The concept and measurement of horizontal inequity

Journal

of Public

Economics

THE

17 (1982) 373-391.

Publishing

Company

CONCEPT AND MEASUREMENT OF HORIZONTAL INEQUITY Robert

Dartmoutlt

Received

North-Holland

November

PLOTNICK”

College, Hanover,

1980, revised

NH 037S5,

version

USA

received

June 1981

This paper discusses the concept of horizontal equity and presents three formal principles as a basis for making comparisons of the extent of horizontal inequity produced by different redistributions. A condition analogous to the Lorenz criterion used for inequality comparisons is established under which the horizontal inequities of redistributions can be ranked without recourse to a cardinal measure of inequity. Ordinal comparisons are not possible in many cases. The three principles suggest three properties that ‘reasonable’ measures of horizontal inequity should possess. A class of measures which satisfies the properties is proposed. Most other existing measures are found not to satisfy them.

1. Introduction In recent years a number of quantitative indicators of the extent of ‘horizontal inequity’ produced by a tax and transfer system have been proposed and applied.’ Each index attempts to measure how seriously a redistribution violates the maxim of ‘equal treatment of equals’. However. with the exception of work by King (1980), their underlying theoretical rationales have not been systematically considered. In this sense, the extant measures are quite ad hoc. During the same period in which these contributions have appeared, a sophisticated literature on the measurement of inequality has developed. Drawing upon Atkinson’s seminal work, economists have analyzed the social welfare functions implied by various measures of inequality and the implications of certain classes of social welfare functions for making inequality comparisons.* Recently, Fields and Fei (1978) suggested three axioms for * I wish to thank Frank Cowell, Robert Hutchens and an anonymous referee for comments on an earlier version, and Robert Z. Norman for help in simplifying the proof of the theorem. This study was supported by the Lewis H. Haney 1903 Endowment in Economics of Dartmouth College and the Institute for Research on Poverty, University of Wisconsin-Madison, ’ For earlv treatments, see Johnson and Mayer (1962), White and White (1965) and Brennan (1971). Recent papers include those by Rosen (1978), Habib (1979), Hettich (1979). Strauss and Berliant (1979), Atkinson (1980). King (1980) and Plotnick (1981). ‘See Atkinson (1970), Dasgupta, Sen and Starrett (1973), Rothschild and Stiglitz (1973). Sen (1973). Kolm (1976). Cowell (1977) and Fields and Fei (1978). 0047-2727/82/0000-0000/$02.75

@ 1982 North-Holland

371

R. Plotnick.

Horizontul

inequity

Inequality comparisons and demonstrated that the Lorenz criterion satisfied them. From the axioms, three corresponding properties that a ‘good’ index of inequality should possess were suggested. Several commonly used cardinal inequality measures were shown to exhibit these properties. This paper melds these two lines of analysis to develop a more systematic foundation for measuring horizontal inequity. The paper first discusses the concept of horizontal equity to determine which consequences of a redistribution are properly viewed as horizontal inequities.’ I draw upon this analysis and other considerations to argue for three formal principles of inequity comparisons. In a manner analogous to Fields and Fei’s work on inequality measures, conditions are established under which the extent of horizontal inequity of two redistributions can be ordinally ranked without recourse to a specific cardinal measure of inequity. The three principles imply the ‘reasonable’ measures of horizontal inequity should have three properties. Section 4 proposes a class of measures based on this analysis and assesses whether other measures possess these properties. Section 5 briefly summarizes and concludes the paper. The concept of horizontal equity may be applied to both the tax and transfer sides of redistributive policy. For convenience, I assume that government is engaged solely in redistributive activities. All taxes collected are paid as transfers. Furthermore, it is assumed that such actions have no efficiency impacts.4 The distribution of economic welfare before taxes and transfers are enforced will be referred to as the initial or pre-distribution. Similarly, the distribution after taxes and transfers will be the final or post-distribution. 2. The concept of horizontal

equity

The principle of horizontal equity is usually stated as ‘equal treatment of equals’.5 Units with the same level of well-being should be liable for identical taxes or transfers.’ As stated, this principle is conceptually unsatisfactory. Consider an economy in which each unit has a unique level of pre-distribution welfare, but is otherwise indistinguishable from others. A redistribution which randomly assigned tax liabilities and transfer income would treat equals equally. So too would one that, for example, imposed equal transfers for all, zero taxes on the poorest 80 percent and such high rates on the richest 20 percent that, on a post-tax and transfer basis, the taxed units were scattered ‘See Atkinson (1980) and King (1980) for recent considerations of this concept. ‘These assumptions can be relaxed without affecting the results of this paper. ’ For a recent statement see Musgrave (1976). ‘This paper ignores how one determines levels of economic well-being and which units halt equal levels of well-being. For a review of many of the relevant issues, see Taussig (1976).

