Calculus I — Introduction to the Gini coefficient The Gini coefficient (or Gini index) is a commonly-used measure of inequality devised by Italian economist Corrado Gini in 1912. In this assignment, we study income inequality in the United States using the Gini coefficient. The Gini coefficient is defined using the Lorenz function (), which describes the distribution of some resource in society (in our case, income): () = percent of a society's total income received by the bottom % of households The Lorenz function has domain 0 ≤ ≤ 100. The graph of the Lorenz function is called the Lorenz curve. 1. What is (0)? 2. What is (100)? 3. What is the greatest possible value of (25)? (35)? ()? 4. Sketch the Lorenz curve for the following societies: a) A “perfectly equal” society, in which every household has the same income. b) A “perfectly unequal” society, in which one household receives all of the income. c) A two-class society in which half of the households receive no income and the other half split the total income evenly. d) A two-class society in which 70% of the households split 20% of the total income and the other 30% of households split the remaining 80% of the income. 5. Between societies c) and d), which do you think is more unequal? The Gini coefficient, a number between 0 and 1, measures the inequality in the distribution of a resource (in our case household income): = area between Lorenz curve and "line of perfect equality" = area under = for 0 ≤ ≤ 100 6. Compute the Gini coefficient for the societies in problem 4. 7. Suppose the Lorenz curve for a society is given by () = 100 ( 100 ) 2 . Graph the Lorenz curve, and find the Gini coefficient for this income distribution.
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Calculus I — Introduction to the Gini coefficient
The Gini coefficient (or Gini index) is a commonly-used measure of inequality devised by Italian
economist Corrado Gini in 1912. In this assignment, we study income inequality in the United
States using the Gini coefficient.
The Gini coefficient is defined using the Lorenz function 𝐿(𝑥), which describes the distribution of
some resource in society (in our case, income):
𝐿(𝑥) = percent of a society's total income received by the bottom 𝑥% of households
The Lorenz function has domain 0 ≤ 𝑥 ≤ 100. The graph of the Lorenz function is called the
Lorenz curve.
1. What is 𝐿(0)?
2. What is 𝐿(100)?
3. What is the greatest possible value of 𝐿(25)? 𝐿(35)? 𝐿(𝑥)?
4. Sketch the Lorenz curve for the following societies:
a) A “perfectly equal” society, in which every household has the same income.
b) A “perfectly unequal” society, in which one household receives all of the income.
c) A two-class society in which half of the households receive no income and the other half
split the total income evenly.
d) A two-class society in which 70% of the households split 20% of the total income and
the other 30% of households split the remaining 80% of the income.
5. Between societies c) and d), which do you think is more unequal?
The Gini coefficient, a number between 0 and 1, measures the inequality in the distribution of a
resource (in our case household income):
𝐺 =area between Lorenz curve and "line of perfect equality" 𝑦 = 𝑥
area under 𝑦 = 𝑥 for 0 ≤ 𝑥 ≤ 100
6. Compute the Gini coefficient for the societies in problem 4.
7. Suppose the Lorenz curve for a society is given by 𝐿(𝑥) = 100 (𝑥
100)
2. Graph the Lorenz
curve, and find the Gini coefficient for this income distribution.
8. Using the data from the Congressional Budget Office found in the spreadsheet titled
“US_Income_Spreadsheet” on our class webpage, sketch the Lorenz curves for household
income in the United States for the years 1967 and 2010.
9. Compute the Gini coefficients for 1967 and 2010 using your graphs from question 7. Has
household income distribution in the United States become more equal or less equal during
this period? (Recall that the area of a trapezoid with height ℎ and parallel bases 𝑏1 and 𝑏2 is
𝐴 =1
2(𝑏1 + 𝑏2)ℎ. )
10. Looking at the data in the spreadsheet, but without sketching graphs or doing
computations, say whether household income distribution in the U.S. shows any general
trends during the years from 1967 to 2010. Explain.
Assignment
Part 1: Computation of Gini coefficients (3% of final course grade, due on Nov. 25)
Do numbers 7-10 above, showing all of your work. Sketch precise graphs on graph paper. (You may
use a computer or graphing calculator to generate the graph in #7. You may sketch the two graphs
for #8 on separate axes or on the same set of axes, whichever you prefer.) Turn in your work
during class on Tuesday, November 25 . (You may work together on this, but each person must
turn in his or her own assignment, reflecting his or her own work and understanding. Don’t turn
something in if you don’t understand it or couldn’t reproduce it on your own!)
