Quantifying inequality
GINI COEFFICIENTS USING CALCULUS TO DISCUSS INEQUALITY
Author: Zack Beamer
HOW SEVERE IS INCOME INEQUALITY? • How can this be measured? • Ratio of CEO to Worker pay†: 354 to 1
• Workers:
$34,645
• S&P 500 CEOs: $12,259,894 • Is this a rigorous method of measuring inequality? Source: AFL-CIO (2013)
HOW SHOULD WE MEASURE INEQUALITY? • A good measurement of inequality would: • Compare individual quantities to the “whole pie” • Have graphical representation • Make comparisons between countries & over time • Apply to income, wealth, or other numerical data
COMPARISONS WITHIN POPULATIONS • Use percentiles to measure relative standing • Individual in 30% income percentile → earns more than 30% of the total population • New York Times income percentile interactive Source: NYTimes (2012)
INCOME PERCENTILES • U.S. household income percentiles Percentile 10th 25th 50th 75th 90th 95th 99th
Household Income $12,154 $25,411 $50,472 $89,125 $140,001 $188,001 $383,001 Source: NYTimes (2012)
U.S. INCOME QUINTILES • Quintiles – use five groups Quintile 1st quintile 2nd quintile 3rd quintile 4th quintile 5th quintile
Mean Household Income $11,651 $30,509 $52,322 $83,519 $185,206
Source: Census(2015)
WAGES IN CONTEXT Income
Occupations with comparable wages
$11,651
Minimum wage ($7.25/hr) - 30 hours/week
$30,509
Full-time office clerks, Bus drivers, Assembly line workers $52,322 Electricians & Plumbers, Executive Secretaries, Healthcare social workers $83,519 Engineers, Postsecondary STEM teachers, Physical Therapists $185,206 Psychiatrists, Chief Executives, Physicians Source: Bureau of Labor Statistics
COMPARING INDIVIDUALS TO THE WHOLE PIE • Suppose that each quintile were one person. • How much money would the five make together? • What would each person’s share of the total income be?
COMPARING INDIVIDUALS TO THE WHOLE PIE Quintile 1st 2nd 3rd 4th 5th Total
Household Income $11,651 $30,509 $52,322 $83,519 $185,206 $363,207
Share of total income 3.21% 8.40% 14.41% 22.99% 50.99% 100%
CUMULATIVE SHARE OF INCOME Quintile
Household Income
Share of total income
Cumulative share
1st
$11,651
3.21%
3.21%
2nd
$30,509
8.40%
11.61%
3rd
$52,322
14.41%
26.02%
4th
$83,519
22.99%
49.01%
5th
$185,206
50.99%
100%
• Let’s graph this cumulative share
VISUALIZING THE CUMULATIVE SHARE • Quintiles are divided by 5 to scale from 0 to 1. US Income Distribution
Cumulative share of income
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
Quintile
0.8
1
LORENZ CURVE • Cumulative share curve is called the Lorenz Curve. • For the Lorenz Curve function 𝐿(𝑥) • 𝐿(0) = 0 • 𝐿(1) = 1 • 𝐿’(𝑥) > 0 • 𝐿’’(𝑥) > 0
Edward Lorenz
One individual earns all income
All individuals earn equal income
Complete inequality
Complete equality
1
1
0.9
0.9
0.8
0.8
Cumulative share of income
Cumulative share of income
EXTREME CASES
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 0
0.2 0.4 Quintile
0.6
0.8
1
0 0
0.2
0.4
0.6
Quintile
0.8
1
MEASURING INEQUALITY • Compare the Lorenz Curve to complete equality numerically 1
Cumulative income percentile
0.9
Find the area between curves
0.8 0.7 0.6 0.5 0.4 0.3
Integration!
0.2 0.1 0 0
0.2
0.4
0.6 Quintile
0.8
1
THE GREATEST POSSIBLE INEQUALITY • Find the area between maximum inequality and maximum equality. Cumulative share of income
1 0.9 0.8
𝐴=
0.7 0.6 0.5
1 𝑏ℎ 2
0.4 0.3 0.2
1 2
𝐴= ×1×1=
0.1 0 0
0.2
0.4 Quintile
0.6
0.8
1
1 2
THE LEAST POSSIBLE INEQUALITY If Lorenz Curve matches the curve of maximum equality. 𝐴 = 0 1
Cumulative share of income
0.9 0.8 0.7
Area ranges between 0 and ½.
