Arthur CHARPENTIER - Welfare, Inequality and Poverty Arthur Charpentier charpentier.arthur@gmail.com http://freakonometrics.hypotheses.org/ Université de Rennes 1, February 2016 Welfare, Inequality & Poverty, # 3 1
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons (2-person Economy) not much to say... any measure of dispersion is appropriate • income gap x 2 − x 1 • proportional gap x 2 x 1 • any functional of the distance � | x 2 − x 1 | graphs are from Amiel & Cowell (1999, ebooks.cambridge.org ) 2
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons (3-person Economy) Consider any 3-person economy, with incomes x = { x 1 , x 2 , x 3 } . This point can be visualized in Kolm triangle. 3
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons (3-person Economy) 1 kolm=function (p=c (200 ,300 ,500) ) { 2 p1=p/sum(p) 3 y0=p1 [ 2 ] 4 x0=(2 ∗ p1 [1]+ y0 ) / sqrt (3) 5 plot ( 0 : 1 , 0 : 1 , c o l=" white " , xlab=" " , ylab=" " , axes=FALSE, ylim=c (0 ,1) ) 6 7 polygon ( c ( 0 , . 5 , 1 , 0 ) , c ( 0 , . 5 ∗ sqrt (3) ,0 ,0) ) 8 points ( x0 , y0 , pch=19, c o l=" red " ) } 4
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons ( n -person Economy) In a n -person economy, comparison are clearly more difficult 5
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons ( n -person Economy) Why not look at inequality per subgroups, If we focus at the top of the distribution (same holds for the bottom), → rising inequality If we focus at the middle of the distri- bution, → falling inequality 6
Arthur CHARPENTIER - Welfare, Inequality and Poverty Inequality Comparisons ( n -person Economy) To measure inequality, we usually • define ‘equality’ based on some reference point / distribution • define a distance to the reference point / distribution • aggregate individual distances We want to visualize the distribution of incomes 1 > income < − read . csv ( " http : //www. vcha r it e . univ − mrs . f r /pp/ lubrano / cours / f e s 9 6 . csv " , sep=" ; " , header=FALSE) $V1 � x F ( x ) = P ( X ≤ x ) = f ( t ) dt 0 7
Arthur CHARPENTIER - Welfare, Inequality and Poverty Histogram of income Densities are usually difficult to com- 0.004 pare, 0.003 1 > h i s t ( income , 2 + breaks=seq (min( income ) − 1,max( Density 0.002 income ) +50,by=50) , 0.001 3 + p r o b a b i l i t y= TRUE) 4 > l i n e s ( density ( income ) , c o l=" red " 0.000 , lwd=2) 0 500 1000 1500 2000 2500 3000 income 8
Arthur CHARPENTIER - Welfare, Inequality and Poverty ecdf(income) 1.0 It is more convenient, compare cumu- 0.8 lative distribution functions of income, 0.6 Fn(x) wealth, consumption, grades, etc. 0.4 1 > plot ( ecdf ( income ) ) 0.2 0.0 0 1000 2000 3000 x 9
Arthur CHARPENTIER - Welfare, Inequality and Poverty The Parade of Dwarfs An alternative is to use Pen’s parade, also called the parade of dwarfs (and a few giants), “parade van dwergen en een enkele reus”. The height of each person is stretched in the proportion to his or her income everyone is line up in order of height, shortest (poorest) are on the left and tallest (richest) are on the right let them walk some time, like a procession. 10
Arthur CHARPENTIER - Welfare, Inequality and Poverty c.d.f., quantiles and Lorenz Pen's Parade 10 8 6 x ( i ) x 1 > Pen( income ) 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 i n 11
Arthur CHARPENTIER - Welfare, Inequality and Poverty c.d.f., quantiles and Lorenz This parade of the Dwarfs function is just the quantile function. 1 > q < − function (u) qua nt i le ( 3000 income , u) 2500 see also 2000 sort(income) 1 > n < − length ( income ) 1500 2 > u < − seq (1 / (2 ∗ n) ,1 − 1/ (2 ∗ n) , 1000 length=n) 500 3 > plot (u , s o r t ( income ) , type=" l " ) 0 plot ( ecdf ( income ) ) 0.0 0.2 0.4 0.6 0.8 1.0 u 12
Arthur CHARPENTIER - Welfare, Inequality and Poverty c.d.f., quantiles and Lorenz To get Lorentz curve, we substitute on the y -axis proportion of incomes to incomes . Lorenz curve 1.0 0.8 1 > l i b r a r y ( ineq ) 0.6 2 > Lc ( income ) L(p) 3 > L < − function (u) Lc ( income ) $L [ 0.4 round (u ∗ length ( income ) ) ] 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 p 13
Arthur CHARPENTIER - Welfare, Inequality and Poverty c.d.f., quantiles and Lorenz x -axis y -axis c.d.f. income proportion of population Pen’s parade proportion of population income (quantile) Lorenz curve proportion of population proportion of income 14
