Equivalizing Incomes: A Normative Approach

International Tax and Public Finance, 7, 619–640, 2000.c© 2000 Kluwer Academic Publishers. Printed in The Netherlands.

Equivalizing Incomes: A Normative Approach

UDO EBERT [email protected] of Economics, University of Oldenburg, D-2611 Oldenburg, Germany

Abstract

The paper deals with the comparison of living standards and investigates two normative methods of derivingequivalizing transformations for a population which has different household types. The first one equates the utilitylevels of representative household members belonging to different households. The second method evaluatesthe well-being of households by a social welfare ordering defined by means of household utility functions. Themethods can determine the implicit normative assumptions involved in conducting distributional analysis usingany equivalizing transformation. In particular income-level dependent equivalence scales can be founded in thisframework. The assumptions underlying both approaches are examined and compared.

Keywords: household composition, living standard, equivalizing transformations, normative comparison, equiv-alence scales

JEL Code: D63, I31, D11

1. Introduction 1

The paper deals with the comparison of living standards. This is a central topic of economicswhich is quite well understood as long as only one individual is concerned (see e.g. Diewert(1981) and the literature cited therein). In the following, different individuals, householdsand household types are considered. Then the problem becomes much more complicatedsince interpersonal or interhousehold comparisons are required. As households in generaldiffer with respect to their size and composition, different needs have to be taken intoaccount. This kind of analysis is fundamental for many areas of theoretical and appliedeconomics: e.g. it forms the basis for the measurement of inequality or poverty, and it isa necessary ingredient whenever tax systems or benefit systems have to be designed in asystematic way.

It is common practice to employequivalizing transformationsof income in order tocompare the living standards attained by the members of households of a different size orcomposition: an equivalizing transformation converts the income level of a given type (e.g.a couple) to the equivalent income in terms of income of a reference type (e.g. a singleadult). The equivalent income can be interpreted as an indicator of welfare which allowscomparisons between different household types.

There is, however, no unanimity among economists concerning the way equivalizing trans-formations are to be derived. Probably a general agreement will never be reached becauseall methods are ultimately based on value judgements. Nevertheless for empirical work onehas to find an operational method. In practice equivalence scales are often applied when theliving standards of households with different needs have to be compared; but they can unfor-

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tunatelynotbe determinedpreciselyfrom market data or other empirical observations. Thesame problem occurs with other equivalizing procedures like e.g. those using equivalencedifferences or more complex forms of scales. In any case normative judgements are required,even if scales are to be inferred from demand patterns and consumers’ behavior. Further-more, even if some scales can be derived the question remains whether they are appropriatefor social judgements. Therefore ana priori ethical approach seems to be justifiable.

The present paper presents and investigates two methods of deriving equivalizing trans-formations. The first one equates the utility levels of household members. Householdincomes are defined as being equivalent if they lead to the same level of utility of the in-dividuals belonging to the households under consideration. The approach is based on acorresponding utility profile describing the utility attained by a representative individual ofeach household. The second method evaluates the well-being of households by a social wel-fare ordering defined by means of household utility functions. This ordering is employedto reveal the value judgement concerning an optimal income distribution. It is assumed thathousehold incomes are equivalent if they are part of the same optimal income distribution.If households were not equally well-off social welfare could be increased (given a fixedamount of total income).

Both approaches will be introduced below. Since the choices of a utility profile and a so-cial welfare function are influenced by value judgements the equivalizing transformationsderived can be called ethical. It turns out that both methods allow us to generate everyequivalizing transformation known. Insofar nothing new seems to be accomplished. Onthe other hand the properties of the utility profile and the welfare ordering, respectively,can be determined. Therefore the methods proposed are able to reveal the implicit nor-mative assumptions involved in conducting distributional analysis using any equivalizingtransformations whatever their basis is (see on). Moreover, examples will demonstratethat income-level-dependent equivalence scales can have a normative foundation. The as-sumptions of both approaches concerning measurability and comparability of individualand household utility functions are clarified. The examination necessitates a precise defi-nition of the notions used. Thus the paper presents a thorough investigation of the basis ofequivalizing transformations.

