Progressivity, vertical and horizonal
equity
Abdelkrim Araar, Sami Bibi and Jean-Yves Duclos
Workshop on poverty and social impact analysis in Sub-Saharan Africa
Kampala, Uganda, 23-27 November 2009
Progressivity and equity
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Checking the
progressivity of taxes
and transfers
Progressivity and
poverty
Concentration curves
Checking the
progressivity
of taxes and transfers
Tax Redistribution
(TR)
Income
Redistribution (IR)
Checking the progressivity of
taxes and transfers
The measurement of
progressivity
Progressivity and equity
Kampala – 2 / 11
Progressivity and poverty
How do the poor benefit from the redistribution of national wealth?
Do the poor benefit more than the non poor from different types of
monetary and in-kind transfers?
Is the tax burden on the poor relatively low ?
Studying the progressivity of taxes and transfers can help answer these
questions.
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Concentration curves
An important descriptive and normative tool for capturing the impact of tax
and transfer policies is the concentration curve.
Suppose pre-tax incomes (gross incomes) X are ranked in ascending
order such that: X1 ≤ X2 ≤ ... ≤ Xn .
Suppose that taxes Tj (or transfers) are ranked according to the size of
their associated gross income.
The concentration curve of a tax T at percentile p is:
Pi
j=1 Tj
CT (p = i/n) = Pn
j=1 Tj
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Concentration curves
i
–
1
2
3
4
Progressivity and equity
Table 1: Illustrative Example
pi
Xi
Ti
L(pi ) C(pi )
–
0.00 0.00 0.00
0.00
0.25 100
10
0.10
0.04
0.50 200
30
0.30
0.16
0.75 300
70
0.60
0.44
1.00 400 140 1.00
1.00
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Concentration curves
1
The concentration curve shows the proportion of total taxes paid by the
bottom p proportion of the population.
45° line
L(p)
.6
.4
0
.2
L(p) and C(p)
.8
C(p)
0
.25
.5
.75
1
Percentiles (p)
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Concentration curves
When the concentration curve of a tax is below the Lorenz curve, the
poor pay less taxes than the non-poor, relative to their income: the tax is
said to be progressive.
When the concentration curve of a transfer is above the Lorenz curve, the
poor receive more transfers than the non poor, relative to their incomes:
the transfer is progressive.
What about inequality of net income N ?
There is a close link between the progressivity of taxes and transfers and
inequality in net income. If a tax is progressive, then the net income share
of the poor will be higher than the poor’s share of gross income.
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Concentration curves
The concentration curve of net incomes N is:
Pi
j=1 Nj
CN (p = i/n) = Pn
j=1 Nj
We can thus compare the concentration curve of N to the Lorenz curve
for X to assess the net progressivity of the tax and transfer system:
µT
CN (p) − LX (p) =
[LX (p) − CT (p)] .
µN
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Concentration curves
When reranking is not observed, we also find:
µT
LN (p) − LX (p) =
[LX (p) − LT (p)] .
µN
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Checking the progressivity
of taxes and transfers
There are two approaches to making progressivity comparisons:
Tax Redistribution : TR approach
Income Redistribution : IR approach.
Using Lorenz and concentration curves, the following rules can be used
to check progressivity.
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Tax Redistribution (TR)
1. A tax T is TR-progressive if:
LX (p) − CT (p) > 0 for all p ∈]0, 1[.
2. A transfer B is TR-progressive if:
CB (p) − LX (p) > 0 for all p ∈]0, 1[
3. A tax T 1 is more TR-progressive than a tax T 2 if:
CT 2 (p) − CT 1 (p) > 0 for all p ∈]0, 1[
4. A transfer B1 is more TR-progressive than a transfer B2 if:
CB1 (p) − CB2 (p) > 0 for all p ∈]0, 1[
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Income Redistribution (IR)
1. A tax or a transfer T is IR-progressive if:
CN (p) − LX (p) > 0 for all p ∈]0, 1[
2. A tax or a transfer T 1 is more IR-progressive than a tax (and/or a
transfer) T 2 if:
CN 1 (p) − CN 2 (p) > for all p ∈]0, 1[
Progressivity and equity
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Checking the
progressivity of taxes
and transfers
The measurement of
progressivity
Quantifying
progressivity
Indices of
progressivity
Redistributive Equity
Progressivity and equity
The measurement of
progressivity
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Quantifying progressivity
1. Lorenz and concentration curves may cross.
2. It may be useful to provide summary quantitative indices of progressivity.
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Indices of progressivity
Musgrave and Thin (1948) propose to measure progressivity by the ratio
N
of Gini “equality” of net income to Gini “equality” of gross income: 1−I
1−IX
This ratio will be greater than one if the tax is progressive.