R. Plotnick,

Horizontal

inequity

315

among the remainder. Intuitively, both schemes violate our sense of what is equitable treatment of units with similar, but not identical levels of initial well-being. In view of these problems, Feldstein (1976) and others suggest that a horizontally equitable redistribution must satisfy two conditions: 1. Equal treatment of equals. 2. The redistribution must not alter the rank order of units. The second condition is more general than the first. It may be thought that the rank order condition is really a principle of vertical equity. However, when one carefully distinguishes between the concepts of horizontal and vertical equity, this conclusion does not follow. The idea of vertical equity is perhaps best interpreted, in Nozick’s (1974, ch. 7) terms, as an ‘end state principle’. One compares an observed distribution of economic well-being to an optimal one. (Whether the optimal distribution was derived from a utilitarian, Rawlsian or other perspective does not matter for this discussion.) If they differ, vertical inequity exists the distances between some or all positions in the distribution are too large or too small. A redistribution reduces vertical inequity if it moves the actual distribution ‘closer’ to the optimum. It is clear that this interpretation of vertical equity does not include a rank order condition. Useful measures of inequality satisfy the ‘symmetry’ or ‘anonymity’ princip1e.7 They are independent of which unit occupies each position in the distribution. Thus, inequality comparisons ignore rank reversals - the core of the notion of horizontal inequity. In contrast, the principle of horizontal equity addresses the fairness of a process of redistribution rather than the final distribution it produces. It does not, and should not, judge whether the initial or final distribution is optimal and whether redistributive policy moves the distribution towards an optimum. Instead, given the pre- and post-distribution, its concern is if the means used to transform the distribution were equitable. If they were not, social welfare has been reduced relative to its level under a process producing the same final distribution without reranking.’ Conceivably, one could argue that a particular final distribution is not just, yet agree that it was attained by a horizontally equitable process.’ This emphasis on process brings out an important implicit assumption in the discussion of horizontal equity-the initial ranking is taken to be fair. Yet there are many reasons to reject this judgement. Arguments for distributive justice based on end state principles place no normative impor‘See Sen (1973) or Fields and Fei (1978). ’ This emphasis is central in King’s (1980) analysis. ’ It is more unlikely, though, that one could regard a final distribution as fair while agreeing that the redistribution that produced it was horizontally inequitable. In this sense, horizontal equity is a prior concept to vertical equity.

376

R. Plotnick.

Horizontul

inequity

tance on possible rerankings. Nozick’s entitlement view gives greater weight to the pre-distribution, but also admits that a rectification of holdings, which would probably rerank units, may be needed.“’ If no normative value is attached to the initial ranking, a reranking cannot be unquestionably judged inequitable. In such a case the norm of horizontal equity has no role. It may be of interest to have a measure of the mobility in the distribution produced by the tax and transfer laws, but such a measure would not necessarily indicate horizontal inequity, nor would one want to view larger values of this measure as less desirable than smaller values.” For the remainder of this paper, then, I assume that the initial ranking deserves to be preserved.” For assessing the horizontal inequity of an existing tax-transfer regime, the relevant initial ranking will clearly be the pre-tax, pre-transfer one. For measuring the horizontal inequity of changes in tax (or transfer) policy, the appropriate initial ordering may be, depending on the situation, either the same pre-current policy one or the post-current policy. pre-change, one. If the policy change seeks to reduce existing horizontal inequities, the precurrent policy ranking must be used. A successful reform would necessarily alter the post-current policy ranking. Thus, if the post-current tax ranking were taken as the initial ranking, one would erroneously conclude that the reform generates horizontal inequity, while the status quo creates none. The proper procedure would be to assess the horizontal inequity of both current and reformed policy against the pre-current policy ranking, and then determine if, in fact. the reform creates less inequity. However, if the postcurrent policy ranking is deemed horizontally equitable, so that the new policy is motivated by, for example, revenue needs, then the pre-change, post-current policy order would provide the appropriate initial ranking. 3. Horizontal Consider P=(p,,p,

,...>

inequity

comparisons:

the following

situation.

P,,),

O
of unit i. The government Y described in a similar

Principles

and a theorem

The pre-distribution is given by a vector < pq, where pi is the level of well-being transforms P into one particular post-distribution manner.13 There are q! ways of redistributing ...

“‘Government policies which alter rankings in the income distribution. such as equal I,pportunity policies. can be understood as attempts to rectify past injustices. ” Atkinson (1980) has offered observations similar to those in this and the prececding paragraph. “Altcrnativcly, suppose the fairness of the initial ranking is questioned. Let the analyst specify the fair initial ranking.. This can be compared to the actual final ranking to assess horizontal inequities. ” Discrete distributions, aside from being more realistic than continuous ones, must be used when dealing with rank order and reranking. I assume each value of p, or y, is unique, so there are no tics.

R. Plotnick, Horizontcll inequity

377

economic welfare to obtain Y. All introduce reranking except the one which gives y, to the unit with p,,y2 to the unit with p2, etc. A method is sought for ordering these redistributions in terms of the seriousness of the horizontal inequities each introduces. Thus, the final distribution Y is to be held constant (except for a scalar multiple - see principle 1 below) in comparing redistributions.‘” (Analysis of a more general method that compares the horizontal inequities of any two redistributions, regardless of the final distributions, lies beyond the scope of this paper.) Given P and Y_a redistribution mapping P into Y may be described by a q xq permutation matrix M. In this representation, if the unit of rank i III the pre-distribution is shifted to post-rank j. rni, = 1. That is, the unit with initial level of welfare of p, now has yi. The perfectly horizontally equitable redistribution is represented by an identity matrix, 1.

