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 smccathern@apu.edu, by Friday

December 5.

Resources

Gini Coefficient

Wikipedia: Gini coefficient

http://en.wikipedia.org/wiki/Gini_coefficient

Gini index map

http://www.sitesatlas.com/Thematic-Maps/Gini-index-(distribution-of-family-income).html

CBO Summary (government report)

Congressional Budget Office Summary: “Trends in the Distribution of Household Income Between

1979 and 2007.”

http://www.cbo.gov/ftpdocs/124xx/doc12485/WebSummary.pdf

Wikipedia list of Countries by Income Inequality

http://en.wikipedia.org/wiki/List_of_countries_by_income_equality (Note that in this list,

the least equal countries are at the top)

Christian Web Resources

Sojourners Blog: “Poverty: By the Numbers”

http://blog.sojo.net/blogs/2011/09/14/poverty-numbers

Compassion: “What the Bible Says About Poverty”

http://www.compassion.com/poverty/what-the-bible-says-about-poverty.htm

Scripture

Isaiah 58.6-11

"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."

Matthew 25.34-40

"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.

The budget office laid out some of that stark reality in a recent report, which documented a sharp decline

in the share of total income going to lower- and middle-income Americans. We still like to think of

ourselves as a middle-class country. But with the bottom 80 percent of households now receiving less

than half of total income, that’s a vision increasingly at odds with reality.

In response, the usual suspects have rolled out some familiar arguments: the data are flawed (they aren’t);

the rich are an ever-changing group (not so); and so on. The most popular argument right now seems,

however, to be the claim that we may not be a middle-class society, but we’re still an upper-middle-class

society, in which a broad class of highly educated workers, who have the skills to compete in the modern

world, is doing very well.

It’s a nice story, and a lot less disturbing than the picture of a nation in which a much smaller group of

rich people is becoming increasingly dominant. But it’s not true.

Workers with college degrees have indeed, on average, done better than workers without, and the gap has

generally widened over time. But highly educated Americans have by no means been immune to income

stagnation and growing economic insecurity. Wage gains for most college-educated workers have been

unimpressive (and nonexistent since 2000), while even the well-educated can no longer count on getting

jobs with good benefits. In particular, these days workers with a college degree but no further degrees are

less likely to get workplace health coverage than workers with only a high school degree were in 1979.

So who is getting the big gains? A very small, wealthy minority.

The budget office report tells us that essentially all of the upward redistribution of income away from the

bottom 80 percent has gone to the highest-income 1 percent of Americans. That is, the protesters who

portray themselves as representing the interests of the 99 percent have it basically right, and the pundits

solemnly assuring them that it’s really about education, not the gains of a small elite, have it completely

wrong.

If anything, the protesters are setting the cutoff too low. The recent budget office report doesn’t look

inside the top 1 percent, but an earlier report, which only went up to 2005, found that almost two-thirds of

the rising share of the top percentile in income actually went to the top 0.1 percent — the richest

thousandth of Americans, who saw their real incomes rise more than 400 percent over the period from

1979 to 2005.

Who’s in that top 0.1 percent? Are they heroic entrepreneurs creating jobs? No, for the most part, they’re

corporate executives. Recent research shows that around 60 percent of the top 0.1 percent either are

executives in nonfinancial companies or make their money in finance, i.e., Wall Street broadly defined.

Add in lawyers and people in real estate, and we’re talking about more than 70 percent of the lucky one-

thousandth.

But why does this growing concentration of income and wealth in a few hands matter? Part of the answer

is that rising inequality has meant a nation in which most families don’t share fully in economic growth.

Another part of the answer is that once you realize just how much richer the rich have become, the

argument that higher taxes on high incomes should be part of any long-run budget deal becomes a lot

more compelling.

The larger answer, however, is that extreme concentration of income is incompatible with real

democracy. Can anyone seriously deny that our political system is being warped by the influence of big

money, and that the warping is getting worse as the wealth of a few grows ever larger?

