The progressivity of government taxes and benefits in Ireland : a comparison of two measures of redistributive impact

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E U R O P E A N U N IV E R S IT Y IN S T IT U T E , F L O R E N C E DEPARTMENT OF ECONOMICS f ^ ' T O ^ | N G P A P E R No. 88/353 OF GOVERNMENT TAXES ELAND: A COMPARISON )F REDISTRIBUTIVE IMPACT by Niall O ’HIGGINS

Thanks are due to Massimo Marrelli, Peter Lambert and John Hutton for comments on previous drafts. Any remaining errors are, of course, my own.

BADIA FIESOLANA SAN DOMENICO (F I)

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permission of the author.

(C) Niall O ’Higgins Printed in Italy in June 1988 European University Institute

Badia Fiesolana - 50016 San Domenico (Fi) -

Italy © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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-1-Abstract.

In this paper we consider some issues in the

measurement of the progressivity of State taxes and

benefits. In particular, two indices based on the concept of

redistributive impact are analysed and compared. The first

measure was suggested by Blackorby and Donaldson (1984) and

is based on Atkinson's (1970) index of inequality. The

second is an extension of an index employed by Reynolds and

Smolensky (1977) based on the generalisation of the Gini

Coefficient suggested by Donaldson and Weymark (1980). The

major points of difference identified relate to: a) the

income base to which the indices are applied; and, b) the

basis of the weighting system used. It is demonstrated that

these differences may lead to alternative rankings of taxes

and benefits in terms of progressivity. Implications of the differences between the two indices are also considered with

respect to the problem of horizontal inequity. It is

suggested that the Blackorby and Donaldson index is to be

preferred. The issues are further illustrated by applying

the indices to Irish data.

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-2-Introduction

In recent years, increasing interest has been

expressed in the measurement of the progressivity (or

regressivity) of government taxation and expenditure. In

this paper we consider theoretical and empirical issues in

the use of what Keifer terms 'distributional progressivity

indices' (Keifer, 1985) which view progressivity as

determined both by the nature of the tax or benefit

structure (for example the marginal income tax rate) and the

existing distribution of income1 . The first part of the

paper compares in detail the properties of two measures of

progressivity based on redistributive effect. In the second

we go on to apply these indices to Irish data for 1973 and 1980.

Progressivity

A tax may be considered progressive (regressive) if

2

the average rate rises with income . However, this

uncontroversial definition of progressivity may be (and,

indeed, has been) used to imply three different types of

progressivity measure. Specifically, progressivity has been measured in terms of:

iJRevenue Responsiveness: a tax structure may be considered progressive if the elasticity of total revenue with respect to an equiproportionate growth in individual

incomes is greater than one. Such a measure has been

considered and applied by Hutton and Lambert (1979,1982).

ii) Deviation from Proportionality: under this

definition, a tax structure is said to be progressive if tax

payments are less equally distributed than pre-tax income.

This notion has been used by, for example, Khetan and Poddar

(1976), Kakwani (1977) and Suits (1977) in suggesting

progressivity indices^.

iii) Redistributive Impact: in a similar way

to deviation from proportionality indices, taxes may be

considered progressive if post-tax incomes are more equally

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distributed than pre-tax incomes. Indices of progressivity based on redistributive impact have been suggested by, for

example, Reynolds and Smolensky (1977) and Balckorby and

Donaldson (1984).

On a priori considerations, we feel that indices

based on redistributive impact are the most appropriate to

apply to combined tax and benefit structures. Whilst being

suitable for application to single categories of taxes (or

benefits), revenue responsiveness presents severe problems of measurement when applied to combined tax and benefit

structures given the different income bases under which

different types of tax and benefit are administered.

Deviation from proportionality measures are also

problematic. In particular, the insensitivity of such

measures to the average rate of taxation may lead to

perverse results.

Consider, for example. the application of the

Kakwani index to a linear income-tax, T(y)= t(y-a), where y

is income, a the tax-free allowance (so that below a

income-tax is negative) and t the marginal tax-rate, the

Kakwani index is given by

aGQ/(y-a) where y is average income4 . For a < p, K

increases unambiguously with a (3K/3a>0), however at a = y, K is undefined and for a > y, K < 0 (although 3K/3a>0). That

is, if the allowance is larger than average income, the tax

is judged to be regressive although it remains progressive

as defined above,the average tax rate rising with income.

Such an example may seem a little unlikely when considering a single tax since a > y implies negative revenue, however,

when the overall tax and benefit structure is being

considered a broadly linear structure with a budget deficit (implying a > y) is not such an unrealistic situation.

One might also argue that it is the effect of

taxation and/or benefits on the distribution of income which is of more interest to both empirical researchers and

policymakers rather than the distribution of tax payments

per se.

Indices based on redistributive impact are not

subject to the difficulties with the other types of measure

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-4-outlined above. Furthermore, they are explicitly based on

the effect of taxation and benefits on the distribution of

income. Having selected redistributive impact as a suitable

basis for the measurement of progressivity we are still left with a number of possible indices to choose from. In this

paper we compare two such indices, the first is based on

Atkinson's index of inequality and was suggested by

Blackorby and Donaldson (1984). The second is an extension

of an index employed by Reynolds and Smolensky (1977) based

on the Gini coefficient. This is extended using Donaldson and Weymark's (1980) generalisation of the Gini coefficient.