3. I. Three principles

for horizontal

inequity

comparisons

Drawing upon analyses of properties commonly regarded as desirable for inequity comparisons, I propose three principles as a basis for inequity comparisons. First, the degree of horizontal inequity should be solely related to a redistributive process and hence independent of the level of aggregate or mean welfare. Consider the vectors P and Y as defined above and two other vectors, P’ and Y’, that are obtained by multiplying by a common scalar. Because M= M’, I require the two redistributions to be equally inequitable. This is denoted R(M) = R(M’). P1. Independence if for some a>O,

from Mean Y’=aY

Welfare

Level: M=M’),

(and. hence,

Given

the initial

then

R(M)-R(M’)

rank order,

“The precise shape of the initial distribution, P, is not important. The importance of P is that it establishes the fair ranking of units. Given that ranking. assessments of horizontal inequity need only consider the post-distribution and any reranking within it. For example. the two redistributions A and B shown here are equally inequitahle because the pattern of reranking is identical: Pre-welfare

Unit X Unit Y Unit 2

Post-welfare

A

B

A

B

12 S 3

9 I 4

6 9 5

6 9 5

Any concern because the differences between pre- and post-welfare levels vary in the two situations reflects uertical equity judgements on the appropriate pattern for altering welfare diKerences via redistribution.

Acceptable measures of inequality satisfy the anonymity condition. If two distributions of welfare differ solely because the identities of the units receiving each share of welfare differ, the distributions have the same level of inequality. Such a property is clearly inappropriate for interests in horizontal inequity. A unit’s identity, i.e. its rank in the pre-distribution, is fundamental. However, the basic idea that an inequality measure should not distinguish among units by anything except their shares of welfare does have an analogue for comparisons of horizontal inequity. It requires such comparisons to be independent of all characteristics of the units cxccpt their preand post-redistribution levels of welfare. Thus. the second principle is: P2. Anonymity: If the pair of vectors identities of the units occupying each

P,Y differ from P’.Y’ only in the rank, and M = M’. then R(M)=

R(M’).

That is to say, if two pairs of initial and final distributions differ only because of the identity of the units occupying the initial ranks, but the reordering of ranks and, hence, welfare levels, is identical, the redistributions create the same degree of horizontal inequity. Pl and P2, like the similar ones for inequality issues, are formal axioms which pose little controversy.” They enable one to recognize when two redistributions are equally inequitable. but provide no guidance when comparing two fundamentally different redistributions and deciding which creates less horizontal inequity. The third property addresses this crucial issue. Useful inequality indices satisfy one form or another of Dalton’s ‘principle of transfers’, which asserts that a positive transfer from one unit to a poorer one should reduce measured inequality. An analogous principle for comparing redistributions can be developed. Consider two units with initial ranks i and j, i
ij-i’(
(1)

and sgn( i - j’) = sgn( i - i’);

sgn(j-

i’) = sgn(j-

j’).

(2)

I define this change as an inequity reducing reversal (IRR). An IRR moves both units closer to their equitable initial ranks. without overshooting. (Further discussion of this definition appears below in subsection 3.3.) An “Postulating that arouses some debate.

the level if inequality is independent of the mean level of well-being See Kolm ( 1976). For horizontal inequity, this is less likely.

R. Plotnick,

Horizontal

inequity

379

IRR is possible if i 5 j’< i’s j. Any other ordering will fail (l), (2), or both. Let M represent a redistribution. Perform an IRR and denote the resulting matrix by M’. (Corollary 2 below implies that an IRR is always possible unless M = 1.) Then write M’ = IR(M). The third principle is: P3. Principle of Inequity Comparisons: If M’= IR(M), where < means ‘less inequitable than’.

then R(M’)<

R(M),

3.2. A theorem on ordinal comparisons of horizontal inequity Given this definition of IRR, one can compare and order the extent of horizontal inequity among redistributions within limited subsets of the q ! possible redistributions without adopting a specific cardinal index. This result parallels the familiar theorem on Lorenz domination for inequality comparisons. One definition is needed before this theorem can be presented. Definition. Given or is interior to X (Ii C,(j)5Cw(j)5j where C,(j) is the (2) For at least

two permutation matrices, W and X, W lies inside of X, if, for each row j, or j5Cw(j)5Cc,(j), column of matrix Z in which row j’s one appears, and two rows C,(j) < C,(j) or C,(j) < C,(j).

In other words, W is interior to X if all of its positive entries lie on the same side of the diagonal as X’s, but not further away from it, and some of them lie closer to it. For example, of the three matrices below, S is interior to Q, but T is not (rows 2 and 4 of T do not satisfy the definition):

[;

fi

;j;

[i

i;

;j;

[i

jT;

;j.

LLR The major result developed is:

from

which

implications

for inequity

comparisons

are

Theorem. The redistribution described by matrix N can be obtained from that described by M by a series of IRR’s if and only if N is interior to M. Proof. The necessary condition (that N obtained from M by a series of IRR’s implies N inside of M) is trivial. Consider rows i and j of N with i < j and l’s in columns i’ and j’, respectively. For an IRR to be feasible, one