Equivalence scales, used in practice, are often estimated from demand systems. Pollakand Wales (1979) were the first to prove that there is a serious identification problem. Thereare several solutions to this problem (cf. Coulter, Cowell and Jenkins (1992) for a detaileddiscussion of possibilities and limitations). One is imposing an additional hypothesis whichallows to fix an appropriate preference ordering. An example is given by Base LevelIndependenceIB (Lewbel (1989)) or Equivalence Scale ExactnessESE (Blackorby andDonaldson (1993)). Then equivalence scales can in general be recovered uniquely. Butanother problem arises: One cannot falsify the hypothesis. An escape from this dilemma isusing other types of equivalence scales not being based on demand systems. The Leydenapproach determinesindividual welfare of income functions by direct questions. Theyimply ‘subjective scales’. A third type of solution utilizes some a priori information: Onemay ask experts, look at social assistance (in order to compare needs), or use pragmaticscales (e.g. those introduced by OECD).

The equivalence scales or, more precisely, the equivalizing transformations generatedbelow are normative. There is, of course, no identification problem with these approaches

EQUIVALIZING INCOMES 621

since the respective norm is specified a priori and does not have to be identified.The paper is organized as follows: Section 2 introduces the framework and the notation

and defines equivalizing transformations. The first approach based on a utility profile ispresented in section 3, the second one employing a social welfare ordering in section 4.Section 5 investigates the problem if household types can be ranked by needs. Section 6makes an attempt at revealing the relationship between both methods and at combiningthem. In section 7 a number of examples are discussed and an application of equivalizingtransformations to the design of tax systems is provided for the case that horizontal equityis imposed. The last section offers some concluding remarks.

2. Basic Framework

At first we have to introduce some notation and to define equivalizing procedures. Weconsider a heterogeneous population. There aren ≥ 2 types of households2 having differentsizes, differing composition and/or needs. In this paper exactly one household of each typeis taken into account. This assumption, made for simplicity, can be dropped by admittingn homogeneous subpopulations. Then the model becomes more complicated, but theextension does not contribute anything essential to the problem to be investigated. At thispoint no condition on the numbering of households or the needs ranking of types is imposed.Xi ∈ D denotes householdsi ’s income (i = 1, . . . ,n) where the set of feasible incomesD is defined as an open interval(d,∞) for d ∈ R or is equal toR; i.e. D can containnegative values. It must comprise negative incomes in some cases since otherwise somestandards of living attained by different household types would be incomparable (see on).The choice ofD as an interval(d,∞), bounded from below, or as the entire set of realnumbers seems to be arbitrary at first sight, but it has severe consequences for the possibleform of an equivalizing procedure as will be demonstrated below. IncomeXi representsthe total income of a household. It is assumed that there is no information about the incomedistribution within households and, therefore, that all members of a household attain thesamestandard of living. Of course, more elaborated models of household behavior couldbe used (cf. Bourguignon (1989) or Chiappori (1992)). But usually the corresponding datais missing and thus the parsimonious approach, employed here, is justified. An incomedistribution is given by

X = (X1, . . . , Xn) ∈ Dn.

