gini in a bottle: the mathematics of income inequality in a bottle: the mathematics of income...

Gini in a Bottle:The Mathematics

of Income Inequality

Rich BeveridgeClatsop Community College

[email protected]://www.clatsopcc.edu/rich-beveridges-homepage

Statistics and Social Justice

• In 1999, I took my high school math class to the library and asked them each to find a data set that they thought was interesting and to make a graph of it.

• They were then to write a paragraph explaining why they chose the data set and what their graph showed.


• I was looking through the Statistical Abstract of the United States and came across the budget summary.

• There are some interesting things you can do with the information in the budget summary.


• I decided to look at how much the federal government collects from the corporate income tax.

• The 1999 budget contains revenue information from 1997. Corporate income taxes collected were $182,293,000,000 and the total revenues were $1,579,292,000,000.


• This comes out to about 11.5% of total federal revenue .

• I wondered what this percentage looked like over time.

• I collected the data going back to about 1920 and produced a graph very much like the next slide.


• About five years ago, my interest was piqued again by the next graph.

• Share of income for the top 10%(1917-2007) Piketty/Saez


• There are many factors that affect this income distribution.

• One of them is tax policy, so I decided to look at the top tax rates during the 20th century.

• These are the marginal rates charged just on income over a certain (very high) threshold.

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

80.0%

90.0%

100.0%

1913

1916

1919

1922

1925

1928

1931

1934

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1958

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2009

Highest Marginal Tax Rate

US Top Marginal Tax Rate (Federal Individual Income Tax)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1913

1918

1923

1928

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1988

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Top

MTR

(Fed

eral

Indi

vidu

al In

com

e Ta

x)

Source: statistics computed by the author

FIGURE 1AThe Top Decile Income Share in the United States, 1917-2010

Source: Piketty and Saez (2003), series updated to 2010. Income is defined as market income including realized capital gains (excludes government transfers).

25%

30%

35%

40%

45%

50%

1917

1922

1927

1932

1937

1942

1947

1952

1957

1962

1967

1972

1977

1982

1987

1992

1997

2002

2007

Shar

e of

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ome

goin

g to

Top

10%


• Inequality is rising – we’ll examine the Gini Index in detail later.

• Corporate profits have skyrocketed in the last 10-15 years, but this hasn’t translated into increased income for all Americans.


• The next graph shows corporate profits as a percentage of Gross Domestic Product.

• GDP is defined as the “market value of all officially recognized final goods and services produced in the United States.”


• As corporate profits have increased, CEO compensation has also increased.

• Calculated as multiple of the median wage for each corporation.


• CEO multiple web page


• While corporate profits and CEO compensation have increased dramatically, the income growth for other groups has been stagnant.


• As the median income has stagnated, Americans have increased their credit card debt.


• There has also been a disconnect between worker productivity and worker compensation in the last 30 years.

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Unemployment

• High unemployment is also a problem.

• The next graph examines the Beveridge Curve for the last 10 years.

• The Beveridge Curve compares the unemployment rate with the job openings rate.

Unemployment

• The Beveridge Curve was developed in 1958 by British economists J.C.R. Dow and L.A. Dicks-Mireaux and named for British economist William Beveridge.

Unemployment

• As the job openings increase, the unemployment decreases.

• In the graph of the Beveridge Curve we see a disconnect in the way businesses are hiring now as compared with the period before 2008.

4

Source: Bureau of Labor Statistics, Current Population Survey and Job Openings and Labor Turnover Survey, February 12, 2013.

This graph plots the job openings rate against the unemployment rate. This graphical

representation is known as the Beveridge Curve, named after the British economist William Henry

Beveridge (1879-1963). The economy’s position on the downward sloping Beveridge Curve reflects the state of the business cycle.

During an expansion, the unemployment rate is low and the job openings rate is high.

Conversely, during a contraction, the unemployment rate is high and the job openings rate is

low. The position of the curve is determined by the efficiency of the labor market. For example, a greater mismatch between available jobs and the unemployed in terms of skills or location would

cause the curve to shift outward, up and toward the right.

From the start of the most recent recession in December 2007 through the end of 2009, each month’s point on the curve moved lower and further to the right as the job openings rate

declined and the unemployment rate rose. From 2010 to the present, the point moved up and to the left as the job openings rate increased and the unemployment rate decreased.

In December 2012, the job openings rate was 2.6 percent and the unemployment rate was 7.8

percent.

Wealth Inequality

• In 2005 behavioral economists Dan Arielyand Michael Norton conducted a survey that asked 5,000 Americans what they thought the distribution of wealth was in America, and what they thought it should be.