Both of these measures incorporate an explicit

weighting system for the comparison of income transfers at

different levels of income and were both developed in

response to criticism of the undesirable properties of the

Gini coefficient as a basis for comparisons of income

distributions. Specifically, in arriving at any index of

redistributive effect implicit or explicit values are

involved in the weighting of income transfers at different levels of income. Gini-based indices implicitly attach most weight to income transfers at the mode of the distribution.

Atkinson's index of inequality, on which the

Blackorby- Donaldson ( hereafter BD) index is based, employs

the notion of equally distributed equivalent income.

Consider the additively separable 'utilitarian' social

welfare function:

(1) W = ₂U.

' ' i i

where n is the number of income recipients and where utility depends only on income.

Equally distributed equivalent income (yecj e ) is

defined as, "the level of income per head which if equally

distributed would give rise to the same level of social

welfare as the present distribution." (Atkinson, 1970,

p. 250 ) . T h u s : (2) _{u<yede>= w/"} © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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rearranging:

(3) yede= U - 1 (W/n)

In order to derive an empirically measurable index,

we must specify each individual's utility function (assumed to be the same for everyone). Here we follow Atkinson in using the specification:

u(y) r

d - e ) £ / 1

u(y) = ln(y) e =1

with e t- 0 for concavity.

Combining (3) and (4) we arrive at an estimable

definition of y , : 2ede

V

=

( l

.? v!i-)\TI-eT

£ » 0

yede In i*l y i j e * 1

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yede= exp(s iiiln(yi ) ^ e = 1

The parameter e represents the weight attached

different levels of income (i.e. the level of inequality

aversion). Atkinson's index of inequality is then

straightforwardly defined as:

(6) K e ) = 1 - (yede/w)

An increase in 1(e) representing an increase in

inequality. Thus, for example, y=0 implies constant marginal

utility of income (3U/3y=l) and zero inequality aversion,

yede= 11' anc* therefore l(e)=0 whatever form the income

distribution takes. For e>0 there is diminishing marginal

utility of income (32U/3y2<0) and therefore positive

inequality aversion. Thus, for example, for e=l, an

increment in income accrueing to someone with income x

increases 'social welfare' by twice the amount an equal

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-

6-increment accruing to someone with income 2x would do (i.e. 3W/y1=23W/3y2 , where 2y1=y2 ).

Increasing e increases the relative importance of

income transfers at the lower end of the income scale until

as €-*•<» , the Rawlsian maximin solution is obtained, the

value of W (and therefore I) is only affected by the income

of the poorest member of the community.

The BD index is defined for taxes as:

(7) BD. (I - I t )/(l - I )

' o p t " ' o'

, . ede , , , ede , .,,, ede , .

[(yp t /uP t )_(yo / “o )]/(yo /uo>

where subscripts o and pt denote the original and post-tax income distributions respectively.

Thus, BDfc measures the proportionate increase in (yede/u) occasioned by the tax. A tax schedule may be said

to be progressive if BD > 0, proportional if BD = 0, and

regressive if BD < 0. Similarly, one may state that tax

schedule tl is more progressive than tax schedule t2 if

BDtl>

BDt2-The BD index may be extended to measure the

progressivity of benefits and the progressivity of taxes and benefits combined:

(8) b d l

= (Ipt- W

^ 1

V

(9) 1 + BDnet= (1 + BDt )(l + BDb )

1 + (I, I _{n e t " '}t )/(l - I ) _o'

The second index we wish to examine is an extension

of the Gini-based index used by Reynolds and Smolensky

(1977) employing the generalisation of the Gini coefficient

suggested by both Donaldson and Weymark (1980). This

generalisation may be defined (in discrete terms) as:

(1 0) G (v ) = 1 - (l/unv )i|1 [(n-i+l)v-(n-i)v ]y.

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with individuals being ranked by income from lowest to highest5 .

As with the Atkinson index an increase in G

represents an increase in inequality and the parameter v

plays an analogous role to e. G (1) represents the equivalent

case to e=0 above,with no distinction between income

distributions being possible in terms of progressivity. G(2)

is simply the Gini coefficient whilst higher values of v

reflect increasing weight being attached to lower rankings

in the income distribution. Once again, as v-*-®, G approaches the maximin outcome.

The extended Reynolds-Smolensky index (RS) is then:

(1 1) RS(v)fc = G(v)o- G(v)

Once again this index may also be employed to

consider the progressivity of benefits and the combined

tax/benefit system:

(12) _{RSb = Go - Gpb}

(13) RS . = G - G .

net o net

= [(l-t)RS + (1+b)RSb ]/(1-t+b)

where subscripts pb and pt represent income distribution

once benefits have been added or taxes subtracted from

original income respectively; b and t are the average rates

of benefit and tax. Once again progressivity,

proportionality and regressivity are indicated by values for

RS > 0, = 0, and < 0, respectively.

