Gini in a Bottle: The Mathematics of Income Inequality

Gini in a Bottle:
The Mathematics
of Income Inequality
Rich Beveridge
Clatsop Community College
[email protected]
https://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.
Statistics and Social Justice
• 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.
Statistics and Social Justice
• 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.
Statistics and Social Justice
• 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.
Statistics and Social Justice
• About five years ago, my interest was
piqued again by the next graph.
• Share of income for the top 10%
(1917-2007) Piketty/Saez
Statistics and Social Justice
• 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.
2009
2006
2003
2000
1997
1994
1991
1988
1985
1982
1979
1976
1973
1970
1967
1964
1961
1958
1955
1952
1949
1946
1943
1940
1937
1934
1931
1928
1925
1922
1919
1916
1913
Highest Marginal Tax Rate
100.0%
90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%
US Top Marginal Tax Rate (Federal Individual Income Tax)
100%
Top MTR (Federal Individual Income Tax)
90%
80%
70%
60%
50%
40%
30%
20%
10%
Source: statistics computed by the author
2008
2003
1998
1993
1988
1983
1978
1973
1968
1963
1958
1953
1948
1943
1938
1933
1928
1923
1918
1913
0%
45%
40%
35%
30%
2007
2002
1997
1992
1987
1982
1977
1972
1967
1962
1957
1952
1947
1942
1937
1932
1927
1922
25%
1917
Share of total income going to Top 10%
50%
FIGURE 1A
The 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).
Statistics and Social Justice
• 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.
Statistics and Social Justice
• 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.”
Statistics and Social Justice
• As corporate profits have increased, CEO
compensation has also increased.
• Calculated as multiple of the median wage
for each corporation.
Statistics and Social Justice
• CEO multiple web page
Statistics and Social Justice
• While corporate profits and CEO
compensation have increased
dramatically, the income growth for other
groups has been stagnant.
Statistics and Social Justice
• As the median income has stagnated,
Americans have increased their credit card
debt.
Statistics and Social Justice
• There has also been a disconnect between
worker productivity and worker
compensation in the last 30 years.
Figure A
Growtth of reall hourly compens
c
ation forr producttion/nonssupervisoory
workers and prroductivity, 1948–2011
Note: Hourly
H
co
ompensattion is of productioon/nonsuppervisoryy workerss in
the priv
vate secto
or and pro
oductivity
y is for thhe total ecconomy.
Sourcee: Authorr's analysiis of unpu
ublished ttotal econnomy data from
Bureau
u of Labo
or Statisticcs, Labor Productiivity and Costs proogram andd
Bureau
u of Econ
nomic Anaalysis, Naational Inncome andd Productt Accounts
public data seriees
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.
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.
4
Wealth Inequality
• In 2005 behavioral economists Dan Ariely
and 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.
out of balance
A 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.
top 20% n
second 20% n
third 20% n
fourth 20% n
bottom 20% n
actual distribution of wealth
what americans think it is
what they would like it to be
0
20
40
60
80
100
Source: Michael I. Norton, Harvard Business School; Dan Ariely, Duke University
45%
40%
35%
30%
2007
2002
1997
1992
1987
1982
1977
1972
1967
1962
1957
1952
1947
1942
1937
1932
1927
1922
25%
1917
Share of total income going to Top 10%
50%
FIGURE 1A
The 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).
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, Gini
worked 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
Income
Cumulative
Share
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%
Percentile
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
A1  12 * .2 * .042  .0042
A2  12 * .2 * (.042  .144)  .0186
A3  12 * .2 * (.144  .312)  .0456
A4  12 * .2 * (.312  .559)  .0871
A5  12 * .2 * (.559  1)  .1559
A1  A2  A3  A4  A5  0.3114
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
Income
Cumulative
Share
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%
Percentile
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
A1  12 * .2 * .032  .0032
A2  12 * .2 * (.032  .116)  .0148
A3  12 * .2 * (.116  .259)  .0375
A4  12 * .2 * (.259  .489)  .0748
A5  12 * .2 * (.489  1)  .1489
A1  A2  A3  A4  A5  0.2792
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%
Source data: Piketty/Saez
Graph Source: Slate/Catherine Mulbrandon
http://www.slate.com/slideshows/news_and_politics/the-great-divergence-inpictures-a-visual-guide-to-income-inequality.html
Top marginal tax rates
Source data: Piketty/Saez
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-incomepercentiles-diverged-in-the-1970s-with-real-incomes-flattening-in-the-20th-percentileand-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-jobopening-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-indexpoverty-wealth-income-equality-51413/
Gini index – income disparity since WWII
Source:
http://en.wikipedia.org/wiki/Gini_coefficient#Gini_coefficient_of_income_distributio
ns