Source: Michael I. Norton, Harvard Business School; Dan Ariely, Duke University0 20 40 60 80 100

top 20% n second 20% n

third 20% n fourth 20% n

bottom 20% n

out of balanceA Harvard business prof and a behavioral economist recently asked more than 5,000 Americans how they thought wealth is distributed in the United States. Most thought that it’s more balanced than it actually is. Asked to choose their ideal distribution of wealth, 92% picked one that was even more equitable.

what americans think it is

what they would like it to be

actual distribution of wealth

FIGURE 1AThe Top Decile Income Share in the United States, 1917-2010

Source: Piketty and Saez (2003), series updated to 2010. Income is defined as market income including realized capital gains (excludes government transfers).

25%

30%

35%

40%

45%

50%

1917

1922

1927

1932

1937

1942

1947

1952

1957

1962

1967

1972

1977

1982

1987

1992

1997

2002

2007

Shar

e of

tota

l inc

ome

goin

g to

Top

10%

Income Inequality

• The problem of income inequality has been in the news more in the last few years than it has previously.

• One way to measure the distribution of income in a population is through the use of the “Gini Index” of an income distribution.

Income Inequality

• The Gini Index was developed by the Italian statistician Corrado Gini in 1914.

• Gini’s index was based on the work of Max O. Lorenz, an American statistician of the same era.

Max O. Lorenz

• Max Lorenz was born in Iowa in 1876 and received his Ph.D. in Economics from University of Wisconsin at Madison in 1906.

• Lorenz published a paper in 1905 while he was still a student called “Methods of Measuring the Concentration of Wealth.”

Measuring Income Inequality

• Prior to the work of Lorenz and Gini, measures of income inequality had relied on the work of the Italian economist Vilfredo Pareto.

• Both Lorenz and Gini were unsatisfied with Pareto’s method because they felt it was ill-suited to detailed analysis of income distributions.

The Lorenz Curve

• In Lorenz’s 1905 paper, he outlined the development of what is known as the “Lorenz Curve.”

• In Lorenz’s original curve, he graphed the percentages of income along the horizontal axis and the percentiles of the population along the vertical axis.

The Lorenz Curve

• In a modern Lorenz Curve of income distribution, the percentiles of the population are graphed on the horizontal axis and the percentage of the income they receive is graphed on the vertical axis.

The Lorenz Curve

• The straight diagonal line in the graph is the graph of “perfect equality.” This is an idealized graph which represents 20% of the population earning 20% of the income, 40% of the population earning 40% of the income and so on up to 100%.

The Lorenz Curve

• Although this is not a realistic expectation for income distribution it does provide a convenient and stable comparison for all of the actual income distributions.

Lorenz Curve

• The Lorenz Curve measures the dispersion/concentration within a data set and can be used for data other than income.

• Here is a Lorenz Curve for the share of business done by auctioneers in 1820s New York.

Lorenz Curve• The Lorenz Curve is a convenient way to

look the dispersion within a statiscal data set.

• While an improvement over previous methods, it still is somewhat complicated.

• Corrado Gini’s contribution was to simplify this graph and express it as a single number –the Gini Index or Gini Coefficient.

Corrado Gini

• Corrado Gini was born in 1884 near Treviso in northeast Italy.

• Gini was from a wealthy family and studied law, biology, mathematics and statistics.

Corrado Gini

• He began to teach statistics in 1909 and ended up at the University of Rome where he remained until 1955.

• Gini founded the international journal of statistics Metron in 1920 and was president of the Italian Central Institute of Statistics from 1926-1932.

Corrado Gini

• In 1932 Gini and Mussolini had a disagreement about the Central Institute of Statistics and Gini resigned, but remained in his position at the University of Rome.

Corrado Gini

• During the period from 1912-15, Giniworked on creating an effective measure of dispersion in a data set.

• He would apply this to income distributions as what became known as the Gini Index.

Gini Index

• Gini’s idea was to take the Lorenz Curve for an income distribution and measure the area between “idealized” perfectly equal distribution and the actual distribution data.

Gini Index

• In Calculating the Gini index, we first look at the area under the idealized perfect equality portion of the graph.

• This corresponds to the sum of sections A and B.

• This is a triangle with base=1 and height=1 so the area is one half.

Gini Index

• Then after the Lorenz curve for an income distribution is plotted, the area under this curve is calculated.

• This is the section marked B.

Gini Index

• To find the area of the section marked A, we simply subtract the area under the Lorenz Curve from the area under the triangle – this gives us the difference between the actual income distribution and the idealized distribution.

Gini Index

• This value for A is then turned into a percentage by dividing it by the area under the idealized distribution.

Gini Index

As an example, let’s calculate the

Gini Index for the U.S. household

income data from 1980 and 2011

1980

Percentile Income

Share

Cumulative

Share

20% 4.2% 4.2%

40% 10.2% 14.4%

60% 16.8% 31.2%

80% 24.7% 55.9%

100% 44.1% 100%

Gini Index

The graph for this is below:

Gini Index

Now, we calculate the area under

each of the five regions.