must have i 5 j’< i’s j. Reverse the final ranks (i’ and j’) of the units and denote the new matrix by M’. This, in effect, interchanges rows i and j of M. In M’, the 1 in row i lies in column j’, which, because j’< i’, is closer to the diagonal (column i) and on the same side of it as is the 1 in M. The same is true of row j, while all other rows are unchanged. Thus, M’ is interior to M. A second IRR performed on M’ must also, by the same logic, result in some M” inside of M’. Thus, any matrix resulting from a series of IRR’s, including N, will always be interior to M. The sufficient condition is harder to establish. I proceed by showing that, given N interior to M, one can always perform an IRR on M that yields a new matrix that either lies ‘between’ M and N (defined in the obvious sense) or coincides with N. Because the demonstration that such an IRR exists is perfectly general, it follows that, given this new matrix, another IRR exists that moves the matrix ‘closer’ to N. Hence, after a finite number of IRR’s N will always be obtainable from M. Let C,(i) be the column of M that contains the unity in row i of the matrix. Let C,(i) be similarly defined. Label each column of M with an R if iZC,(i)
(That is, the 1 in row i of M lies to the right of the 1 in row i of N, and i 5 C,(i) because of the assumption that N is interior to M.) Label with an L (for left) those columns of M where C,(i)
If C,(i) = C,(i), the column is not labeled. I claim the rightmost labeled column must be labeled R. To see this, find row k in M such that C,(k) = q (= number of units). Then in row k of N, C,(k) s C,(k). If the strict inequality holds, column q, the rightmost column in M, is labeled R. If the equality holds, column q is unlabeled. Proceed to row k’ of M such that C,(k’)= q- 1. Then C,(k’) 5 C,(k’). As above, either column q -- 1 is an R-column, or unlabeled. If it is unlabeled, find row k” where C,(k”) = q -2. Continuing in this fashion, eventually a row must be found for which the strict inequality holds. If this were not so, every column would be unlabeled and M = N, contrary to assumption. A parallel argument demonstrates that the leftmost labeled column of M is an L-column. We now know that some columns of M are labeled R and some are labeled L, and that some R-columns lie to the right of some L-columns. Hence, we can locate an L-column, say column e’ and an R-column, say r’, such that 4” < r’ and any intervening columns are not labeled. Suppose c,,,,(e)= +5”, CM(r) = r’. Interchange rows t and r in matrix M. T claim this

R. Plotnick,

Horizontal

3x1

inequity

represents an IRR and the resulting matrix, M’ with C,,(r) = f? C,,(e) = r’, lies between M and N or equals N. To verify this claim, observe that, by definition of an R-column, rSC,(r)
and

(3)

Then note that C,(r)Sk”. This is true because, by assumption columns F+ 1, F-t 2, . . , r’- 1 are unlabeled and, hence, the rows corresponding to these columns [i.e. Cd(e’+ l), etc.] do not include row r. Thus, from (3) and because .!? < r’ we have rSt?
Similarly,

(4)

because

e’ = c,(e)

4’ is an L-column: < c,(e)

5 4.

(5)

And C,..,(e) 2 r’ because, by assumption, columns labelled. Hence, from (5) and because e’< r’:

C’+ 1,. . . , r’- 1 are un-

+!‘
Combining

(6)

(4) and (6) gives rSP
(7)

.

Thus, rows r and e and (2). To verify that M’ necessary condition that N is inside or that C,,,(r) 5 P’:

will satisfy the definition

of IRR given by expressions

(I)

M and Nor equals N, observe first that the of this theorem guarantees M’ will be inside M. To see coincides with M’, note that from (3) and the argument lies between

rSC,(r)S4’=C,Jr)

(8)

Similarly = r’s C,(t)5

C,,(e)

k?

(9)

Expressions

(8) and (9) satisfy the first condition of insidedness. If either or C,,(t) < C,(e), condition 2 also holds and row r or & of N lies inside of the corresponding row of M’. If neither strict inequality hold, the two matrices coincide in rows r and C. If they coincide in all other rows, N = M’. If not, because M’ only differs from M in rows r and JY,and because N is inside M, some rows of N also lie inside those of M’. Q.E.D. C,(r)

< C&(r)

Corollary

1.

If

N is interior to M, then R(N)<

R(M).

The proof is immediate, given the principle of inequity comparisons. This result tells us that inequity comparisons are possible in some cases without resort to a specific cardinal measure.

R. Plotnick,

382

Corollary 2.

For all Mf

I, R(I)<

Horizontal

inequit)

R(M).

The identity matrix lies inside of any other. Thus, any redistribution represented by M creates more horizontal inequity than the complete rank-preserving redistribution, represented by I. The theorem provides a partial ordering of redistributions in terms of the horizontal inequity each creates. However, in comparing many pairs of redistributions, neither matrix will lie inside the other and, consequently, it would be impossible to decide which is the more inequitable. It appears that cardinal measures will still be needed for most comparisons of redistributions. A simple example illustrates the process for finding IRRs described in the proof of the theorem. Let Q and S be the permutation matrices shown above, with S inside of Q. Inspecting row 1 of Q and S, we label column 4 with R. From row 2, column 1 is unlabeled. Columns 2 and 3 are both labeled L. The labels tell us to interchange rows 4 and 1 of Q. The new matrix Q’ (shown below) lies between Q and S. Repeating the process for Q’ gives no label for columns 1 and 4, an L in column 2 and an R in column 3. The procedure now leads us to interchange rows 1 and 3 of Q’, which yields S. In general the sequence of interchanges is not unique because many pairs of R and L columns may be interchangable at any step:

Q'

L

3.3.

