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 |