Thus, the two measures outlined above are, in some

respects, similar. They are both based on notions of

redistributive effect and both incorporate explicit weights

which may be varied to reflect different levels of

'inequality aversion1. However, there are significant

differences between the two which imply that the BD and RS indices may produce different orderings of tax and/or

benefit schedules in terms of progressivity. Indeed,

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-8-different values of the weighting coefficient may also

alter the ranking of taxes and benefits even using the same

index. We will discuss differences between the indices in

greater detail below. However, to illustrate the different possibilities, a hypothetical example is given in table 1 .

(table 1 about here)

In this simple example, there is a society

comprising of four individuals i (i=l,2,3,4) with income

before taxes and benefits X . . Consider the application of

orig

two alternative balanced budget tax/benefit schedules

resulting in incomes X tl and Xnet2 resPecti-vely • Tlle

net effect of schedule 1 is to redistribute $10 from person 4 to person 2, whilst schedule 2 results in a redistribution

of $4 from person 4 to person 1. Applying BD(1) and RS(2 )

indices to the two schedules (the first two rows of the

lower part of the table) we are faced with conflicting

results. BD(1) suggests that schedule 2 is more progressive

than schedule 1, whilst RS(2) suggests the contrary to be

true. Furthermore, as the value of v is increased the

ordering of the two schedules is once again reversed. We might look at this in terms of social welfare6 .

For schedule 1 to be more progressive than schedule 2 it is

necessary that BD .,> BD .-(similarly with RS). This

1 netl net2' '

implies that the change in 'social welfare' resulting

from schedule 1 (i.e. the sum of the increase in utility to

person 2 and the decrease in welfare to person 4 resulting

from the transfer of the $10 from 4 to 2) must be greater

than the change in 'social welfare' arising from schedule 2

(i.e. the sum of the increase in utility to person 1 and

the decrease in utility to person 4 of the transfer of $4

from 4 to l)7 .

Differences between the indices

Having outlined the possibility of different

orderings arising from the use of different weights and

different indices, it is worth now consider ing in more

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detail the differences between the two indices BD and RS which may give rise to contradictory replies to questions of

the form, 'is tax schedule 1 more progressive than tax

schedule 2 ?' or, 'are taxes more progressive than benefits v 1

1) Income Base.

This point of difference relates to the income

distributions which are compared to arrive at the measure of

progressivity. That is, the income bases to which the two

measures are applied. Specifically, refering to the

definitions given above, taking taxes and benefits

separately, RSfc compares post-tax income with the original

income distribution (equation 11), and RS ^compares post

benefit income (i.e excluding taxation) with the original

income distribution (equation 12). On the other hand,whilst

BDt (in the form given above) also compares the original

and post-tax income distributions (equation 7), BD^ compares

the final and post-tax distributions (equation 8). Thus, RS

indices always use original income as their base, whereas BD indices are applied cumulatively so that in this case the redistributive effect of benefits is measured in terms of

its effect on the post-tax income distribution and not (as

is the case with RS) on original income. So long as taxes

are not proportional, the basis of comparison is therefore

different for RS and BD indices. As Lambert (1985a) points

out, taxes (or benefits) may be regressive when applied to

original income whilst being progressive when applied to a

more equally distributed post-benefit (or post-tax)

distribution.

These points are illustrated in table 2, the top

half of which is taken from Lambert (1985a, p.45). This

gives an example of a hypothetical tax/benefit system. In

the top half of the table, column 1 shows the original

income distribution (again in a four person society), column

2 shows the income distribution once taxes have been

subtracted from original income, column 3 gives the

distribution after the ad addition of benefits to original

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1 0

-income, and finally, column 4 shows the distribution

following the imposition of both taxes and benefits.

The lower half of the table shows the progressivity

of taxes and benefits (both individually and taken together)

in three different ways. Firstly, applying RS(2), secondly

using B D (1) taking taxes first, and finally employing BD(1) taking benefits first.

With regard to the RS(2) index we can see immediately

the apparent paradox observed by Lambert (1985a): whilst

taxes appear to be regressive, the progressivity of taxes

and benefits taken together is greater than that of benefits

by themselves. This 'paradox' arises because taxes are

regressive when applied to original income but progressive

when applied to the more equally distributed post-benefit income.

Turning to the BD indices, the aforementioned paradox is avoided because the indices are computed cumulatively.

However, the consequence of this is that the order in which

taxes and benefits are evaluated affects the size of the index.

sign and the

This immediately raises the questions: a) which is

the appropriate base for the measurement of progressivity;

and, b) with respect to the BD index, which is the

appropriate order in which the progressivity of taxes and

benefits should be considered?

We would argue that progressivity should measured so far as possible with respect to the income base on which the

tax or benefit itself is assessed. Thus, for example, if

social security benefits are subject to income taxation,

then the appropriate bases of comparison for the measurement of the progressivity of income tax are original income plus social security payments and income after the imposition of social security and income tax. Such an approach seems most reasonable intuitively. It is difficult to see the relevance of a measure of, for example, income tax progressivity which

is not related to the specific concept of income used to

determine the level of taxation paid by each person. Thus,

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in this respect we would argue that the BD index, which allows the analyst to choose to some extent the income bases

employed in the measurement of progressivity, is to be

preferred to the RS index.