The first is a triangle

and the rest are trapezoids

3114.0

1559.)1559(.*2.*

0871.)559.312(.*2.*

0456.)312.144(.*2.*

0186.)144.042(.*2.*

0042.042.*2.*

54321

21

5

21

4

21

3

21

2

21

1

AAAAA

A

A

A

A

A

Gini Index

The area of the triangle was 0.5, so

the area A between the idealized line

and the Lorenz Curve is

0.5-0.3114=0.1886

Then,

0.1886/0.5=0.3772

for a Gini Index of about 38.

Gini Index

2011

Percentile Income

Share

Cumulative

Share

20% 3.2% 3.2%

40% 8.4% 11.6%

60% 14.3% 25.9%

80% 23.0% 48.9%

100% 51.1% 100%

Gini Index

The graph for this is below:

Gini Index

Now, we calculate the area under

each of the five regions.

The first is a triangle

and the rest are trapezoids

2792.0

1489.)1489(.*2.*

0748.)489.259(.*2.*

0375.)259.116(.*2.*

0148.)116.032(.*2.*

0032.032.*2.*

54321

21

5

21

4

21

3

21

2

21

1

AAAAA

A

A

A

A

A

Gini Index

The area of the triangle was 0.5, so

the area A between the idealized line

and the Lorenz Curve is

0.5-0.2792=0.2208

Then,

0.1886/0.5=0.4416

for a Gini Index of about 44.

Income Inequality

• Former Chairman of the Federal Reserve Bank Alan Greenspan on income inequality:

• “As I’ve often said, this is not the type of thing which a democratic society – a capitalist democratic society – can really accept without addressing.” (2005)

Graph Sources

Individual and corporate income taxes as a percent of all federal

revenues

Source data: Office of Management and Budget

Graph source:

http://nationalpriorities.org/budget-basics/federal-budget-101/revenues/

Sources of federal revenue

Source data: Federal Budget

Graph source:

http://www.taxpolicycenter.org/briefing-book/background/numbers/revenue.cfm

Share of income for the top 10%

Source data: Piketty/Saez

Graph source: Paul Krugman

http://krugman.blogs.nytimes.com/2007/09/18/introducing-this-blog/

Increasing income share top 1%/0.1%


Graph Source: Slate/Catherine Mulbrandon

http://www.slate.com/slideshows/news_and_politics/the-great-divergence-in-

pictures-a-visual-guide-to-income-inequality.html

Top marginal tax rates


Graph source:

http://elsa.berkeley.edu/~saez/course/Labortaxes/taxableincome/taxableincome_att

ach.pdf

Corporate profits to GDP

Source data: FRED (Federal Reserve Economic Data)

Graph Source:

http://research.stlouisfed.org/fred2/graph/?g=cSh

CEO pay ratio

Source data: Economic Policy Institute

Graph source:

http://www.epi.org/blog/ceos-distance-average-worker/

Low, middle and high income growth 1973-2010

Source: Economic Policy Institute

http://stateofworkingamerica.org/charts/real-income-growth-for-different-income-

percentiles-diverged-in-the-1970s-with-real-incomes-flattening-in-the-20th-percentile-

and-the-median-and-increasing-in-the-95th-percentile/

Real mean household income by quintile

Source data: Census Bureau

Graph source:

http://www.businessinsider.com/us-household-incomes-a-42-year-perspective-2011-3

Real median household income

Source data: Census Bureau

Graph Source:

http://blogs.cfr.org/lindsay/2012/01/26/can-americans-afford-college/

Growth in credit card debt vs growth in real wages

Source data: innovestgroup.com

Graph source:

http://www.washingtonmonthly.com/archives/individual/2008_10/015301.php

Productivity vs. hourly compensation

Source data: Econommic Policy Institute Lawrence Mishel

Graph source:

http://www.epi.org/publication/ib330-productivity-vs-compensation/

Beveridge curve

Source data: Bureau of Labor Statistics

Graph source: Economic Populist

http://www.economicpopulist.org/content/job-jolts-there-are-432-unemployed-job-

opening-july-2011

Wealth distribution survey

Source data: Dan Ariely Michael Norton

Graph Source:

http://www.motherjones.com/files/outofbalance.pdf

Top 1% share of income more than doubled

Source data: Congressional Budget Office

Graph source:

http://www.cbpp.org/cms/?fa=view&id=2789

Lorenz curve

Source: #1

http://www.nssl.noaa.gov/users/brooks/public_html/feda/papers/lorenz1905%28R

OC%29.pdf

Source: #2

http://edecon.wordpress.com/2011/06/19/poverty-and-inequality/

Inequality among auctioneers in New York City 1820s

Source:

http://clioviz.wordpress.com/65-2/

Gini coefficient illustrations

Source:

http://www.psmag.com/magazines/january-february-2013/gini-coefficient-index-

poverty-wealth-income-equality-51413/

Gini index – income disparity since WWII

Source:

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

ns

gini in a bottle: the mathematics of income inequality in a bottle: the mathematics of income...

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