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
Quintile
0.8
1
DEFINING THE GINI INDEX • Maximum area is ½, so… •𝐺 =
1 2 0
𝑥 − 𝐿 𝑥 𝑑𝑥
• “Perfect Equality”: G = 0. • “Perfect Inequality”:G = 1. • This number is called the Gini Index, or the Gini Coefficient.
FINDING THE U.S. LORENZ CURVE • We need to model the Lorenz Curve. • To satisfy 𝐿(0) = 0 and 𝐿(1) = 1, • Use a Power Function of the form: 𝐿(𝑥) = 𝑥𝑏 • Desmos allows us to find a regression for our data. • Best fit curve: 𝐿 𝑥 = 𝑥 2.79
CALCULATING THE U.S. GINI INDEX 1
𝐺 = 2 න 𝑥 − 𝐿(𝑥)𝑑𝑥 0 1
𝐺 = 2 න 𝑥 − 𝑥 2.79 𝑑𝑥 0
1
1 2 1 3.79 𝐺=2 𝑥 − 𝑥 อ 2 3.79 1 1 𝐺=2 − ≈ 0.47 2 3.79
0
HOW DOES THE U.S. COMPARE? • The World Bank has calculated Gini Indices for countries around the globe. • Note: These numbers have been multiplied by 100 to scale between 0 and 100.
GINI INDEX ACROSS THE GLOBE
Source: World Bank (2014)
U.S. GINI INDEX OVER TIME 0.500
U.S. Income Gini Coefficient
0.480
0.460
0.440
0.420
0.400
0.380
0.360 1965
1970
1975
1980
1985
1990
1995
Year Source: US Census (2014)
2000
2005
2010
2015
US INEQUALITY IN THE LAST CENTURY
Source: UNU-Wider (2008)
THE GINI INDEX IN OTHER CONTEXTS • Lorenz Curves and Gini Coefficients apply to other contexts • The data journalism website fivethirtyeight.com uses Gini Coefficients to describe: • Income inequality among professional sports. • Win inequality between NFL teams.
CALCULATING WEALTH INEQUALITY • There are ten people. Eight people each have one apple. One person has five apples, and one person has 87 apples. • Calculate the Gini index to assess the level of inequality
SHARES OF APPLE 1
Area between 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.0045 0.0135 0.0225 0.0315 0.0405 0.0495 0.0585 0.0675 0.0745 0.0385
Total area
0.401
Gini Coefficient
0.802
0.9
Cumulative % of apples
Cumulative % Cumulative population % of apples Equal share 0 0 0.1 0.01 0.2 0.02 0.3 0.03 0.4 0.04 0.5 0.05 0.6 0.06 0.7 0.07 0.8 0.08 0.9 0.13 1 1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
Cumulative % population
1
U.S. WEALTH INEQUALITY • U.S. Wealth Gini Coefficient is 0.801. • This wealth inequality is higher than: • Japan (0.547), China (0.550), Italy (0.609), Canada (0.688), UK (0.697), France (0.730), Saudi Arabia (0.737),
Mexico (0.749), South Africa (0.763) • Few countries (Switzerland at 0.803) rank higher. Source: United Nations University World Institute for Development Economics Research
TAKEAWAYS • The Lorenz Curve illustrates a cumulative distribution. • The area between a Lorenz Curve and the curve of equality gives a numerical measurement. • The resulting number is called the Gini Coefficient, or Gini Index.
TAKEAWAYS • Gini Coefficients rank from 0 to 1. • Bigger Number → Higher Inequality
• Gini Coefficients can be used for: • Income • Wealth • And more!
QUESTIONS? • Thank you for your attendance!
SOURCES • "CEO-to-Worker Pay Ratios Around the World." AFL-CIO. N.p., 2013. Web. 18 Mar. 2015. • "What Percent Are You?" The New York Times. The New York Times, 14 Jan. 2012. Web. 13 Mar. 2015. • “Household Income Quintiles.” Tax Policy Center, Urban Institute & Brookings Institution, 28 Jan 2016. Web. 16 Aug. 2016 • "May 2013 National Occupational Employment and Wage Estimates." U.S. Bureau of Labor Statistics. U.S. Bureau of Labor Statistics, 1 Apr. 2014. Web. 10 Mar. 2015. • World Bank Development Research Group. "GINI Index (World Bank Estimate)." GINI Index (World Bank Estimate). World Bank, 2014. Web. 13 Mar. 2015. • Davies, James B., Susanna Sondstrom, Anthony Shorrocks, and Edward N. Wolff. “World Distribution of Household Wealth. United Nations University World Institute for Development Economics Research, Feb. 2008. Web. 13 Mar. 2015. • Berruyer, Olivier. "Income Inequality in the US." The Crises. N.p., 29 Dec. 2012. Web. 18 Mar. 2015.