Some pundits are still trying to dismiss concerns about rising inequality as somehow foolish. But the truth

is that the whole nature of our society is at stake.

http://www.washingtonpost.com/opinions/three-ways-to-combat-rising-

inequality/2011/11/20/gIQAvGb5fN_story.html?hpid=z3

Three ways to combat rising inequality

By Lawrence Summers, Published: November 20, 2011

There has been a strong and troubling shift in market rewards for a small minority relative to the rewards

available to most citizens. A recent Congressional Budget Office study found that incomes of the top 1

percent of the U.S. population (adjusted for inflation) rose 275 percent from 1979 to 2007, while income

for the middle class grew only 40 percent. Even this dismal figure overstates the fortunes of typical

Americans. In 1965, only one in 20 men ages 25 to 54 was not working; by the end of this decade, it is

likely to be one in six, even if a full cyclical recovery is achieved.

Another calculation suggests that if the income distribution had remained constant from 1979 to 2007,

incomes of the top 1 percent would be 59 percent, or $780,000, lower and that incomes among the bottom

80 percent would be 21 percent, or more than $10,000, higher.

Those looking to remain serene in the face of these trends or who favor policies that would

disproportionately cut taxes at the high end — and exacerbate inequality — assert that snapshot inequality

is all right as long as there is mobility within people’s lifetimes and across generations. In fact, there is

too little of both. Inequality in lifetime incomes is only marginally smaller than inequality in a single year.

And intergenerational mobility in the United States is now poor by international standards.

Why has the top 1 percent done so well relative to the rest? The answer lies substantially in changes in

technology and in globalization. When George Eastman revolutionized photography, he did very well,

and because he needed a large number of Americans to carry out his vision, the city of Rochester, N.Y.,

had a thriving middle class for two generations. When Steve Jobs revolutionized personal computing, he

and Apple shareholders did very well, but those shareholders are all over the world, and a much smaller

benefit flowed to middle-class American workers, both because production was outsourced and because

the production of computers and software was not terribly labor-intensive.

The market system distributes rewards increasingly inequitably. On one side, the debate is framed in zero-

sum terms, and the disappointing lack of income growth for middle-class workers is blamed on the

success of the wealthy. Those with this view should consider whether it would be better if the United

States had more, or fewer, entrepreneurs like those who founded Apple, Google, Microsoft and Facebook.

Each did contribute significantly to rising inequality. It is easy to resent the level and extent of the

increase in CEO salaries, but firms that have a single owner, such as private equity firms, pay successful

chief executives more than public companies do. And for all their problems, American global companies

have done very well compared with those headquartered in more egalitarian societies over the past two

decades. Where great fortunes are earned by providing great products or services that benefit large

numbers of people, they should not be denigrated.

Meanwhile, those who call concerns about rising inequality misplaced or a product of class warfare are

even further off base. The extent of the change in the income distribution is such that it is no longer true

that the overall growth rate of the economy is the principal determinant of middle-class income growth —

how the growth pie is distributed is at least equally important. The observation that most of the increase in

inequality reflects gains for those at the very top at the expense of everyone else further belies the idea

that simply strengthening the economy will reduce inequality. Focusing on American competitiveness, as

many urge, could easily exacerbate inequality while doing little for most Americans if the focus is placed

on measures such as corporate tax cuts or the protection of intellectual property for the benefit of

companies that are not primarily producing in the United States.

We need more and better responses to rising inequality. Here are three places to start.

First, government must not facilitate increases in inequality by rewarding the wealthy with special

concessions. Where governments dispose of assets or allocate licenses, preference should be on the use of

auctions to which all have access. Where government provides implicit or explicit insurance, premiums

should be based on the market rather than in consultation with the affected industry. Government’s

general posture should be standing up for capitalism rather than for well-connected capitalists.

Second, there is scope for pro-fairness, pro-growth tax reform. The moment when more great fortunes are

being created and the federal deficit is growing is hardly the time for the estate tax to be eviscerated. And

there is no reason tax changes in a period of sharply rising inequality should reinforce the trends in pretax

incomes produced by the marketplace.