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion The variance for a sample X = { x 1 , · · · , x n } is n Var( X ) = 1 � [ x i − x ] 2 n i =1 n where the baseline (reference) is x = 1 � x i . n i =1 1 > var ( income ) [ 1 ] 34178.43 2 problem it is a quadratic function, Var( α X ) = α 2 Var( X ). 15
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion An alternative is the coefficient of variation, � Var( X ) cv( X ) = x But not a good measure to capture inequality overall, very sensitive to very high incomes 1 > cv < − function ( x) sd ( x ) /mean( x) 2 > cv ( income ) [ 1 ] 0.6154011 3 16
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion An alternative is to use a logarithmic transformation. Use the logarithmic variance n Var log ( X ) = 1 � [log( x i ) − log( x )] 2 n i =1 1 > var_log < − function ( x) var ( log ( x ) ) 2 > var_log ( income ) [ 1 ] 0.2921022 3 Those measures are distances on the x -axis. 17
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion Other inequality measures can be derived from Pen’s parade of the Dwarfs, where measures are based on distances on the y -axis, i.e. distances between quantiles. Q p = F − 1 ( p ) i.e. F ( Q p ) = p e.g. the median is the quantile when p = 50%, the first quartile is the quantile when p = 25%, the first quintile is the quantile when p = 20%, the first decile is the quantile when p = 10%, the first percentile is the quantile when p = 1% 1 > qua nt ile ( income , c ( . 1 , . 5 , . 9 , . 9 9 ) ) 10% 50% 90% 99% 2 3 137.6294 253.9090 519.6887 933.9211 18
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion Define the quantile ratio as R p = Q 1 − p Q p In case of perfect equality, R p = 1. The most popular one is probably the 15 90/10 ratio. 1 > R _p < − function (x , p) qua nt ile (x 10 ,1 − p) / q ua nt il e (x , p) R 2 > R _p( income , . 1 ) 5 90% 3 4 3.776 0 0.0 0.2 0.4 0.6 0.8 1.0 probability This index measures the gap between the rich and the poor. 19
Arthur CHARPENTIER - Welfare, Inequality and Poverty E.g. R 0 . 1 = 10 means that top 10% incomes are more than 10 times higher than the bottom 10% incomes. Ignores the distribution (apart from the two points), violates transfer principle. An alternative measure might be Kuznets Ratio, defined from Lorenz curve as the ratio of the share of income earned by the poorest p share of the population and the richest r share of the population, L ( p ) I ( p, r ) = 1 − L (1 − r ) But here again, it ignores the distribution between the cutoffs and therefore violates the transfer principle. 20
Arthur CHARPENTIER - Welfare, Inequality and Poverty An alternative measure can be the IQR, interquantile ratio, IQR p = Q 1 − p − Q p 4 Q 0 . 5 3 IQR 2 1 > IQR_p < − function (x , p) ( qua nt ile (x,1 − p) − qua nt ile (x , p) 1 ) / qua nt ile (x , . 5 ) 2 > IQR_p( income , . 1 ) 0 90% 0.0 0.1 0.2 0.3 0.4 0.5 3 probability 4 1.504709 Problem only focuses on top (1 − p )-th and bottom p -th proportion. Does not care about what happens between those quantiles. 21
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion Pen’s parade suggest to measure the green area, for some p ∈ (0 , 1), M p , 1 > M _p < − function (x , p) { a < − seq (0 ,p , length =251) 2 b < − seq (p , 1 , length =251) 3 ya < − qua nt ile (x , p) − qua nt ile (x , 4 a ) a1 < − sum (( ya [1:250]+ ya [ 2 : 2 5 1 ] ) 5 /2 ∗ p/ 250) yb < − qua nt ile (x , b) − qua nt ile (x , 6 p) a2 < − sum (( yb [1:250]+ yb [ 2 : 2 5 1 ] ) 7 /2 ∗ (1 − p) / 250) 8 return ( a1+a2 ) } 22
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion Use also the relative mean deviation n � � M ( X ) = 1 � x i � � � x − 1 � n i =1 1 > M < − function ( x) mean( abs ( x/mean( x) − 1)) 2 > M( income ) [ 1 ] 0.429433 3 in case of perfect equality, M = 0 23
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard statistical measure of dispersion Finally, why not use Lorenz curve. It can be defined using order statistics as n 2 i · x i : n − n + 1 � G = n ( n − 1) x n − 1 i =1 1 > n < − length ( income ) 2 > mu < − mean( income ) 2 ∗ sum ( ( 1 : n) ∗ s o r t ( income ) ) / (mu ∗ n ∗ (n − 1)) − (n 3 +1)/ (n − 1) [ 1 ] 0.2976282 4 Gini index is defined as the area below the first diagonal and above Lorenz curve 24
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