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Measuring Inequality. A practical workshop On theory and technique. San Jose, Costa Rica August 4 -5, 2004. Panel Session on: The Mathematics and Logic of The Theil Statistic. by James K. Galbraith and Enrique Garcilazo. The University of Texas Inequality Project. - PowerPoint PPT Presentation
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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