Measurement of Income Inequality and Redistribution
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Transcript of Measurement of Income Inequality and Redistribution
Measurement of Income Inequality and Redistribution
Ivica UrbanInstitut za javne financije
Croatian Quants Day, Zagreb, 9 April 2010
The distribution of income
“Capitalism has solved the problem of production, but failed to solve the problem of distribution.”
Economics: not an easy science – many diverging views on how to organize society
Egalitarians, interventionsts, socialists, libertarians...
Libertarians
“There is no problem of ‘distribution’ at all.” “On the free market, everybody earns
income in accordance with services provided to consumers.”
Example: Ice-cream producer
Egalitarians
“There is too much of inequality around, as a result of market process.”
“It is an unfair situation and the government must be called to redistribute income.”
Redistribution: taking from ones and giving to others
If the “ones” are higher-income, and the “others” are lower-income people → reduction of income inequality
Socialism
Abolishment of private property Central planning of production and
income distribution No economic and political freedoms But still, we can hear: “It was a just
society, everybody had a job, noone was left behind...”
Today, we’re all egalitarians!
Modern democracies: huge “social security” systems
A massive income redistribution apparatus: taxes, benefits and regulation
How to estimate the impact of government?
Lorenz curvesIncome
unitsCumulative
shares of units p
Incomes
X
Cumulative shares of income LX(p)
0.0 0.0000
#1 0.1 50 0.0097
#2 0.2 80 0.0253
#3 0.3 100 0.0448
#4 0.4 200 0.0838
#5 0.5 250 0.1326
#6 0.6 300 0.1910
#7 0.7 350 0.2593
#8 0.8 800 0.4152
#9 0.9 1000 0.6101
#10 1.0 2000 1.0000
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0p
_
_
XL
O
Gini coefficient
= double the area between the Lorenz curve and the line of absolute equality
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0p
_
_
XL
O
s
iiXiX pLp
sG
1
)(2
Income redistribution
X = income before fiscal action Tp = amount of pth tax
Bq = amount of qth benefit N = income after taxes and benefits
N = X - Σ Tp + Σ Bq
Income redistributionPre-fiscal income
X LX(p)
Taxes
T
Benefits
B
Post-fiscal income
N
LN(p)
0 0
50 0.01 11 120 50 0.03
80 0.03 17 120 60 0.07
100 0.04 21 120 80 0.10
200 0.08 43 120 100 0.16
250 0.13 54 120 110 0.22
300 0.19 64 120 150 0.28
350 0.26 75 120 200 0.36
800 0.42 172 120 300 0.50
1000 0.61 214 120 400 0.68
2000 1.00 429 120 800 1.00
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0p
_
_
_
NL
O
Income redistributionPre-fiscal income
X
Taxes
T
Benefits
B
Post-fiscal income
N
Net fiscal gain
B-T(B-T)
/ X
50 11 120 50 109 2.19
80 17 120 60 103 1.29
100 21 120 80 99 0.99
200 43 120 100 77 0.39
250 54 120 110 66 0.27
300 64 120 150 56 0.19
350 75 120 200 45 0.13
800 172 120 300 -52 -0.06
1000 214 120 400 -94 -0.09
2000 429 120 800 -309 -0.15
Horizontal equality / inequality
0
20
40
60
80
100
120
0 20 40 60 80 100 120
Post
-fisc
alin
com
e(0
00 H
RK)
Pre-fiscal income (000 HRK)
Sample data, Croatia 2005
Horizontal equality / inequality
the principle of horizontal equality: “equals” should have equal treatment
people with equal incomes should obtain equal net fiscal benefits
Horizontal equality / inequality
0
20
40
60
80
100
120
0 20 40 60 80 100 120
Post
-fisc
alin
com
e(0
00 H
RK)
Pre-fiscal income (000 HRK)
Horizontal equality / inequality
0 20 40 60 80 100 120
Con
ditio
nal d
ensi
ty, f(
y|x)
Post-fiscal income (000 HRK)
p=0.1
p=0.3
p=0.5
p=0.7
p=0.9
p=0.95
NKDE-2D
Normal Kernel Density Estimation in 2 dimensions:
hn = optimal bandwidth
S = sample covariance
wi = sample weights
N = n x wi,
n = number of observations
n
i i
i
i
i
ni
nYy
XxS
Yy
Xx
hw
ShNx,yf
1
122 2
1exp
)det(2
11ˆ
Conditional density
Normal KDE in 2 dimensions, to obtain estimates of f(x,y), from which f(y|x) are calculated using:
dyyxf
yxf
xf
yxfxyf
yx
,
,,|
18
Measurement model
Utility of income function U(y) with inequality aversion parameter ε:
Equally distributed equivalent income (income that, if obtained by every individual, would deliver the same welfare as actual income):
1)(
1yyU
1
1
| 1x
Uxy
Measurement modelD
ensi
ty
Post-fiscal income (y)
xyf |
x
dyxyyfxy ||
x group of 'equals', all having pre-fiscal income x
conditional density of post-fiscal income
xy |
mean (conditional) post-fiscal income
xy|
equally distributed equivalent income
1
1
| 1x
Uxy
1)(
1yyU
xyf |