slides_02_measures_continued.pdf

slides_02_measures_continued.pdf - Econ 473 Class 2...

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Econ 473 Class 2 Numerical inequality measures
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Today Numerical inequality measures I Gini coefficient I Equalizing transfers Desirable properties of inequality measures Stata introduction Next: Link measurement to theory
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Numerical inequality measures
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Common inequality measures variance coefficient of variation Gini coefficient log variance quantile-based measures Later: theory-based measures
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The variance V = 1 N N X i =1 ( y i - ¯ y ) 2 An obvious candidate. Problem: It rises with average income. For example, doubling all incomes, V 0 = 1 N N X i =1 (2 y i - y ) 2 = 1 N N X i =1 4( y i - ¯ y ) 2 = 4 V . Two remedies.
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The coefficient of variation cv = sd The previous problem disappears: cv 0 = sd 0 0 = 2 sd / (2 μ ) = cv .
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The log variance is actually the variance of log income: v = 1 N N X i =1 (ln y - ln y ) 2 . Again, this is insensitive to scale: v 0 = 1 N N X i =1 (ln(2 y ) - ln(2 y )) 2 = 1 N N X i =1 (ln 2 + ln y - ( ln 2 + ln y )) 2 = 1 N N X i =1 (ln 2 + ln y - ln 2 - ln y ) 2 = v .
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Sensitivity to equalizing transfers A desirable feature of an inequality measure is that it decreases if an equalizing transfer is made: Reduce y i by 1 and raise y j by 1, i > j in ordered data. Where necessary, assume that this does not affect rankings. Δ V = V y j - V y i ¯ y constant = 1 N [2( y j - ¯ y ) - 2( y i - ¯ y )] = 2 N ( y j - y i ) < 0 . Δ c = 1 sd 1 N y j - y i μ < 0 . Both of these depend only on y i - y j , not on the position of i and j in the distribution. This appears desirable.
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Sensitivity to equalizing transfers – 2 The log variance: Δ v = 2 N ln y j y j - ln y y j - ln y i y i - ln y y i Given y i - y j , this increases in y i , which is probably desirable, but: it can turn > 0! That is undesirable. (It happens at very high incomes. Try it out. ) Since Δ v increases in y i , redistribution in the lower tail affects this inequality measure most.
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Other aspects V ignores the skewness of empirical income distributions. The log variance captures the distribution perfectly if income is actually log-normally distributed. A skewness statistic is an interesting additional aspect, but not a measure of inequality per se .
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The Gini coefficient Probably the most popular “fancy” inequality measure. Developed by Corrado Gini, 1912. Defined as (twice) the area between the 45 degree line and the Lorenz curve. Ranges between 0 (full equality) and 1 (one person has it all). (If all incomes are positive.) There are alternative definitions of what it captures.
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The Gini coefficient 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Proportion of population Fraction of income Gini coe ffi cient
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The Gini coefficient A definition based on the graph: G = 1 - 2 Z Φ( y )d F ( y ) There are also alternative definitions. For example, in the discrete case with N individuals, G = 1 2 N 2 N i =1 N j =1 | y i - y j | ¯ y , the relative mean difference over mean income.
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The Gini coefficient: Sensitivity to transfers Reduce y i by 1 and raise y j by 1, i > j in ordered data, without changing the ranking.
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