Progressivity, vertical and horizonal...

Progressivity and equity Kampala – 1 / 11

Progressivity, vertical and horizonalequity

Abdelkrim Araar, Sami Bibi and Jean-Yves Duclos

Workshop on poverty and social impact analysis in Sub-Saharan AfricaKampala, Uganda, 23-27 November 2009

Checking the progressivity oftaxes and transfers

Checking theprogressivity of taxesand transfers

Progressivity andpoverty

Concentration curves

Checking theprogressivityof taxes and transfers

Tax Redistribution(TR)

IncomeRedistribution (IR)

The measurement ofprogressivity


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 typesofmonetary and in-kind transfers?

� Is the tax burden on the poor relatively low ?

Studying the progressivity of taxes and transfers can help answer thesequestions.



An important descriptive and normative tool for capturing the impact of taxand transfer policies is the concentration curve.

� Suppose pre-tax incomes (gross incomes)X are ranked in ascendingorder such that:X1 ≤ X2 ≤ ... ≤ Xn.

� Suppose that taxesTj (or transfers) are ranked according to the size oftheir associated gross income.

� The concentration curve of a taxT at percentilep is:

CT (p = i/n) =

∑i

j=1Tj

∑n

j=1Tj



Table 1: Illustrative Examplei pi Xi Ti L(pi) C(pi)– – 0.00 0.00 0.00 0.001 0.25 100 10 0.10 0.042 0.50 200 30 0.30 0.163 0.75 300 70 0.60 0.444 1.00 400 140 1.00 1.00



The concentration curve shows the proportion of total taxespaid by thebottomp proportion of the population.

0.2

.4.6

.81

L(p)

and

C(p

)

0 .25 .5 .75 1Percentiles (p)

45° line L(p)

C(p)



� When the concentration curve of a tax is below the Lorenz curve, thepoor pay less taxes than the non-poor, relative to their income: the tax issaid to beprogressive.

� When the concentration curve of a transfer is above the Lorenz curve, thepoor receive more transfers than the non poor, relative to their incomes:the transfer isprogressive.

� What about inequality of net incomeN?� 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 shareof the poor will be higher than the poor’s share of gross income.



� The concentration curve of net incomesN is:

CN(p = i/n) =

∑i

j=1Nj

∑n

j=1Nj

�

� We can thus compare the concentration curve ofN to the Lorenz curvefor X to assess the net progressivity of the tax and transfer system:

CN(p)− LX(p) =µT

µN

[LX(p)− CT (p)] .



� When reranking is not observed, we also find:

LN(p)− LX(p) =µT

µN

[LX(p)− LT (p)] .

Checking the progressivityof 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 rulescan be usedto check progressivity.

Tax Redistribution (TR)


1. A taxT is TR-progressive if:

LX(p)− CT (p) > 0 for all p ∈]0, 1[.2. A transferB is TR-progressive if:

CB(p)− LX(p) > 0 for all p ∈]0, 1[3. A taxT1 is moreTR-progressive than a taxT2 if:

CT2(p)− CT1(p) > 0 for all p ∈]0, 1[4. A transferB1 is moreTR-progressive than a transferB2 if:

CB1(p)− CB2(p) > 0 for all p ∈]0, 1[

Income Redistribution (IR)


1. A tax or a transferT is IR-progressive if:

CN(p)− LX(p) > 0 for all p ∈]0, 1[

2. A tax or a transferT1 is moreIR-progressive than a tax (and/or atransfer)T2 if:

CN1(p)− CN2(p) > for all p ∈]0, 1[


Checking theprogressivity of taxesand transfers


Quantifyingprogressivity

Indices ofprogressivity

Redistributive Equity


Quantifying progressivity


1. Lorenz and concentration curves may cross.2. It may be useful to provide summary quantitative indices of progressivity.

Indices of progressivity


� Musgrave and Thin (1948) propose to measure progressivity by the ratioof Gini “equality” of net income to Gini “equality” of gross income:1−IN

1−IX

� This ratio will be greater than one if the tax is progressive.



� The Kakwani index of progressivity is based on theTR approach andequals twice the area between the Lorenz curve and the concentrationcurve of a tax.

� This is also the difference between the concentration indexof the tax(ICX) and the Gini index of gross income:ICT − IX



� The Reynolds-Smolensky index of progressivity is based on theIRapproach and equals twice the area between the concentration curve ofnet incomes and the Lorenz curve of gross incomes.

� This is also the difference between the Gini index of gross income andthe concentration index of net income:IX − ICN



0.2

.4.6

.81

L(p)

& C

(p)

0 .2 .4 .6 .8 1Percentiles (p)

0.5(Kakwani Index) 0.5(Reynolds−Smolensky Index)

Lorenz Curve & Concentration curves

R-S Index=µT

µN

(Kakwani Index). (1)



Example: Calculating progressivity indices

Ranki Xi Ti Ni

1 100 10 902 200 30 1703 300 70 2304 400 140 260

Total 1000 250 750

IX = 0.25 //IN = 0.19 //ICT = 0.43 //ICN = 0.19

Musgrave and Thin index= (1− IN)/(1− IX) = 1.08Kakwani index = ICT − IX = 0.18

R-S index = IX − ICN = 0.06



� Does redistribution compress the distribution of post-taxincomes?(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 nonreranking).



Ranki X NA NB NC

1 100 90 90 1002 100 90 100 1003 150 100 90 904 150 100 100 905 200 140 140 1406 200 140 140 140

Average 150 110 110 110

IX = 0.148; IN = 0.101



Ranki X NA NB NC

1 100 90 90 1002 100 90 100 1003 150 100 90 904 150 100 100 905 200 140 140 1406 200 140 140 140

Average 150 110 110 110

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.



Ranki X NA NB NC

1 100 90 90 1002 100 90 100 1003 150 100 90 904 150 100 100 905 200 140 140 1406 200 140 140 140

Average 150 110 110 110

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.



Ranki X NA NB NC

1 100 90 90 1002 100 90 100 1003 150 100 90 904 150 100 100 905 200 140 140 1406 200 140 140 140

Average 150 110 110 110

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.



Ranki X NA NB NC

1 100 90 90 1002 100 90 100 1003 150 100 90 904 150 100 100 905 200 140 140 1406 200 140 140 140

Average 150 110 110 110

One can use the following decomposition of the redistributive effect oninequality:

IX(ρ)− IN(ρ) = IX(ρ)− ICN(ρ)︸︷︷︸

Vertical equity

− (IN (ρ)− ICN(ρ))︸︷︷︸

Reranking

.



020

000

4000

060

000

Obs

erve

d &

Exp

ecte

d P

C−

Net

Inco

me

0 10000 20000 30000 40000 50000 60000PC−Gross Income in 1994

line_45° E(N|X)

Observed

Canada 1994Gross and Net Per Capita Incomes

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