Prices are supposed to be fixed.For a given income distribution the incomesXi andXj of two householdsi and j (1≤ i ,

j ≤ n, i 6= j ) can be compared in monetary units, but they do not directly reflect thecorresponding living standards since the types (and needs) are different. Our main objectiveis to introduce equivalizing proceduresSallowing comparisons of the well-being of differenthouseholds. For such comparisons the fundamental concept generating an equivalizingprocedure is crucial. Two different fundamental concepts will be employed: either a utilityfunction of a representativeindividual for each household type or a social welfare functionbased on evaluation functions of income which reflecttotal welfare of households. Detailswill be given in section 3 and 4. An equivalizing procedure is then a vectorS= (S1, . . . Sn)

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of equivalizing transformations

Si : D→ D

for an (a priori chosen) reference typer (1≤ r ≤ n). HereSi (t) is equal to the income thereference householdr needs in order to be as well off as householdi possessing incomet .Si (t) is called equivalent income. Of course,Sr (t) = t . Using this device living standardscan be compared; i.e. for a given equivalizing procedureS we define: householdi is(weakly) better off than householdj if and only if Si (Xi ) ≥ Sj (Xj ). In other words, theliving standard is defined as equivalent income (measured for a reference typer ). Of course,equivalent income depends on the type chosen for reference. But comparisons of livingstandard should not. In practice often a single adult person is used as reference. In principleany other type could be employed instead.

In order to be precise we introduce a very weak definition.3

Definition. A vector of equivalizing transformationsS= (S1, . . . , Sn), whereSi : D→ Dfor i = 1, . . . ,n, is called an equivalizing procedure for reference typer (1≤ r ≤ n) if thefollowing conditions are satisfied:

(i) Si (t) is continuous onD, i = 1, . . . ,n.

(ii) Sr (t) = t for all t ∈ D.

(iii) Si (t) is strictly increasing int for i = 1, . . . ,n and allt ∈ D.

(iv) Im(Si ) = D for i = 1, . . . ,n, where Im(Si ) denotes the image ofSi .

The first property is a regularity condition. (ii) reflects the selection of a reference type. Theincome of a typer household has not to be transformed. (iii) requires that the equivalentincome (or the living standard) is strictly increasing in household income. The last condition(iv) means that the equivalizing transformations are surjective; i.e. any equivalent incomecan in principle be obtained by a household of typei . (iii) and (iv) imply that equivalizingtransformations are invertible. Here the selection ofD matters as mentioned above: IfDis bounded from below (i.e. is equal to(d,∞)) the equivalent income of all types tends tod whenever the respective household income approachesd. In this case the living standardof all types is approximately the same for household incomes close tod. Then differencesin needs are not (really) relevant. (This is e.g. the case for equivalence scales defined forstrictly positive incomes when household incomes are close to zero.) It should be stressedthat this property is an immediate implication of the above definition and the choice ofD.If D = R the unpleasant property is no longer implied: An equivalizing transformationmay possess e.g. the formSi (t) = t − ei , being based on the equivalence differenceei .Here the living standards of householdi andr are never the same (not even approximately)when the same household incomet is considered.4

We denote the set of equivalizing procedures for a reference typer by EP(r ). It iseasy to see that there is a one-to-one correspondence betweenEP(r ) and EP(s): Forevery S ∈ EP(r ) the procedureS = (S1, . . . , Sn) defined bySi := S−1

s ◦ Si for i =


1, . . . ,n is an element ofEP(s). (S−1s denotes the inverse ofSs and◦ the composition of

transformations.) Since the transformationS−1s is strictly increasing, comparisons of living

standards do not depend on the choice of the reference type (but equivalent incomes do ofcourse).

Now the question arises how to choose equivalizing procedures. Before answering it insection 3 and 4 we will define another concept which is in correspondence with the notionof an equivalizing procedure: an income expansion path. It also defines an equivalencebetween household incomes of different household types. Suppose that an amount of incomen · α is available and that this amount is to be distributed to the households considered ina way which guarantees thesameliving standard of all households. At this stage we donot know how to solve this distribution problem. But we would obtain a vector functionX(α) = (X1(α), . . . , Xn(α)) describing the result of the process, the income expansionpath. We propose the

Definition. A vector functionX: D → Dn defined byX(α) = (X1(α), . . . , Xn(α)) iscalled an income expansion path,5 if the following conditions are satisfied:

(i) Xi (α) is continuous onD, i = 1, . . . ,n.