R

Discussion of the definition of an LRR

Concern for horizontal inequity often rests on the premise that it reduces social welfare [see Atkinson (1980) and King (1980); Atkinson discusses other interpretations, also.] In this subsection I consider functions that measure this loss of social welfare and discuss the relationship between such functions and the IRR defined earlier. Holding constant the final distribution, the fundamental arguments of a loss function would be the degrees of horizontal inequity suffered by each unit. A unit’s ‘degree of horizontal inequity’ could plausibly be a function of either i ~ i’, where i is the unit’s initial rank and i’ its final rank [e.g. King (1980)], or by y,-y,,, where yi is the final welfare it would have if its rank were not changed by the redistribution, and yiZ is its actual final welfare

R. Plotnick,

Horizontal

inequity

3x3

If i - i’ = yi = yiS= 0, its degree of horizontal resulting from redistribution.” inequity is assumed to be zero. The rank difference approach appears suspect because, depending on the shape of the post-distribution, a given difference between actual and fair (rank-preserving) post ranks may represent a wide range of differences in final welfare. For example, the case where i = 100, i’ = 110, yi (expressed as a fraction of total welfare) = 0.01, yiC= 0.0100001 intuitively would seem a less serious violation of horizontal equity than were i = 100, i’= 110, y, = 0.01 and yiS = 0.011. While ranks have been equally altered in both cases, in the first case the unit’s actual welfare is rather close to its fair value, but in the second the divergence is much larger. Hence, assume that arguments of the loss function L(.) are welfare differences, di = yi - y,,.” Then we have

L(k.,(d,L . . . >~L,(d,,)) = Uw,,

. . >w,),

(10)

where the p, are functions that transform the di into ‘degrees of horizontal inequity’.” If all di = 0, L(.) = 0. The function L should be an increasing function of the w,: g>O, I

for all i.19

The wi could be equally weighted or not, depending on one’s choice of specific loss function.20 Thus, preservation of rank order ensures that one’s observed final level of well-being is also one’s ‘fair’ level and gives L( .) = 0. The definition of an IRR is consistent with the loss function approach. Unless one assigns to all units specific weights for comparing the degrees of their horizontal inequities, one can conclude that reversal of final ranks ‘“Thus, a unit whose initial and final incomes are identical may, nonetheless, be subject to horizontal inequity. For example, if initial rank were j, final rank j’ and the levels of final or v, -y,, even if, by welfare at ranks j and j’ were y, and y,,, one would examine j-j’ coincidence, the unit’s actual final welfare y, equalled its welfare in the initial distribution. ” The reason we care about preserving rank order, it seems to me, is that rerankinp means a unit’s actual post-welfare diverges from its rank-preserving value. It is this divergence which is troubling and which I seek to capture in the loss function. (In a society in which relative position - rank - was all, this analysis would be less relevant and one based entirely on the rank differences would apply.) “L clearly cannot be a function of the utility or welfare of each unit. When horizontal inequity exists, some units’ levels of welfare are too high while others’ are too low compared to what they would be if ranks were preserved. Any deviation from the rank-preserving final distribution creates horizontal inequity and, hence, a loss of social welfare. So one cannot compare changes in units’ welfare levels to determine if gains for some offset losses to others. Both gains and losses of welfare indicate horizontal inequity; they cannot offset each other. 19A priori, it is not clear whether one would want ?I”L/&J: to be positive or negative. The inequality measure literature assumes the standard negative case for the welfare function it analyzes. Here, when dealing with a loss function, my preference is for positive (or perhaps non-negative) second derivatives. “‘Note that P2, the anonymity principle, does not require equal weights. It simply requires that, whatever the identity of the unit in rank i, its impact on L is always F,(d,).

7x4

R. Plotnick,

Horizontcrl

inequirv

between two units reduces the loss of social welfare only if it reduces the horizontal inequity suffered by both units. Note that any reversal that satisfies expressions (1) and (2) will necessarily reduce both units’ welfare differences.” Thus, an IRR will decrease the value of any loss function having the properties discussed above, while imposing few conditions on the functions L and ~~~~~ It appears difficult to find an alternative definition of IRR that permits an equivalence theorem such as the one above. A logical approach might change the rank difference inequalities to inequalities of the differences between units’ rank preserving final welfare levels and their observed final levels before and after a rank reversal, drop the sign condition, and establish the corresponding definition of ‘insidedness’. However, simple counterexamples can be found which contradict a conjecture parallel to the earlier theorem. 4. Cardinal

indices of horizontal

inequity

Based upon the considerations raised in sections 2 and 3, this section offers a new class of measures of horizontal inequity and evaluates several measures proposed by others. Principles l-3 suggest that ‘good’ measures of horizontal inequity should exhibit three properties. Let H(P,Y) denote the index value for the redistribution described by initial and final welfare vectors P and Y. Then: Property 1. If P, Y, P’ and H(P, Y) = H(P’, Y’).

Y’ are

Property 2.

If P. Y, P’ and

Property 3. M’=IR(M),

If matrix M corresponds then H(P’,Y’)
These 4.1.

properties A new class

as in the

Y’ are in P2, then

will be denoted

PRl-3

to

discussion

H(P,Y)

P,Y;

and

to correspond

of Pl,

then

= H(P’.Y’). M’ to

P’,Y’;

and

with Pl-3.

of measures

Any specific measure of horizontal inequity embodies an implicit choice of both CL,in (10) and the weights attached to the pi at different positions in ‘I A decrease in the absolute value of the rank difference - condition (1) - in general need not imply a decrease in Id,/ if overshooting is allowed (i.e. moving from a rank below one’s initial value to a rank above it). If Id,1 d oes not fall, the degree of horizontal inequality measured by L may not decrease. The sign requirement of condition (2) prevents overshooting and ensures that Id,1 declines for both units involved in the TRR. ” The theorem would hold if condition (11 of the IRR definition were stated in terms of welfare differences, as well.