Third, the public sector must ensure greater equity in areas of the most fundamental importance. It will

always be the case in a market economy that some will have mansions, art, etc. More troubling is that

middle-class students’ ability to attend college has been seriously compromised by increasing tuitions and

sharp cutbacks at public universities, and that, over the past generation, a gap has opened between the life

expectancy of the affluent and the ordinary.

Neither the politics of polarization nor those of noblesse oblige on behalf of the fortunate will serve to

protect the interests of the middle class in the post-industrial economy.

The writer, a professor and past president at Harvard University, was Treasury secretary in the Clinton

administration and economic adviser to President Obama from 2009 through 2010.

http://www.washingtonpost.com/blogs/wonkblog/wp/2013/12/05/the-government-is-the-only-

reason-u-s-inequality-is-so-high/

December 5, 2013

The Government is the Only Reason U.S. Inequality is so High

By Dylan Matthews

While the rhetoric in President Obama's big inequality speech Wednesday was characteristically soaring,

the policy proposals were largely rehashes of past administration initiatives. What's more, a surprising

number of them had little to do with tax or transfer programs. Things like increasing exports, reducing

certain regulations, boosting spending on scientific research and other investments, and raising the

minimum wage are intended to reduce inequality before taxes or transfer programs like Social Security

and the Earned Income Tax Credit come into the picture.

That's a totally valid way of tackling the problem, and there are plenty of other initiatives, such as patent

reform or reducing occupation licensing requirements, that would reduce inequality before taxes and

transfers. Dean Baker at the Center for Economic Policy Research has a long list of worthwhile proposals

along these lines.

But it's worth noting that you can get U.S. inequality down to the levels seen in extremely egalitarian

societies like Sweden by doing nothing but changing tax and transfer policies. Pre-tax/transfer inequality

in the U.S., as the above chart by the Luxembourg Income Study’s Janet Gornick shows, is about equal to

that of Sweden, Norway, and Denmark. Finland, Germany, and Britain actually have higher pre-

tax/transfer inequality than the U.S. does. The only reason these countries enjoy such low levels of

inequality is that their tax and transfer systems reduce inequality much, much more than the U.S.

system does.

And, interestingly, the way they do that is by using relatively regressive but highly efficient consumption

taxes to fund very generous welfare states. That's a marked contrast to the U.S. approach of having

relatively progressive income taxes but stingier spending programs.

http://www.nytimes.com/2011/11/20/opinion/sunday/reading-between-the-poverty-

lines.html?src=recg

November 19, 2011

Reading Between the Poverty Lines

By TERESA TRITCH

A new and improved gauge of poverty, released this month by the Census Bureau, shows that 49.1

million Americans are poor, and that the ranks of those just above poverty are larger than previously

believed. The middle class is under pressure, too, battered by stagnating incomes and unavoidable

expenses like medical bills.

The older, official poverty line is still used to determine eligibility for government benefits, but the new

formula offers a broader view of life both in and out of poverty. These numbers bear directly on issues of

joblessness, budget cuts and health care costs — and more broadly on the question of whether

government policies to help the poor and boost the economy do any good. The answer is an emphatic yes.

If only lawmakers were paying attention. Instead, they are fixated on budget cutting, generally

downplaying the good that government programs do while ignoring the consequences when they fall

short.

However you slice it, the definition of poverty is abysmally low. In the new formula, called the

supplemental poverty measure, the poverty line, based on the cost of food, shelter, clothing and utilities,

is $11,282 for an individual and $24,343 for a family of four, slightly higher than the official threshold.

To determine who is poor, both the official and the supplemental formula count cash income, generally

from work or welfare, unemployment benefits, Social Security and investments. The supplemental

measure also includes the value of tax credits for low-income workers and noncash government benefits,

like food stamps, subsidized school lunches and housing aid, while subtracting necessary outlays, like

medical spending, taxes, commuting and child care.