(ii) 1n

∑ni=1 Xi (α) = α for all α ∈ D.

(iii) Xi (α) is strictly increasing inα for i = 1, . . . ,n and allα ∈ D.

(iv) Im(Xi ) = D for i = 1, . . . ,n.

The path is defined onD = (d,∞) or onR otherwise.

ObviouslyXi (α) is the amount householdi receives. It has to be a feasible income. Theaverage income(1/n)

∑Xi (α) has to be equal toα (ii). If α increases all incomesXi (α)

have to be increased as well in order to keep living standards equivalent (iii). (i) is againa regularity condition. Finally, (iv) implies that every level of income inD is attainablewhenα is chosen appropriately. LetIEP denote the set of income expansion paths. For themoment we are able to establish

LEMMA 1

a) Every equivalizing procedure S∈ EP(r ) defines a unique income expansion pathX ∈ IEP, for 1≤ r ≤ n.

b) Every income expansion path X∈ IEP defines a unique equivalizing procedure S∈EP(r ), for 1≤ r ≤ n.

Proof:a) ChooseS∈ EP(r ) and defineα(t) := 1

n

∑ni=1 S−1

i (t) for t ∈ D andXi (α) := S−1i (t)

if α = α(t).We have to prove thatX(α) satisfies the properties of an income expansion path. (i) and

(ii) are obvious. (iii) is satisfied since an increase inα is equivalent to an increase int whichimplies an increase ofXi (α) for i = 1, . . . ,n. (iv) is implied by the fact that Im(S−1

i ) = D.

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b) ChooseX ∈ IEP andr , 1≤ r ≤ n, and defineS= (S1, . . . , Sn) by

Si (ti ) := Xr(X−1

i (ti ))

for i = 1, . . . ,n.

It is obvious thatS∈ EP(r ).

In other words there is a one-to-one correspondence between the set of equivalizingprocedures and the set of income expansion paths. This relationship will be employedbelow.

3. Fundamental Concept: Individual Utility

Now we turn to a first approach to derive an equivalizing procedure. The well-being of anindividual depends on the type of household it belongs to and on its income. By assumptionthere is no distribution problem within a household: All members of a household areequally well-off.6 Therefore it seems to be reasonable to consider the utility of a typical (orrepresentative) individual for each household type.

In the following it is supposed that a social decision maker is able to assign a socialevaluation of incomeUi (t) to each typei , 1 ≤ i ≤ n. Ui (t) then denotes the utility levelof an individual belonging to a household of typei with household incomet . Of course,Ui has to satisfy some properties. It is assumed that it is an element ofU = {u: D→ R |u continuous, strictly increasing}. Continuity and monotonicity are minimal requirements.Concavity (or decreasing marginal utility of income) is not a necessary assumption, as itwill be demonstrated below. Thus the starting point of the analysis will be a utility vector

U = (U1, . . . ,Un)

belonging to the set of utility profilesUP = {U = (U1, . . . ,Un)∣∣Ui ∈ U and ImUi ) =

Im(Uj ); for i, j = 1, . . . ,n}. I.e. all functionsUi of a vectorU are continuous andstrictly increasing (monotonic) in income and possess the same image. The last propertyis indispensable for the comparison of well-being. Otherwise some levels of utility cannotbe compared with one another.

At first we have to define the measurability and comparability of utility functions. Wepostulate ordinal level comparability (cf. Roberts (1980), Blackorby and Donaldson (1991)):Two utility profiles U ∈ UP and U ∈ UP are informationally equivalent if there is acontinuous, strictly increasing function8 such that

Ui (t) = 8(Ui (t)) for all t ∈ D, i = 1, . . . ,n, (1)

where8 is independent ofi . Since8 may be an arbitrary strictly increasing function,concavity of an evaluation functionUi is not necessarily preserved. Therefore it does notmake sense to impose it.