R. Plotnick,

Horizontul

inequity

the final distribution. I suggest that all units’ ‘extent of inequity’, however measured, should be equally weighted. If one accepts the initial ranking as fair, a reranking of any unit increases horizontal inequity. Holding the extent of inequity constant, violations of horizontal equity would seem to reduce social welfare by the same amount, regardless of the initial ranks of the affected units. For analysis of vertical equity, it may well be appropriate to have variable weights, but a similar conclusion for horizontal inequity seems less warranted. The options in measuring a unit’s ‘extent of inequity’ - the wLiin (10) are more varied. For reasons similar to those in the preceding paragraph, the ~~ should be identical. I will restrict attention to ki such that (see footnote 19) d2L/dwf 2 0.

(11)

Normalize all final levels of well-being to sum to one. This ensures that PRl is satisfied. Let there be N units. Denote their observed level of post-redistribution well-being by yp and their rank-preserving final level by yr. A plausible class of measures H(P,Y,s) is H(P,Y,s)

(12)

= f lY,-y:l’, i=l

where sz 1. Let H(P, Y,s) = L( .). Clearly, H(I: P,s) meets (11). If yb= yf for all i. H(Y,P,s)=O. Such measures satisfy PR3 because, for an IRR to be feasible, y: sy”, < yj, 5~5. If the final welfare (hence, ranks) of units j and k are interchanged, (12) will decrease in value. While this measure has a lower bound of zero for all s, it does not have a uniform upper bound. For convenience and easier interpretation, it may be preferable to use as an index HH(Y,P,s)EH(Y,P,s)/max(H(I:P,s)), which ranges between zero and one. It can be shown that the maximum is always associated with the redistribution giving the unit with initial rank i a final rank N + 1 -i, a complete reversal of ranks.2’ This is intuitively appealing. Alternatives to H are, of course, available. One need not be restricted to additive loss functions, nor accept the argument for equal weights or uniform pi. However, it is clear that any measure must select some p, and embody value judgements about the importance attached to rank changes and differences in well-being at different positions in the final distribution. The judgements implicit in other possible measures of horizontal inequity should be identified by their proponents. ” A proof of this claim is available shape

of the final distribution

being

upon request. analyzed.

Of course,

max (N(Y,P,s))

depends

on the

386

R. Plotnick,

Horizontal

inequity

4.2. Evaluation of other indices of horizontal inequity A number of other measures of horizontal inequity have been proposed and applied in papers by Johnson and Mayer (1962), White and White (1965), Brennan (1971), Rosen (1978), Strauss and Berliant (1979), Habib (1979), Hettich (1979), Atkinson (1980), King (1980) and Plotnick (1981). This subsection evaluates them based upon the arguments developed in this paper. From the perspective of this paper, there are reasons for rejecting most of them because they either fail PRl or do not deal effectively with rank reversal and fail PR3. Some mix norms of vertical equity into what are proferred as measures of departures from horizontal equity. For those which satisfy PRl-3 - the Spearman rank correlation used by Rosen (1978), an index based on concentration curves developed separately by Atkinson (1980) and Plotnick (1981), and King’s (1980) index, which builds on the approach to inequality measurement in Atkinson (1970) - the choice of kZ and the weighting scheme implicit in each are examined. Johnson and Mayer (1962) propose two simple measures of horizontal inequity. One examines units which are ‘by definition within the same group, but receiving unequal treatment’ (p. 457) and counts the number of inequities. If there are N units with N, receiving treatment 1 and N2 receiving differing treatment 2, the number of inequities is N, . NZ. The calculation is easily extended to t treatments. The lower the count, the less horizontal inequity exists. This simple approach satisfies PRl - independence from the mean and PR2 anonymity. It badly fails PR3 inequity comparisons because it only addresses differences within a group of equals and is completely silent on the crucial reranking issue. Brennan (1971, pp. 440-441) levies further valid criticisms, which need not be repeated here. Johnson and Mayer’s second measure sums the money value of the inequities. They also consider sums that would not weight inequities in strict proportion to the money value. These indices violate PRl. Brennan (1971) modifies this approach in several ways which satisfy both PRl and PR2. He sees ‘a strong case’ for the coefficient of variation V = M/t, where t is the average tax payment and M the standard deviation of the distribution of tax payments among equals, but favours the kurtosis of the distribution. White and White (1965) also use V in an empirical study. Such measures have the same major defect as the numbers approach - they ignore possible reranking across different groups of equals. Brennan later (pp. 450-453) recognizes that the Johnson and Mayer framework and his modifications only apply when all units in the population are economic equals. To incorporate differences in initial income across units into the analysis, alternative indices are proposed. These are unsatisfactory because they combine notions of horizontal and vertical equity. The