2)Basis of the Weighting System.

A second difference between the two indices is the

O

weighting system employed . I based measures such as BD

apply the weight e to income, y , whereas using G based

measures the weight v is applied to the individual's

ranking in the income distribution, i. A quick glance back

at equations 6 and 10 above should be sufficient to confirm this.

This implies that, whilst the BD index has the

possibly desireable property of attaching more weight to

income transfers to individuals low in the income rankings

if the income distribution is highly skewed, treating income transfers above and below the mean more symetrically if

income is more equally distributed, the RS does not. Since

the weight in the RS index is applied to the rank in the

distribution and not income itself the weighting of income

transfers is independent of the income distribution and

therefore treats such transfers equally however skewed the

distribution is.

This might be used as another justification for

prefering the BD to the RS index in the measurement of

progressivity. In any case, of perhaps more importance in

empirical applications are the implications of the nature of the weighting system for the error of measurement arising from horizontal inequity.

A common problem with empirical applications of

progressivity indices is that such analyses often assume

that attributing taxes and benefits to individuals (or in

the application below to households) does not alter their

ranking in the income distribution. If the level of taxes

paid and/or benefits received are functions of any

individual attribute other than original income (e.g in the

case of households, household size is likely to be an

important factor) then the ranking of individuals in the

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1 2

-income distribution is likely to change and estimates of RS

will tend to overstate the redistributive impact of taxes

and benefits.

An illustrative example may help make this clear.

Table 3 (based on Atkinson, 1979, p .6) shows the income

distribution for our now augmented five person society

before and after tax. The first post-tax distribution is

based on the original income ranking whilst the second is

reranked according to the level of post-tax income.

(Table 3 about here)

Applying RS(2) to the unaltered ranking suggests that

the tax is progressive (RS = .034) whereas the correctly

ranked post-tax income distribution shown in the final

column of the table clearly indicates that the tax is

strictly proportional.

This error does not arise in the case of the BD index

since it is based on income itself and is not affected by

income rank per se. This again offers support for prefering the BD over the RS index. In practice, however, when grouped data is employed (as in the application below) the BD index will be affected since changes in the ranking of individuals (or households) changes the mean income in each group and it

is on this mean income which provides the basis for the BD

index.

3)Absolute Vs Proportionate.

The final and least important distinction between the

two indices that we wish to draw attention to is the fact

that the RS index is an absolute measure whereas BD is proportionate. That is, RS is related to the absolute change

g

in 'Social Welfare' (AW) while BD relates to the

proportionate change, (AW/W) brought about by the tax and/or benefit.

This difference implies that whilst RS must lie in the range, GQ> RS > (G - 1), BD may be any real number. This should be bourne in mind when comparing the results from the two indices. In particular, it serves to remind one that a

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direct comparison between the values of RS and BD is meaningless. Thus, the statement RS > BD of itself tells us nothing about the progressivity of the tax.

Irish Tax and Benefit Progressivity, 1973-1980.

In this section we present the results of applying BD

and RS indices outlined above to Irish data for 1973 and

198010. The data employed comes from the Irish Household

Budget Surveys of those years. The data is grouped by income

decile11. The income concepts and categorisation of

government taxes and benefits are adhered to. In particular, taxation and expenditure (benefits) are classified into four

groups to which RS and BD indices were applied. These are:

i) cash benefits ; ii) direct taxation; iii) non-cash

12 benefits; and, iv) indirect taxation

In applying the BD index we follow the order (given above) in which taxes and benefits are attributed to income

in the H B S . That is, cash benefits are added to direct

(original) income to arrive at gross income. Direct taxation

is then applied to gross income to produce disposable

income. Thirdly, non-cash benefits and then indirect

taxation are added to arrive at final (net) income. Thus,

the progressivity of cash benefits (using BD) is assessed

using a comparison of the direct and gross income

distributions. Similarly, estimation of the progressivity of

direct taxation involves a comparison of the gross and

disposable income distributions and so on.

There are two reasons for adopting this ordering.

Firstly, as far as is possible with taxes and benefits at

this level of aggregation, this ordering conforms to the

method suggested above. That is, it was argued that it is

sensible to assess the progressivity of, for example, direct taxation on the basis of the income concept used when the

tax is levied. Thus, cash benefits are usually awarded on

the basis of direct income. Similarly, direct taxation is

largely levied on gross income (as defined above) . This

somewhat imprecise justification becomes purely arbitrary

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-14-with respect to the order of assessment of the progressivity 14

of non-cash benefits and indirect taxation , however, it

is felt that the order of progressivity assessment is at

least as sensible as any of the twenty-three alternatives.

Again, with respect to the BD index, the presence of

zero direct income for the lowest income decile in 1980

meant that this measure of cash benefit and overall tax and

benefit progressivity was undefined for values of e>l.