The chart compares the share of Americans at various income levels. The difference between the

measures reflects the effect of government benefits, taxes and out-of-pocket expenses, as well as the

change in the poverty threshold itself.

The poor make up 16.1 percent of the population under the new formula, compared with 15.2 percent by

the official measure — a difference that works out to about three million people. It’s telling that by the

new measure there were 3.2 million fewer children in poverty and one million fewer people in extreme

poverty precisely because government benefits aimed at those vulnerable groups are indeed helping. But

even taking benefits into account, the poverty rate for children is still 18.2 percent; it’s 22.5 percent under

the official measure. Poverty among the elderly, however, is greater by the supplemental measure — 15.9

percent versus 9 percent officially — largely because out-of-pocket medical costs and other expenses

swamped the noncash benefits, like food and housing assistance, that some people over 65 receive.

The low-income group — 98 million Americans by the new measure — is bigger partly because millions

were lifted above the poverty line by government benefits. For example, without the earned-income tax

credit for the working poor, six million more people would have been poor last year. Another factor is

Americans falling out of the middle class — those who make too much to receive poverty-related benefits

but shoulder expenses that eat into income. The data, on balance, show that the bigger low-income group

owes more to families being reclassified downward than to families being lifted up.

The profile of the well off also changes with the new measure. Families with four times the supplemental

poverty threshold or more — $97,372 for a family of four — were 17.3 percent of the population in 2010

compared with 35.8 percent under the official measure, largely because taxes, medical expenses and

work-related costs eroded income.

There are many lessons to draw from this picture. As long as the economy is weak and joblessness high,

antipoverty programs — like food stamps, jobless benefits, Medicaid and tax credits for the working poor

— must be extended and shielded from across-the-board budget cuts. At the same time, Americans need a

spare-no-expense effort to create jobs and retrain sidelined workers. Tax reform should ask more from the

wealthy and the financial sector to begin to distribute the burden more progressively. Health reform

should be accelerated to reduce crushing out-of-pocket medical expenses.

And as bleak as the census data already are, they tell us nothing about the huge debt burden and lack of

assets facing millions of Americans, problems that can shackle a family’s future. They don’t consider the

loss of trillions of dollars in home equity to foreclosures and plummeting home values. Nor do they

reflect the loss of retirement savings because of rocky markets or early withdrawals by unemployed

workers.

This broader crisis calls for even greater policy changes, like loan modifications and enhanced consumer

and investor protections to restore trust and stability in financial markets.

We have known for a long time what Americans need to weather the bad times. The question since the

recession began is whether Congress will deliver remedies that are big enough, timely enough and

consistent enough to help Americans struggling to hold on.

http://www.nytimes.com/2011/11/16/us/middle-class-areas-shrink-as-income-gap-grows-

report-finds.html?_r=1&hp

November 15, 2011

Middle-Class Areas Shrink as Income Gap Grows, New Report Finds

By SABRINA TAVERNISE

WASHINGTON — The portion of American families living in middle-income neighborhoods has

declined significantly since 1970, according to a new study, as rising income inequality left a growing

share of families in neighborhoods that are mostly low-income or mostly affluent.

The study, conducted by Stanford University and scheduled for release on Wednesday by the Russell

Sage Foundation and Brown University, uses census data to examine family income at the neighborhood

level in the country’s 117 biggest metropolitan areas.

The findings show a changed map of prosperity in the United States over the past four decades, with

larger patches of affluence and poverty and a shrinking middle.

In 2007, the last year captured by the data, 44 percent of families lived in neighborhoods the study

defined as middle-income, down from 65 percent of families in 1970. At the same time, a third of

American families lived in areas of either affluence or poverty, up from just 15 percent of families in

1970.

The study comes at a time of growing concern about inequality and an ever-louder partisan debate over

whether it matters. It raises, but does not answer, the question of whether increased economic inequality,

and the resulting income segregation, impedes social mobility.