R. Plotnick, Horizmtal inequity

387

measures focus upon differences between the actual tax paid by a unit and the ‘conceptually optimal’ tax for units with that level of pre-tax income. Yet it was argued above that horizontal inequity is properly concerned just with the process of redistribution and any rerankings. not with how closely the final distribution approximates an optimal one.‘4 More recently, Strauss and Berliant (1979) construct an index based on ‘the extent to which effective [tax] rates are different among all paired comparisons of taxpayers within each income class’. To implement this index, all units must be placed in specific income and tax rate intervals. Their approach is akin to the ‘number of inequities’ measure, through more sophisticated. It suffers from the same defect - reranking across income classes are not treated. Furthermore, finite income classes and tax rate classes will generally contain units with similar, but not identical, incomes and tax rates. An IRR might be possible between two units categorized into the same income and tax rate class, but if such an IRR were performed, this index would not change in value. Habib (1979, p. 300) uses a similar index with similar flaws. The measure offered in Hettich (1979) is

where n = number of units, ti = actual taxes of unit i and g, = taxes if the preferred tax base of ‘broadly defined income’ is adopted. A value of zero is taken to signify no horizontal inequity. The index fails PRl, but this can be easily corrected. More problematic is that the index does not address the central issue of reranking because nowhere in its construction are comparisons of initial and final ranks needed. Moreover, while deviations of actual taxes from those with an isorevenue alternative are supposed to represent inequity, nothing assures that, in general, this alternative itself preserves horizontal equity.” Rosen (1978) applies two indices in his analysis. One is the Spearman rank correlation R: R=l-6Cc:/(n’-n),

” An appendix available upon request addresses the problems with these measures in more detail. *s Suppose that the g do preserve ranks (as Hettich later does by making g, a linear function of income). The index rlmains problematic, however, for it is then confounding norms of vertical and horizontal equity by comparing actual taxes with some pattern of g, imposed from the outside. Given the initial income distribution, there will be many ways to raise a given sum of revenue, all of which preserve ranks (though, or course, they would vary in how income differences across ranks are altered). None creates horizontal inequity, but the index will be positive except under the unique tax regime in which t, = R, for all i.

JPE-

E

where c is the difference between a unit’s initial and final rank and n is the population size. If RR = 1 - R, then RR meets all three suggested properties. To see that PR3 holds, note than an IRR, by definition, reduces rank differences of both units, Thus, following an IRR, the summation term decreases, and RR will decline. Because RR satisfies the three properties, consider its implict I_L,,which transform differences between unit’s observed final levels of welfare and their rank preserving levels into degrees of horizontal inequity. and the weights on each IA,. With RR. the weights are identical, while pi(dc) = c’. The function is independent of the size of di. This may not appeal intuitively, as our simple example illustrates (see table 1).

fable

1

1

2

3

3

0.10 0. 1 I 0.12 0.21 0.22 0.24

0.12

(1.10 0.2 I 0.12 0.11 0.22 0.24

0.10 0.1 I 0.21 0.12 0.22 0.24

0. I I 0.10 0.21 0.22 0.24

Column 1 shows the rank-preserving levels (normalized to sum to unity) of final welfare in a six unit economy. Columns 2-4 are redistributions with some horizontal inequity. Then RR( 1)= 0; RR(2) = RR(3) = 0.229,and RR(4) = 0.057.Redistributions 2 and 3 are judged equally inequitable, yet in the former, the differences in welfare - the di - due to reranking are slight (c).02), while in the latter they are rather large (0.10). Intuitively. the latter represents a more serious infraction of the norm of horizontal equity since its reranking leaves the reranked units rather far from their fair (rank-preserving) levels of final welfare (see subsection 3.3 and footnote 17, also). Similarly, comparing 2 and 4 using RR suggests that the former is more inequitable. Yet the differences in welfare in redistribution 4 are 0.09. well above those in column 2. Rosen’s second index is derived as follows. Let P and Y be the initial and final distribution vectors. Randomize the elements of P and arrange the elements of Y in the same order. Form vectors DP and DY, with the ith element defined as:

R. Plotnick.

Horizontal

inequity

389

The simple correlation between DP and DY is the measure; a value of one implies no horizontal inequity. This index suffers because its value is a random variable an odd property for measures of this sort. Moreover, a perfectly equitable nonlinear tax rate schedule will yield correlations less than one. Last, results depend on the correlation between income differences in the initial distribution and those in the final one. A judgement that cardinal differences in moving from the initial to final distribution should be correlated is a vertical equity decision, not one of horizontal equity. Atkinson (1980) and Plotnick (1981) independently propose the same measure. A pseudo-Lorenz curve is constructed by ordering units by their initial ranks, but plotting cumulative final levels of income. The resulting curve always lies above the conventional Lorenz curve for final income (and may even cross the diagonal). The area between these two curves, suitably normalized, is the index. This index satisfies all three properties. Suppose there are N units, M of whom are reranked. Let mean level of well-being be Y. Arrange these M in ascending order of their initial rank, 1. Denote their observed final level of well-being by yf and their rank-preserving final level by yn. Plotnick showed the index value is (13) This expression is invariant with respect to the mean level. To see that PR3 holds, let the level of (13) be H and find two units among the M with initial ranks j, k, j < k, for which an IRR is possible. For an IRR to be feasible, yr5 yfk< yf5 y:. Reverse the two units, calculate (13) again and let it be H’. Then : (H-

H’)N*Y

=r;(Yr-Yf)-rj(Yg-y:)+r,(ye-y:)-rk(y~-yj) =r,(yfk-yyf)+k(yf-A) =(rk-I;)(yi-yfk)>O.