Therefore, results are presented for values of e between

(but not including) zero and one. In the case of the RS

index this problem does not arise. Although the application of either direct or indirect tax to original income produces

negative net incomes15 in certain cases; as Yitzhaki (1983)

points out, the generalised Gini coefficient, and therefore

the RS index is still calculable so long as mean income is

positive15. This condition is fulfilled and so results for the RS index are therefore presented for values of v between two and ten.

Table 4 presents the results of applying the RS and

BD indices for different values of v and e to the four

categories of tax and benefit with households ranked by

original income for 1973 and 1980.

It is immediately apparent that the indices all

produce broadly similar results. Thus, the progressivity of 17

each category of tax and benefit has increased between

1973 and 1980. Following directly from this, the

progressivity of taxes and benefits taken as a whole has increased over the period. The ranking of taxes and benefits

in terms of progressivity is the same for each index, cash

benefits being the most progressive then non-cash benefits, followed by direct tax with the regressive indirect taxation

at the bottom. Finally, for both indices,the change in

progressivity with respect to changes in the weighting

coefficient is positive for progressive taxes and benefits

and negative for the regressive indirect tax. That is, as e

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and v are increased so too is the progressivity (or regressivity) of each tax or benefit.

There are, however, differences between the results

produced by the different types of index. Most obviously,

the progressivity of cash benefits relative to the

progressivity of other categories is much larger using BD.

Similarly, the regressivity of indirect tax is (relatively)

much greater when the RS index is employed. These

differences stem largely from the different income base to

which the indices are applied. As emphasised above, RS

measures the progressivity (or regressivity) of taxes and

benefits always using original income as the base for

comparison whereas the BD index assesses progressivity

cumulatively . Thus, we observe, for example, that indirect

taxation is more regressive when applied to original income than when it is applied to the more equally distributed post non-cash benefit income distribution.

Horizontal inequity.

It was noted above that if taxes or benefits alter

the ranking of households in the income distribution then

the estimates of RS and (in practice) BD will tend to

overestimate the progressivity (underestimate regressivity)

of these taxes and benefits. Table 5 shows the tax/benefit

mobility matrix for 1980 in Ireland. This cross-classifies

households by original and final (table 5 about here)

income quintiles. Thus, the top row shows the final income

positions of households in the lowest quintile of original

income. The substantial degree of mobility indicated by the

table suggests that there may be a substantial overestimate of in the calculations of progressivity in table 4.

Using data reranked by gross, disposable and final

income it is possible to produce estimates of the extent of

the error caused by horizontal inequity in at least some of the cases. Table 6 presents the results of this exercise.

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1 6

-The first row relating to each index gives the

revised estimate (where calculable) of progressivity based

on reranked data. The second (in brackets) gives the

original estimate and the third row gives the error as a

percentage of the new figure. It might be noted that, as was

suggested above, the error is much larger for the RS index

than for BD.

Adult Equivalence.

It was pointed out above that the re-ranking of

households after the imposition of taxes and benefits arises because such taxes and benefits are not levied solely on the basis of original household income. One possible source for

this re-ranking might be sought in household size. It is

reasonable to expect a positive relationship certainly

between government benefits (and perhaps taxes) and

household size.

Table 7 provides support for this proposition. It

shows that in each original income quintile the ranking of

households after redistribution is related to household

size. The rather unsurprising implication being that it

tends to be the larger households who gain most from

redistribution.

One possible way of compensating for this is to use

'adult equivalence scales' to convert the incomes of

households to a common individual base. A number of

different methods for the calculation of such scales has

18

been suggested . Nolan (1981) derives four such scales for

Ireland based on two different approaches19.however the

results produced do not appear to be sensitive to the

20

particular scale employed . In this paper we use CSO

estimates based on McClement's (1978) UK scales derived

from the Family Expenditure Survey . These are :

First Adult 1.00 Married Couple 1.74 Persons aged 0-4 .25 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Persons aged 5-13 .38

Persons aged 14-20 .53

Additional Adults .74

The principle of application of the scale is simple. The members of each household are classified into the above

groups, the sum of the appropriate coefficients is

calculated and this sum is used to divide the households

income (of whatever type). Thus, for example, a married

couple with two children aged 3 and 7 would warrant a

coefficient of 2.37. Their income would therefore be

multiplied by 1/2.37 to arrive at the household's 'adult

equivalent' income.

Table 8 shows RS and BD indices applied to adult

equivalent income for 1980. Two points of interest are worth

noting, while the progressivity of direct taxes and both

types of benefit increases (or at least remains roughly the same) for both indices and using different weights. Indirect taxation, however is more regressive according to the RS

index but less regressive (becoming roughly proportional)

according to the BD index.

The second observation is that using adult

equivalence scales does substantially reduce the error

arising from horizontal inequity. The percentage error for

the overall taxes and benefits is given in brackets in the

final column of the table. The effect of employing adult

equivalence scales appears to be a reduction in the

'horizontal inequity' error of roughly one-third for RS and

as much as one-half for BD.

