ARE/ECN 115A
Development
Economics
Discussion Section 2: Inequality
Week 3
Outline
1.
Why Study Inequality?
2.
Section Reading & Discussion
3.
Measuring Inequality
1. Axioms
2. Lorenz Curve
3. Gini Index
4.
Example with Data
The Concept of Inequality
• Unequal distribution of income or assets in a given economy
• Inequality is not poverty
• Economies with few poor people can have high inequality
• Different types of inequality
• Income, wage, wealth, land, etc.
• Different levels of inequality
• Country, rural/urban, regional, by race, by gender
Today, we will focus on income inequality
Why Does Inequality Matter?
• Positive reasons
• Political effects
• Concentration of political power
• Social unrest
• Effect on economic efficiency
• Credit access: there may be minimum level of wealth to gain access to credit
• Social unrest causes inefficient expenditures on private security
• A normative reason
• Social justice
• Gan Li’s team uses representative surveys and census methods
• Collect data on individuals’ income and assets
• Another method is to use administrative data
• Tax records, for example
• For land distribution: agricultural census
• Found a Gini Index of 0.61 in 2012
• Methodology differed from Chinese Bureau of Statistics
• Bureau found a Gini index of 0.474 – big difference!
• Li came under government scrutiny
• Are two weeks of training sufficient for this large of an important
survey? Would his survey receive less criticism if it wasn't done by
students?
• Are there alternative ways in which the data could be collected, such
as regional research centers that can gather accurate data for each
region?
• What does it mean by a Gini coefficient above 0.4 to be "socially
destabilizing"?
MEASURING INEQUALITY
Measuring inequality
Four axioms that a good inequality index should satisfy:
• Anonymity
• Individuals’ identities don’t matter to the value of index
• Independence of scale
• Only relative value of income matters
• Proportional increase in everybody’s income won’t affect the value of the index
• Population Independence
• Size of population doesn’t matter to the value of the index
• Pigou-Dalton Transfer Principle
• A regressive transfer increases the value of the index
• Transfer from a poor to a less poor person
Several inequality measures satisfy these axioms but the Gini index is the most
intuitive among them
Lorenz curve
• The Lorenz curve is a graphical representation
that relates cumulative population shares
(horizontal axis) with cumulative income
shares (vertical axis) in an economy
• A point on the Lorenz curve tell us what
percent of the total income in the economy is
controlled by the poorest X% of the
population
• The 50% poorest control 20% of the total
income in the economy
Lorenz curve
• Perfect equality
• Depicts a situation in which all individuals have the
same income
• Relative equality
• Relative Inequality
• Perfect inequality
• Depicts a situation in which only one individual
controls the total income of the economy
Lorenz Curve and Gini Index
• How do the Gini index and Lorenz
relate to each other?
• Inequality increases as Lorenz curve bows out
• Inequality increase as Gini increases
• It turns out that there’s a direct relationship:
𝐴
𝐺=
𝐴+𝐵
• We won’t use this to compute Gini!
Gini index
•
•
•
•
The most commonly used index of inequality
Lies between 0 (perfect equality) and 1 (perfect inequality)
Two ways to calculate the Gini index
In excel: covariance of income (𝑦𝑖 ) and cumulative population share (𝐹(𝑦𝑖 ))
• Not intuitive, but good for big datasets
• Pg. 121 of Taylor-Lybbert (Ch. 5 “Inequality”):
2 × 𝐶𝑜𝑣(𝑦𝑖 , 𝐹 𝑦𝑖 )
𝐺=
𝜇
• Later: Pairwise Comparisons
• More intuitive, can be computed by hand
Some Clarifications!
• Language differs between the slides and the book
• Other names for “cumulative population share”
• 𝐹(𝑦𝑖 )
• Cumulative % of population
• Cumulative distribution of income – used in the book – why?
• The cumulative distribution of income answers the question: what
percentage of people have at most 𝑦𝑖 income?
• Still about how much of the population has that income
EXAMPLE WITH DATA
Inequality Example – Three Ways
1. Lorenz Curve – on the board
2. Lorenz Curve and Gini Index – in Excel
3. Pairwise Comparison Gini Index – on handout
Drawing a Lorenz Curve
Person
ID (i)
Income (𝒚𝒊 )
Anubhab
1
240
Isaac
2
320
Daniel
3
360
Yaxi
4
480
Steve
5
650
1. Make sure your data is sorted! Sort
individuals from smallest to largest
income level
2. Calculate the cumulative share of
population for each individual
3. Calculate the cumulative share of
income for each individual
4. Draw the Lorenz curve and the
perfect equity line
Lorenz curve – on board
A
B
C
D
1
Person
ID
Income (𝑦𝑖 )
% of Pop.
2
Anubhab
1
240
3
Isaac
2
320
4
Dan
3
360
5
Yaxi
4
480
6
Steve
5
650
Lets take a look on the board
E
F
Cumu. % of Pop. % of Income
G
Cumu. % of income
Lorenz Curve and Gini Coefficient – in Excel
A
B
C
1 Person
ID
Income (𝑦𝑖 )
2 Anubhab
1
240
=1/B$7
3 Isaac
2
320
4 Dan
3
5 Yaxi
6 Steve
7 Total
F
G
% of Income
Cumu. % of income
= D2
= C2 / C$7
= F2
0.2
= E2 + D3
0.16
= G2 + F3
360
0.2
0.6
0.18
0.45
4
480
0.2
0.8
0.23
0.68
5
650
0.2
1
0.32
1.00
=count(B2:B6) = sum(C2:C6)
D
E
% of Pop. Cumu. % of Pop.
1.00
1.00
Let’s compute the Gini coefficient using the covariance method:
2 × 𝐶𝑜𝑣(𝑌𝑖 , 𝐹 𝑌𝑖 )
𝐺=
𝜇
Pairwise Comparisons – on handout
𝐺=
σ𝑛𝑖=1 σ𝑛𝑗=1 𝑛𝑖 𝑛𝑗 |𝑦𝑖 − 𝑦𝑗 |
2𝑛2 𝜇
• Intuition for numerator: making comparisons
• We make “pairwise comparisons” by taking the difference in 𝑖 and 𝑗’s incomes
• When inequality is high, this sum of these pairwise comparisons gets big
• 𝑛𝑖 or 𝑛𝑗 is the number of people in income class 𝑖 or 𝑗.
• If all individuals, 𝑛𝑖 = 𝑛𝑗 = 1
• Intuition for denominator: normalizing
• What does the numerator mean? We need to normalize to find out.
• For: double counting (2), number of comparisons (𝑛2 ), mean income (𝜇)