Hence, this IRR reduced measured horizontal inequity. Note that a reranked unit’s contribution to (13) is weighted by its initial rank. If the argument in subsection 4.1 is accepted, however, equal weights are probably more appropriate. The measure by King (1980) is perhaps the most interesting of all those reviewed here, for it is derived from explicit consideration of the social welfare judgements concerning the tradeoffs among horizontal and vertical

390

equity

K. Plotnick.

and efficiency.

Given

Horizorttul

his assumptions,

inequitv

the index

for N households

is

k#0

k = 0. Here. q is the degree of aversion to horizontal inequity, and s, =w h ere r; is the rank in the final distribution and F, is the rank Ir,-r,il(N-1). in the initial distribution. This index also satisfies PRl-3. Npte the reliance on rank differences in the s,. A unit’s impact on I,, depends only on s,, not the difference between its actual final welfare and rank-preserving welfare. The examples discussed above for RR and their counterintuitive results would apply equally well to 1bl. 5. Summary

and conclusion

This paper has explored the concept of horizontal inequity in redistribution and how the extent of the phenomenon can be appropriately measured. The central tenet of the norm of horizontal equity was seen to be the preservation of rank order. This generalizes the usual ‘equal treatment of equals’ approach. The paper argued that comparisons of horizontal inequity should be based upon three plausible principles closely related to those often suggested for inequality comparisons. Two of them - independence from the mean and the anonymity condition - are formal conditions that impose few restrictions on a mcasurc. The third is a principle of inequity comparisons which rests upon the notion of an ‘inequity reducing reversal’ and is ultimately grounded in a welfare loss function view of horizontal inequity. Under one definition of an IRR, an equivalence theorem was proved showing that the classes of redistributions that can be obtained from one another via IRRs, and, hence, for which ordinal comparisons of horizontal inequity can be made, coincide with the classes that meet a certain ‘inside’ condition. Without further assumptions, there will be many rcdistributions for which degrees of horizontal inequity will be incommensurable. To compare the horizontal inequity of most redistributions, therefore, a cardinal index must be posited. Three properties that follow from the principles were proposed, A new class of indices consistent with them was presented. Of the other proposed measures judged against these properties, only three satisfied them. It was

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Horizontal

inequity

391

emphasized that any measure embodies implicit weighting schemes, and the particular schemes of these three measures and the new one were examined. Applying any such index will be difficult because appropriately measuring economic well-being poses many problems. Also problematic is determining the counterfactual distribution which would exist in the absence of the redistribution being analyzed. It is the ranking of this distribution which is assumed to be fair and forms the basis for assessing departures from horizontal equity, but the theoretical and empirical problems of establishing it are severe.

References Atkinson. Anthony. 1970, On the measurement of inequality, Journal of Economic Thcorq 2. 244-263. Atkinson, Anthony, 1980, Horizontal equity and the distribution of the tax burden. in: H. Aaron and M. Boskin, eds.. The economics of taxation (Brookings Institution. Washington. D.C.) 3-18. Brennan. Geoffrey, 197 I. Horizontal equity: An extension of an extension. Public Finance 26, 437-456. CowelI, Frank, 1977, Measuring inequality (John Wiley and Sons. New York). Dasxupta. Partha, Amartya Sen and Dacid Starrett. 1973. Notes on the measurement of inequality. Journal of Economic Theory 6, 1X0-187. Feldstein, Martin, 1976. On the theory of tax reform. Journal of Public Economics 6, 77-104. Fields, Gary and John Fei. 197X. On inequalit) comparisons, Econometrica 46, 303-316. Habib. Jack. 1979. Horizontal equity with respect to family size. Public Finance Quarterly 7, 2X3-302. Hettich, Walter, 1979. A theory of partial tax reform. Canadian Journal of Economics 12, 693-7 12. Johnson, Shirley and Thomas Mayer. 1962. An extension of Sidpwick’s equity theorem. Quarterly Journal of Economics 76, 454-463. King, M.A., 1980, An index of inequality with applications to horizontal equity and social mobility, Social Science Research Council, Programme on Taxation, Incentives and the Distribution of Income. no. 8. Kolm. Serge-Christophe, 1976. Unequal inequalities I, Journal of Economic Thcorq 12. 416-442. Musgrave. Richard, 1976. ET, OT and SBT. Journal of Public Economics 6. 3-16. Nozick. Robert. 1974, Anarchy. state and utopia (Basic Books, New York). Plotnick. Robert, 19X1, A measure of horizontal inequity, Review of Economics and Statistics 63, 2X3-288. Rosen, Harvey, 1978, An approach to the study of income, utility and horizontal equity. Quarterly Journal of Economics 92. 306-322. Rothschild. Michael and Joseph Stiglitz. 1973. Some further results on the measurement of inequality, Journal of Economic Theory 6, 1X8-204. Scn. Amartya, 1973, On economic inequality (Norton. New York). Strauss, Robert and Marcus Berliant. 1979. The vertical and horizontal equity of recent tax reform and reduction proposals. unpublished. Taussig, Michael. 1976. ‘Trends in inequality of well-offness in the United States since World War 11, in: Conference on the Trend in Income Ineuualitv in the U.S., Special Report 11 (Institute for Research on Poverty, Madison. Wisconsin). . White, Melvin and Anne White, 1965, Horizontal inequality in the federal income tax treatment of homeowners and tenants, National Tax Journal 18, 225-239.