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 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:

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

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:

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:

Theil’s Income Inequality Measure.

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

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

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

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

Partitioning The Theil  Formally:  Where:

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

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

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)

Two Level Hierarchy – Between Theil  The left hand side is the between group component: –Expressed in absolute terms

Two Level Hierarchy – Between Theil –Convert absolute income into average income: –The between expressed in average terms is very intuitive

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

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

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 employed 2.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

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