Conclusion

In this paper we have compared, both theoretically

and empirically two indices of progressivity based on the

notion of redistributive effect. It was shown that the two

indices, indeed even different values of the weighting

coefficient within each index, may produce different

rankings of taxes and benefits in terms of progressivity.

One approach to the problem is to employ different

indices in order to see how robust the results are to

changes in the weighting coefficient and between the indices

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1 8

-themselves. An alternative approach would be to select the

index (and perhaps the value of the weight) on a priori

grounds. We went some of the way towards this in excluding

measures based on revenue responsiveness and deviation from

proportionality. From the analysis above we would suggest

that, if a choice must be made between the two indices considered in detail above, the BD index is to be prefered. It has the desireable property of treating income transfers

differently according to the skewness of the income

distribution which the RS does not. Furthermore, it was

suggested that the cumulative nature of the BD index is

desireable. There seems little relevance in always relating

progressivity to effect of taxes and benefits on original

income. Finally, the BD index is much less subject to the

error arising out of horizontal inequity than is the RS

index. © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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References

Atkinson, A.B., "On the measurement of inequality". Journal

of economic theory, 1970, vol. 2, pp. 243-263.

Atkinson, A.B., "Horizontal equity and the distribution of the tax burden", in The economics of taxation, edited

by H.J. Aaron & M.J. Boskins, 1979, Washington,

Brookings Institute.

Blackorby, C. & Donaldson, D., "Ethical social index numbers

and the measurement of effective tax/benefit

progressivity", Canadian journal of economics, 1984,

vol. 17, pp. 683-694.

Central Statistics Office, Redistributive effects of state taxes and benefits on household incomes in 1973, CSO, 1980, Dublin.

Central Statistics Office, Redistributive effects of state taxes and benefits on household incomes in 1980, CSO, 1983, Dublin.

Donaldson, D. & Weymark, J.A., "A single parameter

generalisation of the Gini indices of inequality". Journal of economic theory, 1980, vol. 22, pp. 67-86.

Hutton, J.L. & Lambert, P.J., "Income tax progressivity and

revenue growth". Economics Letters, 1979, vol. 3, pp. 377-380.

Hutton, J.P. & Lambert, P.J., "Modelling the effects of

income growth and discretionary change on the

sensitivity of U.K. income tax revenues". Economic

journal, 1982, vol. 92, pp. 145-155.

Kakwani, N.C., "Measurement of tax progressivity: an

international comparison". Economic journal, 1977,

vol. 87, pp. 71-80.

Khetan, C.P. & Poddar, S.N., "Measurement of income tax

progression in a growing economy: the Canadian

experience", Canadian journal of economics, 1976,

vol. 9, pp. 613-629.

Kiefer, D.W., "Distributional tax progressivity

measurement". National tax journal, 1985, vol. 37,

pp. 497-513. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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2 0

-Lambert, P.J., "The redistributive effect of taxes and

benefits", Scottish journal of political economy,

1985, vol. 32, pp. 39-54.

McClements, L.D., Table E.l, page 168, Report No. 6 (lower

incomes) of the Royal Commission on the distribution of income and wealth, 1978, HMSO, cmnd 7175, London.

Morris, N. & Preston, I., "Inequality, poverty and the

redistribution of income", Bulletin of economic

research, 1986, vol. 38.

Murphy, D.C., "The impact of state taxes and benefits on

Irish household incomes", 1987, Journal of the

statistical and social inquiry society of Ireland, session 1983-4, vol. 25, pp. 55-120.

Musgrave, R.A. & Thin, T., "Income tax progression, 1929-

48", Journal of political economy, 1948, vol. 56, pp. 498-514.

Nolan, B., "Redistribution of household incomes in Ireland

by taxes and benefits", Economic and social review, 1981, vol.13, No. 1.

O'Higgins, N . , "The changing burden of income taxation and

the progressivity of the Irish tax/benefit system

1973-80", M.Sc. Dissertation, 1985, University of

York.

Reynolds, M. & Smolensky, E., Public expenditures, taxes,

and the distribution of income: the United States,

1950, 1961, 1970. 1977, New York, Academic Press.

Yitzhaki, S. "On an extension of the Gini index",

International economic review, 1983, vol. 24, pp.

617-628. © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Notes :

1 As opposed to 'Structural Indices' which depend only on the structure of tax or benefit rates, as in the work of Musgrave and Thin (1948).

2 Clearly, for a benefit to be progressive the average

rate must decrease as income rises.

3 Thus, the Kakwani index is defined as: G fcax- Gq , where

G fc , and GQ are the Gini coefficients for tax payments and

pre-tax income respectively.

4 See, Keifer (1985) for proof of this.

5 That is, individual 1 is the poorest and individual n

the richest member of the community.

6 Perhaps a more neutral term would be the social

valuation of income. In as much as weights are attached to different levels of income in both indices, it is clear that a judgement is being made concerning the relative importance

of different levels of income. We are not concerned here

with the difficulties involved in the justification of the

use of the concept of 'Social Welfare', however, it should

be remembered that any measure or redistributive impact (or

deviation from proportionality) does involve such a

judgement whether it is explicit or not.