Part 2: Response to the reading (3% of final course grade, due on Dec. 5)
Below, you will find government reports, Christian web resources, scripture, and several opinion
articles from the New York Times and Washington Post, all relating to the issues of income
inequality and poverty, which we have begun to explore via the Gini coefficient. This issue has been
in the news in the last few years as a result of the “Occupy Wall Street” movement, the “one percent
vs. 99 percent” discussions, Mitt Romney’s “47 percenters” comments during the 2012 presidential
campaign, and several of President Obama’s speeches on inequality. Whatever one may think of
politicians and policies, issues of justice and inequality deserve thoughtful Christian engagement
and response.
Write a 3-4 page paper giving a thoughtful response to this material, interacting with as many of
the sources as possible. (You must explain how your personal faith commitments and beliefs play a
role in shaping your views.) Your essay should discuss a Christian view of economic justice and
how that relates to the world we live in, as well as what you think should be done by Christians or
others to move toward a just society – whatever that means to you. You should refer and respond
to as many of the resources as you can in your paper, but in particular you must include at least
some response to the CBO summary, your graphs and computations from Part 1 of this
assignment, the Wikipedia list of countries by income inequality, and two or more scriptural
passages. Turn in your paper as an attachment to an email to [email protected], by Friday
"Is not this the kind of fasting I have chosen: to loose the chains of injustice and untie the cords of the yoke, to set the oppressed free and break every yoke? Is it not to share your food with the hungry and to provide the poor wanderer with shelter — when you see the naked, to clothe him, and not to turn away from your own flesh and blood?
"Then your light will break forth like the dawn, and your healing will quickly appear; then your righteousness will go before you, and the glory of the LORD will be your rear guard. Then you will call, and the LORD will answer; you will cry for help, and he will say: Here am I. If you do away with the yoke of oppression, with the pointing finger and malicious talk, and if you spend yourselves in behalf of the hungry and satisfy the needs of the oppressed, then your light will rise in the darkness, and your night will become like the noonday.
"The LORD will guide you always; he will satisfy your needs in a sun-scorched land and will strengthen your frame. You will be like a well-watered garden, like a spring whose waters never fail."
"Then the King will say to those on his right, 'Come, you who are blessed by my Father; take your inheritance, the kingdom prepared for you since the creation of the world. For I was hungry and you gave me something to eat, I was thirsty and you gave me something to drink, I was a stranger and you invited me in, I needed clothes and you clothed me, I was sick and you looked after me, I was in prison and you came to visit me.'
Then the righteous will answer him, 'Lord, when did we see you hungry and feed you, or thirsty and give you something to drink? When did we see you a stranger and invite you in, or needing clothes and clothe you? When did we see you sick or in prison and go to visit you?'
The King will reply, 'I tell you the truth, whatever you did for one of the least of these brothers of mine, you did for me.'"
Deuteronomy 10:17-19
For the LORD your God is God of gods and Lord of lords, the great God, mighty and awesome, who
shows no partiality and accepts no bribes. He defends the cause of the fatherless and the widow,
and loves the foreigner residing among you, giving them food and clothing. And you are to love
those who are foreigners, for you yourselves were foreigners in Egypt.
Opinion Articles
http://www.nytimes.com/2011/11/04/opinion/oligarchy-american-style.html November 3, 2011
Oligarchy, American Style
By PAUL KRUGMAN
Inequality is back in the news, largely thanks to Occupy Wall Street, but with an assist from the
Congressional Budget Office. And you know what that means: It’s time to roll out the obfuscators!
Anyone who has tracked this issue over time knows what I mean. Whenever growing income disparities
threaten to come into focus, a reliable set of defenders tries to bring back the blur. Think tanks put out
reports claiming that inequality isn’t really rising, or that it doesn’t matter. Pundits try to put a more
benign face on the phenomenon, claiming that it’s not really the wealthy few versus the rest, it’s the
educated versus the less educated.
So what you need to know is that all of these claims are basically attempts to obscure the stark reality: We
have a society in which money is increasingly concentrated in the hands of a few people, and in which
that concentration of income and wealth threatens to make us a democracy in name only.