Author: Zack Beamer
HOW SEVERE IS INCOME INEQUALITY? • How can this be measured? • Ratio of CEO to Worker pay†: 354 to 1
• Workers:
$34,645
• S&P 500 CEOs: $12,259,894 • Is this a rigorous method of measuring inequality? Source: AFL-CIO (2013)
HOW SHOULD WE MEASURE INEQUALITY? • A good measurement of inequality would: • Compare individual quantities to the “whole pie” • Have graphical representation • Make comparisons between countries & over time • Apply to income, wealth, or other numerical data
COMPARISONS WITHIN POPULATIONS • Use percentiles to measure relative standing • Individual in 30% income percentile → earns more than 30% of the total population • New York Times income percentile interactive Source: NYTimes (2012)
INCOME PERCENTILES • U.S. household income percentiles Percentile 10th 25th 50th 75th 90th 95th 99th
Household Income $12,154 $25,411 $50,472 $89,125 $140,001 $188,001 $383,001 Source: NYTimes (2012)
U.S. INCOME QUINTILES • Quintiles – use five groups Quintile 1st quintile 2nd quintile 3rd quintile 4th quintile 5th quintile
Mean Household Income $11,651 $30,509 $52,322 $83,519 $185,206
Source: Census(2015)
WAGES IN CONTEXT Income
Occupations with comparable wages
$11,651
Minimum wage ($7.25/hr) - 30 hours/week
$30,509
Full-time office clerks, Bus drivers, Assembly line workers $52,322 Electricians & Plumbers, Executive Secretaries, Healthcare social workers $83,519 Engineers, Postsecondary STEM teachers, Physical Therapists $185,206 Psychiatrists, Chief Executives, Physicians Source: Bureau of Labor Statistics
COMPARING INDIVIDUALS TO THE WHOLE PIE • Suppose that each quintile were one person. • How much money would the five make together? • What would each person’s share of the total income be?
COMPARING INDIVIDUALS TO THE WHOLE PIE Quintile 1st 2nd 3rd 4th 5th Total
Household Income $11,651 $30,509 $52,322 $83,519 $185,206 $363,207
Share of total income 3.21% 8.40% 14.41% 22.99% 50.99% 100%
CUMULATIVE SHARE OF INCOME Quintile
Household Income
Share of total income
Cumulative share
1st
$11,651
3.21%
3.21%
2nd
$30,509
8.40%
11.61%
3rd
$52,322
14.41%
26.02%
4th
$83,519
22.99%
49.01%
5th
$185,206
50.99%
100%
• Let’s graph this cumulative share
VISUALIZING THE CUMULATIVE SHARE • Quintiles are divided by 5 to scale from 0 to 1. US Income Distribution
Cumulative share of income
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
Quintile
0.8
1
LORENZ CURVE • Cumulative share curve is called the Lorenz Curve. • For the Lorenz Curve function 𝐿(𝑥) • 𝐿(0) = 0 • 𝐿(1) = 1 • 𝐿’(𝑥) > 0 • 𝐿’’(𝑥) > 0
Edward Lorenz
One individual earns all income
All individuals earn equal income
Complete inequality
Complete equality
1
1
0.9
0.9
0.8
0.8
Cumulative share of income
Cumulative share of income
EXTREME CASES
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 0
0.2 0.4 Quintile
0.6
0.8
1
0 0
0.2
0.4
0.6
Quintile
0.8
1
MEASURING INEQUALITY • Compare the Lorenz Curve to complete equality numerically 1
Cumulative income percentile
0.9
Find the area between curves
0.8 0.7 0.6 0.5 0.4 0.3
Integration!
0.2 0.1 0 0
0.2
0.4
0.6 Quintile
0.8
1
THE GREATEST POSSIBLE INEQUALITY • Find the area between maximum inequality and maximum equality. Cumulative share of income
1 0.9 0.8
𝐴=
0.7 0.6 0.5
1 𝑏ℎ 2
0.4 0.3 0.2
1 2
𝐴= ×1×1=
0.1 0 0
0.2
0.4 Quintile
0.6
0.8
1
1 2
THE LEAST POSSIBLE INEQUALITY If Lorenz Curve matches the curve of maximum equality. 𝐴 = 0 1
Cumulative share of income
0.9 0.8 0.7
Area ranges between 0 and ½.
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
Quintile
0.8
1
DEFINING THE GINI INDEX • Maximum area is ½, so… •𝐺 =
1 2 0
𝑥 − 𝐿 𝑥 𝑑𝑥
• “Perfect Equality”: G = 0. • “Perfect Inequality”:G = 1. • This number is called the Gini Index, or the Gini Coefficient.
