Measuring Inequality

A practical workshopOn theory and technique

San Jose, Costa RicaAugust 4 -5, 2004

Panel Session on:

The Mathematics and

Logic of The Theil Statistic

byJames K. Galbraith

and Enrique Garcilazo

The University of Texas Inequality Project

http://utip.gov.utexas.edu

Session 2

Outline

1. Shannon’s Measure of Information

2. Theil’s Measure of Income Inequality at the Individual level

3. Decomposition of the Theil Statistic - Fractal Properties

4. Two Level Hierarchical Decomposition

Shannon’s Measure of Information

Claude Shannon (1948) – developed theory to measure the value of

information. – more unexpected an event, higher yield of

information– information content and transmission channel

formulated in a probabilistic point of view – measure information content of an event as a

decreasing function of the probability of its occurrence

– logarithm of the inverse of the probability as a way to translate probabilities into information

Shannon’s Measure of Information

Formally if there are N events, one of which we are certain is going to occur, each have a probability xi of occurring so that:

The expected information content is given by the level of entropy:

1

1

n

iix

i

n

ii xxxH 1log)(

1

Shannon’s Measure of Information

Level of entropy interpreted as the relative differences of information

Smaller entropy means greater equality:

– The least equal case when one individual has all the income– Spread the income evenly among more people our measure

should increase– n individuals with same income. if we and take away from all

and give it to one our measure should decrease

n Sequence of xi sum y=sum x*ln(1/x) ln(N)1 1.00 . . . . . . . . . 1 0.000 0.0002 0.50 0.50 . . . . . . . . 1 0.693 0.6934 0.25 0.25 0.25 0.25 . . . . . . 1 1.386 1.386

10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 1 1.651 2.3032 0.50 0.50 . . . . . . . . 1 0.693 0.6932 0.60 0.40 . . . . . . . . 1 0.673 0.6932 0.90 0.10 . . . . . . . . 1 0.325 0.693

10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 1 1.651 2.30310 0.91 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 1 0.370 2.303

Theil’s Income Equality Measure

Henry Theil (1967) used Shannon’s theory to produce his measure of income inequality

The problem in analogous by using income shares (y) instead of probabilities (x) thus:

The measure of income equality becomes:

i

n

ii yyyH 1log)(

1

n

iiy

1

1

Theil’s Income Inequality Measure

To obtain income inequality Theil subtracted income equality from its maximum value

Maximum value of equality occurs when all individuals earn the same income shares (yi=1/N) thus:

Income inequality becomes:

NN

NyH

n

i

loglog1)(1

)(log yHN

Theil’s Income Inequality Measure

.

n

i ii yyNyHN

1

)1log(log)(log

n

iii yyN

1

)log(log

n

iii

n

ii yyNy

11

)log(log

n

iiy

1

1

NyyyNy i

n

iii

n

ii

1log)*log(11

Theil’s Income Inequality Measure

.

Calculates income inequality for a given sequence/distribution of individuals

Nyy i

n

ii

1log1

Theil’s Inequality Measure Income inequality (expressed in relative

terms) can be expressed in absolute terms:

where– y(iT) = total income earned by

person I– Y=sum Yi = total income of all people

NYy

Yy iT

n

i

iT 1log1

Partitioning The Theil Statistic If we structure our

sequence/distribution into groups– each individual belongs to one

group The total Theil is the sum of:

– between-group (A,B) and a within- group component

Group A Group B

Partitioning The Theil Mathematically the Theil is expressed as:

Groups (g) range from 1 to k

Individuals (p) within each group range from1 to n(g) First term measures inequality between

groups Second term measures inequality within

groups

gn

p gg

gp

g

gpgggk

gi

g

nYy

Yy

YY

Nn

YY

YY

11

1log*log

Partitioning The Theil Formally:

Where:B WT T T

Nn

YY

YY

T ggk

gi

gB log

1

1

kg

W Wgi

YT T

Y

1

1loggn

gp gpW

p g gg

y yT

Y nY

Partitioning The Theil The between group in now a within group as

well If distribution partitioned into m groups

where n = # individuals in each group:

– income and population relative to larger group

– weighted by income shares of that group– at individual level population equals one

g

gg

gt

gggii

g g

ggi gii

g

gi

iii

iiii

iii

iiiim

i iii

iiiim

i

m

i

m

i

iiim

i n

n

Y

Y

Y

Y

Y

YT

...

...

...

...

1 ....

....

1 1 1

...

1 21

121

21

121...1

1 21

121

1

1

2

1.....121

211

3

log.............

Partitioning The Theil• The Theil has a mathematical property of a

fractal or self similar structure: Partitioned into groups if they are MECE.

Partitioning The Theil Three Hierarchical Levels

– Income weight is group pay of each group relative to the total

– At the individual level population equals to one

1 1

1 1

1 1 1

1 1

ln ln ln

gg gg gn n

gg gn

g g gg g gg gi in n n

g gg gg gg gg g gn n

i i ig g

i i

Y YY Y YY Y Y Y YYn n nY Y Y Y YN n n

1

n n

g

Partitioning The Theil Typically we face one or two

hierarchical levels : Data is aggregated by geographical

units. Each geographical is composed further into industrial sectors (we no longer have individual data)

1

1 1 1

1 1

ln ln

n

i Yn n kiip ip ii i i

n ni i p i ij i

i i ii i

Y Y Y Y YY YTY Y n nY n n

Two Level Hierarchy – Between Theil

The left hand side is the between group component:

– Expressed in absolute terms

n

in

iii

n

iYi

n

ii

iB

nn

YY

Y

YT1

1

1

1

ln

Two Level Hierarchy – Between Theil

– Convert absolute income into average income:

– The between expressed in average terms is very intuitive

i i iy Y n

i

n

ii

n

iii nYY

11

n

in

iii

n

iiiii

n

iii

iiB

nn

nYny

nY

nyT

1

1

1

1

**ln

*

*

n

i ii

in

iiiB Y

yYynnT

1 1

ln

Two Level Hierarchy – Between Theil

– Bounded by zero and Log N– Negative component if group is

below average – Positive component if group above

average– Sum must be positive

n

i ii

in

ii

iB Y

yYy

n

nT

1

1

ln

Two Level Hierarchy – Within Theil

Calculate Theil within each group (among p individuals/groups) weights are relative income of each group i

Sum of all weighted components is the within Theil component

1

ni

W Wii

YT TY

1

lnk

ip ip iWi

p i ij i

Y Y YT

Y n n

Data Collection When our distribution given by groups

that are MECE we need to collect data on two variables:1. Population 2. Income

Income data usually obtained through surveys:– Lack of objectivity (bias associated)– Changing standards of surveys through time– Lack of comparability at country level– Expensive to obtain

– Quality not very reliable Deininger and Squire data

Data Collection Data on industrial wages

– Objectivity – Consistency through time – Easily available (cheaper)– Better quality

Analysis with Theil is perfectly valid variables of interest are:1. number of people employed2. compensation variable such as wages

Obtain a measure of pay-inequality

Advantages of Decomposition and Pay-Inequality

Consistent data through time series:– measure evolution of pay-inequality

through time – other measures (by surveys) are limited

to time comparisons. Consistent data in by different

sectors:– industrial composition a backbone of

the economy

For more information:

The University of Texas Inequality Project

http://utip.gov.utexas.edu

Type “Inequality” into Google to find us on the Web