The basic idea of the approach proposed is to derive an equivalizing procedure by consid-ering those household incomes which lead to thesame level of utilityfor a given utility profileU = (U1, . . . ,Un) ∈ UP. For any reference typer we define a vector of transformations


Si (t): D→ D by the condition

Ui (t) = Ur (Si (t)) , i = 1, . . . ,n and allt ∈ D (2)

or explicitly by

Si (t) := U−1r (Ui (t)) . (3)

The functionsSi (t) can be calculated sinceUi andUr have the same image: Thereforean incomeSi (t) exists for everyt ∈ D such that the utility levels are the same. Thenboth households are equally well-off according to the social evaluation by the decisionmaker andU−1

r (Ui (t)) is always well-defined. Furthermore, all utility profiles which areinformationally equivalent lead to the same vectorS = (S1, . . . , Sn) because of ordinallevel comparability. The vector is denoted byS= S(U, r ). Si (t) can be interpreted7 as thelevel of income a household of typer needs in order to be as well off as a household of typei with incomet , in other words, as equivalent income for reference typer .

Indeed, we are in the position to establish

PROPOSITION2 Every utility profile U defines a unique equivalizing procedure S(U, r ) foreach reference type r, or S(U, r ) ∈ EP(r ) for U ∈ UP and1≤ r ≤ n.

Its proof is almost obvious:

Proof: Continuity and monotonicity ofSi are inherited fromUi andUr .Sr (t) = U−1

r (Ur (t)) = t .Im(Ui ) = Im(U−1

r ) = D since the domain ofUr is D.

The approach allows us to derive an equivalizing procedureS from any utility profileUbelonging toUP. In this senseS(U, r ) possesses a normative basis. The correspondingequivalizing transformations can be called ethical since their characteristics are determinedby the choice of the social evaluation functions of incomeUi , i = 1, . . . ,n. This brings upthe question whether there are further equivalizing procedures which cannot be founded ona utility profile. The answer is negative:

PROPOSITION3 For every S∈ EP(r ) there exists a utility profile U∈ UP such thatS= S(U, r ).

In other words, the approach proposed is sufficiently general to provide a normative basisfor every equivalizing procedureS. Arbitrary equivalizing procedures can be supported bya utility profile. At first sight this result seems to be disquieting; but, on the contrary, it isgood news: Since the normative properties of a generating utility profile can be revealed,the implicit assumptions of every equivalizing procedure can be examined. Again the proofof this assertion is simple:

Proof of Proposition 3: ChooseS ∈ EP(r ) and defineUi (t) := Si (t) for i = 1, . . . ,nand allt ∈ D.

ThenUi is defined onD, continuous and strictly increasing int ∈ D. Moreover,

Im(Ui ) = Im(Si ) = D = Im(Sj ) = Im(Uj ) for i, j = 1, . . . ,n.

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ThusU = (U1, . . . ,Un) ∈ UP.SetS := S(U, r ), thenSi (t) = U−1

r (Ui (t)) = S−1r (Si (t)) = Si (t) sinceSr (t) = t ;

i.e.U ‘generates’S.

On the other hand, the utility profile generating a given equivalizing procedureS∈ EP(r )is not unique. But we obtain

PROPOSITION4 S∈ EP(r ) is generated by U∈ UP andU ∈ UP, if and only if U andU areinformationally equivalent, i.e.Ui (t) = 8(Ui (t)) for a continuous and strictly increasingfunction8, t ∈ D, and i = 1, . . . ,n.

Proof: Suppose thatS(U, r ) = S(U , r ) for U , U ∈ UP, 1≤ r ≤ n.By constructionU−1

r (Ui (t)) = U−1r (Ui (t)).

ThereforeUi (t) = Ur ◦U−1r (Ui (t)) for t ∈ D andi = 1, . . . ,n.

8(t) := Ur ◦U−1r (t) is continuous, strictly increasing and independent ofi .

The converse is obvious.

