Measuring Inequality

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Measuring Inequality

• A practical workshop
• On theory and technique

San Jose, Costa Rica August 4 -5, 2004
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• Panel Session on
• The Mathematics
• and
• Logic of The Theil Statistic

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by James K. Galbraith and Enrique Garcilazo
The University of Texas Inequality Project
http//utip.gov.utexas.edu
Session 2
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Outline

1. Shannons Measure of Information
2. Theils Measure of Income Inequality at the
   Individual level
3. Decomposition of the Theil Statistic - Fractal
   Properties
4. Two Level Hierarchical Decomposition

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Shannons 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

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Shannons 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

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Shannons 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

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Theils Income Equality Measure

• Henry Theil (1967) used Shannons 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

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Theils 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 (yi1/N)
  thus
• Income inequality becomes

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Theils Income Inequality Measure

• .

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Theils Income Inequality Measure

• .
• Calculates income inequality for a given
  sequence/distribution of individuals

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Theils Inequality Measure

• Income inequality (expressed in relative terms)
  can be expressed in absolute terms
• where
• y(iT) total income earned by person I
• Ysum Yi total income of all people

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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

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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

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Partitioning The Theil

• Formally
• Where

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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

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Partitioning The Theil

• The Theil has a mathematical property of a
  fractal or self similar structure
• Partitioned into groups if they are MECE
• .

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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

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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)

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Two Level Hierarchy Between Theil

• The left hand side is the between group
  component
• Expressed in absolute terms

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Two Level Hierarchy Between Theil

• Convert absolute income into average income
• The between expressed in average terms is very
  intuitive

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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

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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

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Data Collection

• When our distribution given by groups that are
  MECE we need to collect data on two variables
• Population
• 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

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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
• number of people employed
• compensation variable such as wages
• Obtain a measure of pay-inequality

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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

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