7 That is, if we consider G as containing a Social

Welfare, or if you prefer, Social Valuation of Income

function, W, such that G = 1 - aW, (a=l/nnv ) then, in the

case of a balanced budget:

RS= Gpre- Gpost= aAW = “ S or, for the BD index:

BD = ‘W W » / ! 1 - W = (W5ost- WSre>/W5re-

Where 8 = 1/(1-e).

8 It might be noted that whilst the difference due to the

weighting system arises because of differences between the

Atkinson index of inequality and the extended Gini, the

divergence due to the income base is a feature of the form BD and RS indices and not the inequality indices underlying them.

9 See note 6 above. 10 CSO (1980,1983).

11 In most cases data was available in thes form from the

CSO (see Murphy, 1987). Where it was not we used logarithmic interpolation to derive decile groupings from the CSO data. For more details, see O'Higgins (1985).

12 For the most part, the types of taxes and benefits

included in each group should be fairly obvious. For a

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2 2

-detailed description of the taxes and benefits covered see 0'Higgins (1985) .

13 There are, of course a number of exceptions. Thus, for example, injury compensation (in the cash benefit category) is not based on original income, and, for the period covered

by the study, unemployment benefit (cash benefit) was not

subject to direct tax.

14 Following the logic used above, since indirect taxation

paid depends on the level of consumption of taxed goods,

then the appropriate base for the application of the

progressivity index would depend on the basis for

consumption decision. That is, whether or not indirect

taxation should be assessed before or after non-cash

benefits should depend on whether consumption expenditure decisions are basecd on disposable income or disposable income plus non-cash benefits.

15 That is, net of direct or indirect tax respectively. 16 This approach differs from that of Morris and Preston

(1986) in their study of UK tax and benefit progressivity.

They adopt the convention of setting all negative incomes to

zero, however, such an approach leads to an overestimate of

the progressivity (or underestimate of the regressivity) of

taxes. Such a convention is unnecessary for the calculation

of the RS index and therefore is not employed here.

17 For indirect tax regressivity has decreased which

amounts to the same thing.

18 See, for example, Muellbauer (1980) for a discussion of

these.

19 Specifically, using: a) surveys of family expenditure;

and, b) social security provisions. 20 See Nolan (1981, p.88). © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE 1: X orig X netl X net2 i 1 L 5 2 10 20 10 3 30 30 30 4 59 49 55 BD (e=l) 1353 ■4693 RS (v=2) .1000 .0600 (v»3) .1125 .0900 (v=4) .1000 .108S (v»5) .0820 .1219 (v=10) .0221 .1510 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE 2 ORIGINAL. INCOME 10 20 30 40 POST-TAX INCOME 4 11 18 25 POST- 8ENEFIT INCOME 31 34 37 40 NET INCOME 25 25 25 25 RS (v=2) -■0517 ■ 1972 .2500 BD (e=l) a) taxes first -.0760 .2224 .1295 b) benefits first .0045 .1244 .1295 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TASLE 3: EXAMPLE DF HORIZONTAL INEQUITY i ax Payer- I N C O M E E l t e r a r e tax p c s ' - c a x (original rank) post t ax (ra— anksd) 1000 9 0 0 <i ) aoo 2. 1000 6 4 0 aoo 3T a o o BOO (2! 640 4 s o o 512. (5) 640 5" 640 640 (4) 5 1 Z © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE

4

.: Tax and Benefit prcgrsssivity 1973“80t calculated for households raided by original income.

' CASH 3ENEFITS | I OIRECT j 1 TAX I NON-CASH | BENEFITS ! jINDIRECT jI iMET 1 EFFECT 1973 i 198011 1973i 1930|1 1973 O O O o > H 1 19731 1980 |1 1973 1980 RS(2) • 0757 .0927 .0092 .0253 .0956 .0860 -.0970 -.0379 .1129 .1683 RS(3) .1191 .1915 .0108 .0206 .0616 .0909 -.0617 -.0505 .1663 .2967 RS(9) .1392 .1731 .0109 .0189 .0702 .1058 -.0690 -.0579 • 1996 .2937 RS(5) .1572 • 1953 .0099 .0151 .0761 .1158 -.0735 -.0616 .2232 3252 RS{ 10) .2027 .2970 .0052 J .0093 .0919 .1378 -.0835 -.0699 .2829 • 3952 BD(.l) .0199 .0189 .0020 .0091 .0038 .0098 -.0019 -.0009- .0189 .0270 BD(.25) • 0933 .0558 .0053 .0105 .0098 .0129 -.0039 -.0009 .0559 •0791 BD(.5) .1268 .1692 .0113 .0222 .0206 .0262 -.0068 -.0 0 17 .1552 .2295 BD(.75) .3209 • 5187 .0182 • 0399 .0329 .0913 -.0099 -.0022 .3797 • 6330 BD( .9) .5852 1-982? .0226 .0928 • 0399 .0509 -.0117 -.0023 .6660 2.2609 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Table

S-

Tax/Benefit Mobility Matrix

FINAL INCOME QUINTILE.

