data of the type (proportion of households, cumulative proportion of aggregate income) is called a Lorenz curve.

5. 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.

6. 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.

Part II: Calculating the Gini Index and Using Calculus to Measure Inequity in the United States Now, lets use the skills from the income distribution activities to create a measure of inequity. We will focus on income inequity in the US in these activities. The data that economists use to quantify distribution of income is often presented in the form of a table (see http://www.census.gov/compendia/statab/2011/tables/11s0693.pdf). The table below gives the percent of the total income of the United States earned by each fifth of the population in 2000, ordered by income.

1. Use the information in the table provided in the table above to create a table for the cumulative percent distribution of aggregate income in the US for 2000.

2. Create a graph of the Lorenz curve using cumulative proportional distribution of aggregate income versus the proportion of the population data from #1. Convert the percentages to decimal numbers. For instance, the lowest one-fifth earning 3.6% of the aggregate income is represented by the point (0.2,0.036) . The points (0,0) and (1,1) should be included because 0% of the households earn 0% of the income and 100% of the households earn 100% of the income. Carefully sketch a curve that fits all six points.

3. The graph you created in #2 provides a visual representation of how the income distribution in the US in 2000 differs from a perfectly equitable economy. We can develop a measure of inequity by comparing the inequitable data to data that would represent perfect equity by considering how far apart the graphs are. One measure employed by the economists is the ratio of the shaded areas A and B shown in Figure 4. This ratio is called the Gini Index.

This ratio can have a value anywhere from 0, representing perfect equity, to 1, representing perfect inequity. The larger the ratio, the more inequitable the distribution of income is. To find the area of the shaded region A, we need to find the area between the line and the Lorenz curve. What is the area of region B (the area under the curve that represents perfect equity)?

What tool from calculus could enable us to calculate the area of A?

4. Using five trapezoids, calculate a numerical approximation for the Gini Index for the US income distribution 1 data from 2000. (Recall the area of a trapezoid is Area (base1 base2 )height . 2

5. Go to the US Census web-page http://www.census.gov/compendia/statab/2011/tables/11s0693.pdf (the table is shown below). Using the method from #4, find the Gini index for each of the following years: 1970, 1980, 1990, 2000, and 2008. Which decade saw the greatest change in the Gini Index?

1970: Lowest Fifth

Second Fifth

Third Fifth

Fourth Fifth

Highest Fifth

Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

1980: Lowest Fifth

Second Fifth

Third Fifth

Fourth Fifth

Highest Fifth

Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

1990: Lowest Fifth

Second Fifth

Third Fifth

Fourth Fifth

Highest Fifth

Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

2000: (you did this one already copy data from above!) Lowest Fifth Second Fifth Third Fifth

Fourth Fifth

Highest Fifth

Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

2008: Lowest Fifth

Second Fifth

Third Fifth

Fourth Fifth

Highest Fifth

Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

Gini Index in the US over time: 1970 1980 1990 2000 2008

Look at your charts, and your Gini indexes over time. How is the income distribution changing over time? What conclusions can you draw?

Part III: Measuring Inequity in the Various Countries

In Part III you will be asked to calculate the Gini Index for various countries. We will also share some additional resources for further exploration of income inequity across the globe. The data shown in the following table gives the distribution of household income for Brazil, Cambodia, Mexico, India and Germany in 2005. How do these indexes compare to each other and to the Gini index in 2000 for the U.S.?

Brazil 2005 Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

Cambodia 2005 Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

Mexico 2005 Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

India 2005 Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

Germany 2005 Lowest fifth

Lowest two fifths

Lowest third fifths

Lowest four fifths

All fifths

Calculation of Gini-Coefficient:

Further Exploration: To find additional data for these countries in other years or for other countries through the Gapminder site, choose time along the horizontal axis and Economy, Poverty and Inequality and then the appropriate income share along the horizontal access for various years and various countries. You will have to choose the individual income share one at a time. For example: Income share of poorest 20%, income share of second poorest 20%, etc. Other Related Data Representations: There is an interesting graph related to income inequity on the WorldMapper website: http://www.worldmapper.org/textindex/text_index.html Choose Income, then for example, choose Poorest Fifth. This graph shows a map of the world in which the territory size shows the earnings of the poorest fifth of the population living there, as a proportion of the earnings of the poorest fifth living in all territories.
Japan is the region with the richest poor people in the world. The average income of the poorest fifth of the population in Japan is at least 7 times more than that of the equivalent group in 8 other regions. The regions with the lowest average incomes for the poorest fifth of the population are Central Africa, Southeastern Africa and Northern Africa. The poorest fifth of the population of South America have especially low relative incomes given the average incomes there. Despite being located in South America, French Guiana and the Falklands / Islas Malvinas share data with France and the United Kingdom respectively, so are resized accordingly.

Exploring Other Related Factors: Choose other data from these sites for the countries in #1 and compare those factors to the Gini Indexes that you calculated. For example, explore health factors, level of education, or employment rates. How are these additional factors related to the Gini Indexes?

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