FINDING THE U.S. LORENZ CURVE • We need to model the Lorenz Curve. • To satisfy 𝐿(0) = 0 and 𝐿(1) = 1, • Use a Power Function of the form: 𝐿(𝑥) = 𝑥𝑏 • Desmos allows us to find a regression for our data. • Best fit curve: 𝐿 𝑥 = 𝑥 2.79
CALCULATING THE U.S. GINI INDEX 1
𝐺 = 2 න 𝑥 − 𝐿(𝑥)𝑑𝑥 0 1
𝐺 = 2 න 𝑥 − 𝑥 2.79 𝑑𝑥 0
1
1 2 1 3.79 𝐺=2 𝑥 − 𝑥 อ 2 3.79 1 1 𝐺=2 − ≈ 0.47 2 3.79
0
HOW DOES THE U.S. COMPARE? • The World Bank has calculated Gini Indices for countries around the globe. • Note: These numbers have been multiplied by 100 to scale between 0 and 100.
GINI INDEX ACROSS THE GLOBE
Source: World Bank (2014)
U.S. GINI INDEX OVER TIME 0.500
U.S. Income Gini Coefficient
0.480
0.460
0.440
0.420
0.400
0.380
0.360 1965
1970
1975
1980
1985
1990
1995
Year Source: US Census (2014)
2000
2005
2010
2015
US INEQUALITY IN THE LAST CENTURY
Source: UNU-Wider (2008)
THE GINI INDEX IN OTHER CONTEXTS • Lorenz Curves and Gini Coefficients apply to other contexts • The data journalism website fivethirtyeight.com uses Gini Coefficients to describe: • Income inequality among professional sports. • Win inequality between NFL teams.
CALCULATING WEALTH INEQUALITY • There are ten people. Eight people each have one apple. One person has five apples, and one person has 87 apples. • Calculate the Gini index to assess the level of inequality
SHARES OF APPLE 1
Area between 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.0045 0.0135 0.0225 0.0315 0.0405 0.0495 0.0585 0.0675 0.0745 0.0385
Total area
0.401
Gini Coefficient
0.802
0.9
Cumulative % of apples
Cumulative % Cumulative population % of apples Equal share 0 0 0.1 0.01 0.2 0.02 0.3 0.03 0.4 0.04 0.5 0.05 0.6 0.06 0.7 0.07 0.8 0.08 0.9 0.13 1 1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
Cumulative % population
1
U.S. WEALTH INEQUALITY • U.S. Wealth Gini Coefficient is 0.801. • This wealth inequality is higher than: • Japan (0.547), China (0.550), Italy (0.609), Canada (0.688), UK (0.697), France (0.730), Saudi Arabia (0.737),
Mexico (0.749), South Africa (0.763) • Few countries (Switzerland at 0.803) rank higher. Source: United Nations University World Institute for Development Economics Research
TAKEAWAYS • The Lorenz Curve illustrates a cumulative distribution. • The area between a Lorenz Curve and the curve of equality gives a numerical measurement. • The resulting number is called the Gini Coefficient, or Gini Index.
TAKEAWAYS • Gini Coefficients rank from 0 to 1. • Bigger Number → Higher Inequality
• Gini Coefficients can be used for: • Income • Wealth • And more!
QUESTIONS? • Thank you for your attendance!
SOURCES • "CEO-to-Worker Pay Ratios Around the World." AFL-CIO. N.p., 2013. Web. 18 Mar. 2015. • "What Percent Are You?" The New York Times. The New York Times, 14 Jan. 2012. Web. 13 Mar. 2015. • “Household Income Quintiles.” Tax Policy Center, Urban Institute & Brookings Institution, 28 Jan 2016. Web. 16 Aug. 2016 • "May 2013 National Occupational Employment and Wage Estimates." U.S. Bureau of Labor Statistics. U.S. Bureau of Labor Statistics, 1 Apr. 2014. Web. 10 Mar. 2015. • World Bank Development Research Group. "GINI Index (World Bank Estimate)." GINI Index (World Bank Estimate). World Bank, 2014. Web. 13 Mar. 2015. • Davies, James B., Susanna Sondstrom, Anthony Shorrocks, and Edward N. Wolff. “World Distribution of Household Wealth. United Nations University World Institute for Development Economics Research, Feb. 2008. Web. 13 Mar. 2015. • Berruyer, Olivier. "Income Inequality in the US." The Crises. N.p., 29 Dec. 2012. Web. 18 Mar. 2015.