1 2 3 4 5 o i q R N U 1 12.6 5-2 1.6 0.5 0.1 20.0 I C I G 0 N 2 6.5 7-7 3-6 1.6 0.5 20.0 I M T N E I 3 0.8 5.8 8 .4 4 .2 0.8 20.0 A L L E A - 0.1 1.1 5-5 9-7 3-5 20.0 5 0.0 0.1 0.8 3-9 15 -1 20.0 20.0 20.0 20.0 20.0 20.0 100.0 Soof2.ce: HofcPH'd ( \ ^ ^ ) © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE TAX AND BENEFIT PROGRESSIVITY 1973*80 BASED ON RE-RANKED DATA.

CASH DIRECT NET

BENEFITS TAX EFFECT

1973 1980 | 1973 1980 || 1973' 1980 |

RS( 2) .0672 .0819 n.a. n.a. .0682 .1289

(.0757) (-0927) ( 1129) (.1683)

12.6 13-2 _65-5 _31-1

as(3) • 0983 .1219 n.a. n.a. .0989 .1707

(-1193) ( -1915) _(-1663) (.2967) 16.3 16 .6 68.1 99.5 RS(9) .1169 .1938 a.a. n.a. .1169 .1972 (-1392) ( .173D (.1996) (-2937) 19-1 20.9 _70-7 _98.9 RS(5) .1291 .1579 n.a. n.a. .1286 .2120 (.1572) (-1953) (_-2232) (-3252) 21.8 29.1 _73-6 _53-9 RS( 10) .1518 .1797 n.a. n.a. .1988 .2299 (.2027) (-2970) _(-2829) _{( -3952)} 33-5 91.9 _90.1 _75-7 BD(.1) .0130 .0165 .0015 .0039- .0130 .0215 (.0199) (.0189) (.0020) (.0091) (.0189) (.0270) 10.8 11.5 _33-3 2a . 6* 95.9 25-6 BD(.25) ■0392 .0509 .0090 •.0088 • 0392. .0638 ( 0933) (.0558 (.0053) (.0105) (.0559) (.0791) 10.5 10.7 32.5 19 .3 91-3 29.0 BD(.5) _{.115 5} .1595 .0088 _{.0 187} _•₁₁₅₅ _.1862 (.1268) (.1692) (.0113) (.0222) (.1552) (-2295) 9.8 _9-5 28.9 18.7 39.6 20.6 BD(.75) .2961 .9836 .0196 .0296. ■ 2951 ■ 5989 (-3209) (-5187) (.0186) (-0399) (-3797) (-6330) 8.9 7-3 27.9 _17-9 _27-0 _15-3 BD(.9) •5997 1.8897 .0185 .0366- .5929 2.0982 (.5852) (1.9827) (.0226) (.0928) (.6660) (2.2609) 7-9 9.9 22.2 16-9 22.8 10.9 19 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE -3: AVERACE NUMBER OF PERSONS PER HOUSEHOLD IN IRELAND, 1980, CLASSIFIED 3Y DIRECT AND FINAL QUINTILE

INCOMES-FINAL INCOME QUINTILE

1 2 3 9 5 MEAN 0 L Q R rr U i 1.71 2.90 5-03 7-57 11.86 2.99 I c I G 0 N 2 1.83 2.63 9.05 5-90 8.89 3-05 i M T N E I 3 2.33 2.65 3-77 5-97 8.13 3-92 A L L E 9 2.69 2.86 3.22 9-. 33 6-33 9.28 5 3-98 3-18 3.26 3-37 5-36 9.87 1-79 2.72 3-75 9.59 5-75 3-72 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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TABLE g: TAX AND BENEFIT PEOCHESSIVITY, 1930. CALCULATED ON DATA A D J U S T S 3Y ADULT EQUIVALENCE SCALE.

CASH BENEFITS DIRECT TAX ION-CASH 3ENEFITS INDIRECT TAX 1 MET 1 EFFECT!a) NET EFFECT(b ) RS(2) .1190

_•0253

.0860

-.0930

_.2273

_.1863

(

22

.

0

) RS<3) . 1876

_•0255

.1220

■

.0599

• 3325 .2639 (26.2) RS(9) .2267

.0215

.1921 -.0689 3961 .2639 (

30

.

6

) RS(5) ■ 2577

.0171

■1551 -.0792 ■ 9392 ■ 3256 (39.9) RS(10) • 3329

.0021

.1822

-.0858

_■₅₃₆₇ _•₃₉₇₁ (59.6) 3D(.1) .0208 .0096 • 0053 .0000

_.0309

.0279 (12.8)

8

D(.25) .0621 .0116 •0135 -.0001 .0889 • 0791 (12.9) 3D(.5)

.1879

■ 0237 .0276 .0000

_.2936

_•₂₁₉₅ (11.0) BD<.75) • 5389 •0359 .0920

.0003

.,6619

.6100

(8.9) BD(.9) 2.0186 .0932 •0507 .0009

2.3101

2.1822

(5-9) notes: (a) net effect calculated using final income ranked by original income,

(b) net effect calculated using final income ranked by final income.

13 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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