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Indices of progressivity
The Kakwani index of progressivity is based on the TR approach and
equals twice the area between the Lorenz curve and the concentration
curve of a tax.
This is also the difference between the concentration index of the tax
(ICX ) and the Gini index of gross income: ICT − IX
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Indices of progressivity
The Reynolds-Smolensky index of progressivity is based on the IR
approach and equals twice the area between the concentration curve of
net incomes and the Lorenz curve of gross incomes.
This is also the difference between the Gini index of gross income and
the concentration index of net income: IX − ICN
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Indices of progressivity
0
.2
L(p) & C(p)
.4
.6
.8
1
Lorenz Curve & Concentration curves
0
.2
.4
.6
Percentiles (p)
0.5(Kakwani Index)
.8
0.5(Reynolds−Smolensky Index)
µT
R-S Index =
(Kakwani Index).
µN
Progressivity and equity
1
(1)
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Indices of progressivity
Example: Calculating progressivity indices
Rank i
1
2
3
4
Total
Xi
100
200
300
400
1000
Ti
10
30
70
140
250
Ni
90
170
230
260
750
IX = 0.25 // IN = 0.19 // ICT = 0.43 // ICN = 0.19
Musgrave and Thin index = (1 − IN )/(1 − IX ) = 1.08
Kakwani index
=
ICT − IX
= 0.18
R-S index
=
IX − ICN
= 0.06
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Redistributive Equity
Does redistribution compress the distribution of post-tax incomes?
(Vertical equity)
Are equals in pre-tax incomes treated equally by the tax system?
(Classical horizontal equity)
Does the redistribution re-rank households? (Horizontal equity as non
reranking).
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Redistributive Equity
Rank i
1
2
3
4
5
6
Average
X
100
100
150
150
200
200
150
NA
90
90
100
100
140
140
110
NB
90
100
90
100
140
140
110
NC
100
100
90
90
140
140
110
IX = 0.148 ; IN = 0.101
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Redistributive Equity
Rank i
1
2
3
4
5
6
Average
X
100
100
150
150
200
200
150
NA
90
90
100
100
140
140
110
NB
90
100
90
100
140
140
110
NC
100
100
90
90
140
140
110
Case A:
VE: Vertical equity, since inequality has decreased.
HE: Horizontal inequity equals zero since equals are treated equally.
RE: Reranking inequity equals zero since no re-ranking is observed.
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Redistributive Equity
Rank i
1
2
3
4
5
6
Average
X
100
100
150
150
200
200
150
NA
90
90
100
100
140
140
110
NB
90
100
90
100
140
140
110
NC
100
100
90
90
140
140
110
Case B:
VE: Vertical equity, since inequality has decreased.
HE: Horizontal inequity since equals are treated unequally.
RE: Reranking inequity equals zero, since no re-ranking is observed.
Progressivity and equity
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Redistributive Equity
Rank i
1
2
3
4
5
6
Average
X
100
100
150
150
200
200
150
NA
90
90
100
100
140
140
110
NB
90
100
90
100
140
140
110
NC
100
100
90
90
140
140
110
Case C:
VE: Vertical equity, since inequality has decreased.
HE: Horizontal inequity equals zero, since equals are treated equally.
RE: Reranking inequity since some households are re-ranked.
Progressivity and equity
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Redistributive Equity
Rank i
1
2
3
4
5
6
Average
X
100
100
150
150
200
200
150
NA
90
90
100
100
140
140
110
NB
90
100
90
100
140
140
110
NC
100
100
90
90
140
140
110
One can use the following decomposition of the redistributive effect on
inequality:
IX (ρ) − IN (ρ) = IX (ρ) − ICN (ρ) − (IN (ρ) − ICN (ρ)) .
|
{z
} |
{z
}
Vertical equity
Reranking
Progressivity and equity
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Redistributive Equity
Gross and Net Per Capita Incomes
60000
Canada 1994
line_45°
E(N|X)
0
Observed & Expected PC−Net Income
20000
40000
Observed
0
Progressivity and equity
10000
20000
30000
40000
PC−Gross Income in 1994
50000
60000
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