Gini Index Student Handout

8/17/2019 Gini Index Student Handout

http://slidepdf.com/reader/full/gini-index-student-handout 1/8

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

Support

$28,369 Public Relations

Specialist

$39,913

President of the

country

$400,000 Advertising $40,424

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 farm worker $2,500

Drafter $37,500 Farm worker $7,500

Mechanic $29,521 College Basketball

Coach

$260,000

Firefighter $27,976 Microbiologist $55,411

Cashier $15,184 Forensic ScienceTechnician $32,864

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 fifths. Since there are 25 people, there will be

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 fifth. For example,

suppose we had a society with the following distribution of income:

Fifths of Households Percent of Income

Lowest fifth

Second fifthThird fifth

Fourth fifth

Highest fifth

11.5

1516.5

20

37

Table 1: Percentage of Total 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:

Fifths of Households Cumulative Percent of Income

Lowest fifth

Lowest two fifths

Lowest three fifths

Lowest four fifths

All fifths

11.5

26.5

43

63

100

Table 2: Cumulative Percentage of Total Income

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

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.

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.

Part II: Calculating the Gini Index and Using Calculus to Measure Inequity in the United States

Now, let’s 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.

Fifth of Households Percent of income

Lowest fifth 3.6

Second fifth 8.9Third fifth 14.8

Fourth fifth 23.0

Highest fifth 49.8Percent distribution of aggregate income for 2000

http://www.census.gov/compendia/statab/2011/tables/11s0693.pdf






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.

Fifths of Households Cumulative Percent of income

Lowest one-fifth 3.6

Lowest two-fifths

Lowest three-fifths

Lowest four-fifths

All fifths 100

Cumulative percent distribution of aggregate income 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.

Note that Lorenz curves always lie somewhere between the two curves

y x and since income distribution must fall somewhere

between perfect equity and perfect inequity. Sketch a curve of the line

y x on the same graph as your Lorenz Curve above.

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.

y 0



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 under the curve that represents

perfect equity? What tools from calculus could enable us to calculate this area?

4.

Using five trapezoids, calculate a numerical approximation for the Gini Index for the US income distribution datafrom 2000.

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?







6. Optional Extension: Finding the Lorenz Curve using Least Squares

There are many ways of finding the equation for the Lorenz curve based on the data from the table in #1. If you

search the Web for Gini indices for various countries, you will find that two estimates for the same year may

differ. This is due to possible differences in their sources of information and to the method used to approximate

the area A. Since we are interested in comparisons over time or among countries, as long as all indices are

computed using the same techniques, comparisons are legitimate.

The Least Squares Method: Since (0, 0) and (1, 1) are always points on the curves, a reasonable model for this

data is a power function of the form y xn

, with . We cannot use a power least squares regression on our

calculator to fit a power function to the data because a Lorenz curve must contain the point (1,1) and a power

regression curve does not necessarily contain (1,1). Also note that we only use the four points

0.2,0.036 , 0.4,0.125 , 0.6,0.273 , and 0.8,0.503 in our calculations. By using y xn

as our model, we

guarantee that (0,0) and (1,1) fall on the curve.

a. The least square method uses the fact that a log-log re-expression linearizes data that is modeled by a

power function. Since y xn

, we take the logarithm of both sides of the equation to obtain ln ln y n x .

We now can use our knowledge of calculus to find a least-squares estimate of n. Consider the linear

equation Y nX (in our case lnY y and ln X x ). Use the methods of calculus to minimize

4

2

1

i i

i

S n Y nX

(remember i X and i

Y are constants). Find a general solution for n for the Lorenz

curve y xn

, given the cumulative aggregate income percentages for the quintiles. Use the general

solution to find the equation for the Lorenz curve for the 2000 data.

b.

Another method is to fit a true least squares model to the data. In this case, we have

2

n

i iS n y x . What equation must be solved to find the value of n that minimizes S? You will not

be able to solve this equation analytically, so feel free to use a calculator or computer to find the value ofn

for 2000.

c.

Graph the Lorenz curves you created in parts a and b over a scatterplot of the data from 2000. In your

opinion, which Lorenz curve fits better, and why?

d.

For the 2000 data, calculate the Gini Index using each version of the Lorenz curves (from parts a and b) byincorporating calculus methods to find Area A. Discuss the differences between these indices, and compare

them with the index calculated with the trapezoid approximations in #4. How accurate would comparisons

among Gini indices for an economy over time (or perhaps comparisons of Gini indices for different

countries) be if the indices were computed using different methods?

n 1



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.?

Fifth Brazil Cambodia Mexico India Germany

Lowest fifth 2.9% 7% 4.8% 9% 8%

Second fifth 6.5 10 9.2 11 14

Third fifth 11 14 14 15 18

Fourth fifth 19 20 21 20 23

Highest fifth 61 49 51 45 37

Reference: GapMinderWorld http://www.gapminder.org/world/#;example=75;

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 theterritory 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 inJapan 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 Africaand Northern Africa. The poorest fifth of the population of South America have especially low relative incomes given theaverage 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.”

http://www.gapminder.org/world/#;example=75








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?

Gini Index Student Handout

Documents

Transcript of Gini Index Student Handout