Measuring Inequality

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 http://utip.gov.utexas.edu Session 2

Outline • Shannon’s Measure of Information • Theil’s Measure of Income Inequality at the Individual level • Decomposition of the Theil Statistic - Fractal Properties • 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 .

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: • 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

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

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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Measuring Inequality