Much of the shift is the result of changing income structure in the United States. Part of the country’s

middle class has slipped to the lower rungs of the income ladder as manufacturing and other middle-class

jobs have dwindled, while the wealthy receive a bigger portion of the income pie. Put simply, there are

fewer people in the middle.

But the shift is more than just changes in income. The study also found that there is more residential

sorting by income, with the rich flocking together in new exurbs and gentrifying pockets where lower-

and middle-income families cannot afford to live.

The study — part of US2010, a research project financed by Russell Sage and Brown University —

identified the pattern in about 90 percent of large and medium-size metropolitan areas for 2000 to 2007.

Detroit; Oklahoma City; Toledo, Ohio; and Greensboro, N.C., experienced the biggest rises in income

segregation in the decade, while 13 areas, including Atlanta, had declines. Philadelphia and its suburbs

registered the sharpest rise since 1970.

Sean F. Reardon, an author of the study and a sociologist at Stanford, argued that the shifts had far-

reaching implications for the next generation. Children in mostly poor neighborhoods tend to have less

access to high-quality schools, child care and preschool, as well as to support networks or educated and

economically stable neighbors who might serve as role models.

The isolation of the prosperous, he said, means less interaction with people from other income groups and

a greater risk to their support for policies and investments that benefit the broader public — like schools,

parks and public transportation systems. About 14 percent of families lived in affluent neighborhoods in

2007, up from 7 percent in 1970, the study found.

The study groups neighborhoods into six income categories. Poor neighborhoods have median family

incomes that are 67 percent or less of those of a given metropolitan area. Rich neighborhoods have

median incomes of 150 percent or more. Middle-income neighborhoods are those in which the median

income is between 80 percent and 125 percent.

The map of that change for Philadelphia is a red stripe of wealthy suburbs curving around a poor, blue

urban center, broken by a few red dots of gentrification. It is the picture of the economic change that

slammed into Philadelphia decades ago as its industrial base declined and left a shrunken middle class and

a poorer urban core.

The Germantown neighborhood, once solidly middle class, is now mostly low income. Chelten Avenue,

one of its main thoroughfares, is a hard-luck strip of check-cashing stores and takeout restaurants. The

stone homes on side streets speak to a more affluent past, one that William Wilson, 95, a longtime

resident, remembers fondly.

“It was real nice,” he said, shuffling along Chelten Avenue on Monday. Theaters thrived on the avenue,

he said, as did a fancy department store. Now a Walgreens stands in its place. “Everything started going

down in the dumps,” he said.

Philadelphia’s more recent history is one of gentrifying neighborhoods, like the Northern Liberties area,

where affluence has rushed in, in the form of espresso shops, glass-walled apartments and a fancy

supermarket, and prosperous new suburbs that have mushroomed in the far north and south of the metro

area.

Lawrence Katz, an economist at Harvard, said the evidence for the presumed adverse effects of economic

segregation was inconclusive. In a recent study of low-income families randomly assigned the

opportunity to move out of concentrated poverty into mixed-income neighborhoods, Professor Katz and

his collaborators found large improvements in physical and mental health, but little change in the

families’ economic and educational fortunes.

But there is evidence that income differences are having an effect, beyond the context of neighborhood.

One example, Professor Reardon said, is a growing gap in standardized test scores between rich and poor

children, now 40 percent bigger than it was in 1970. That is double the testing gap between black and

white children, he said.

And the gap between rich and poor in college completion — one of the single most important predictors

of economic success — has grown by more than 50 percent since the 1990s, said Martha J. Bailey, an

economist at the University of Michigan. More than half of children from high-income families finish

college, up from about a third 20 years ago. Fewer than 10 percent of low-income children finish, up from

5 percent.

William Julius Wilson, a sociologist at Harvard who has seen the study, argues that “rising inequality is

beginning to produce a two-tiered society in America in which the more affluent citizens live lives

fundamentally different from the middle- and lower-income groups. This divide decreases a sense of

community.”