Proposition 4 demonstrates that ordinal level comparability is the appropriate propertyto impose on utility profiles. There is a one-to-one correspondence between equivalizingprocedures and (informationally equivalent) utility profiles.

4. Fundamental Concept: Social Welfare Function

The second approach is not based on the evaluation of the individual well-being, but on asocial evaluation of the well-being of households and on a social welfare ordering. It isassumed that a social decision maker is able to determine social evaluation functions ofincomeVi (t) for eachhousehold(type) i , 1 ≤ i ≤ n, in a first step and to define a socialwelfare function, based on this information, in a second step. Maximization of the welfarefunction will yield a corresponding equivalizing procedure.

Vi (t) denotes the welfare level8 attained by householdi having incomet , for i = 1, . . . ,n.The functionVi must be an element of the set of household welfare functionsV = {v: D→R | v once continuously differentiable, strictly increasing, strictly concave, satisfying(R)} where(R) is defined below. x Differentiability is assumed in order to simplify theanalysis: Then the first-order conditions of social welfare maximization can be exploited.Monotonicity is necessary since household welfare increases with income. Concavityimplies decreasing marginal utility of income and implies concavity of the social welfarefunction defined in a moment. A household welfare function has to satisfy a regularitycondition:

v′(t)→∞ for t → d if D = (d,∞)or t →−∞ if D = R

v′(t)→ 0 for t →∞(R)

As we shall see, this condition will guarantee that the range of living standards (equivalentincomes) is the same for different types and is equal toD.


The starting point of the analysis is a vectorV = (V1, . . . ,Vn) belonging to the set ofhousehold welfare profilesWP= {V = (V1, . . . ,Vn) | Vi ∈ V, i = 1, . . . ,n}. Again wehave to define the measurability and comparability of household welfare functions. Sincewe will introduce additive social welfare functions in a second step we postulate cardinalunit comparability (cf. Roberts (1980)): Two welfare vectorsV ∈ WP and V ∈ WPare informationally equivalent if there are continuous, strictly increasing functions8i ,i = 1, . . . ,n such that

Vi (t) = 8i (Vi (t)) for t ∈ D, i = 1, . . . ,n (4)

where8i (t) = ai + b · t for someai andb > 0 and whereb is independent ofi .Obviously the transformations8i preserve strict concavity.In choosing this framework, notice that we refrain from comparing household welfare

levelsVi : They reflect the size and neediness. Cardinal unit comparability does not allowus to compare levels, but to compare differences and the first derivatives.

Given a welfare profileV we are in a position to introduce a social welfare function

W(X) :=n∑

i=1

Vi (Xi ) (5)

for X ∈ Dn, V ∈WPand the corresponding set of social welfare functions

W = {W | W possesses form(5)}This type of welfare function has already been considered by Atkinson and Bourguignon(1987) in their investigation of the sequential generalized Lorenz ordering. Separability ofW is a restriction; but it turns out that without this condition equivalizing transformationsderived from a social welfare function would depend on the entire income distribution andnot only on the respective income.

Cardinal unit comparability is the property required in this framework. ForV, V ∈ WPwe obtain the following equivalence:[

W(X) ≥ W(Y)⇔ W(X) ≥ W(Y) for all X,Y ∈ Dn]

(6)

if V and V are informationally equivalent. HereW denotes∑

Vi . The proof is obvioussince

Vi (t) = ai + b · Vi (t) (7)

implies

W(X) =∑

ai + b ·W(X), (8)

i.e. in this caseW andW are cardinally fully comparable.In the following we will assume that a decision maker is able to specify a social welfare

functionW ∈ W, which represents a social welfare ordering. FurthermoreW reflects hervalue judgements concerning an optimal income distribution and the neediness of differenthousehold types. These norms can partially be revealed by maximizing social welfare given

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the total income available. Varying the total income we obtain an income expansion pathdescribing the optimal distributions. It can be employed to define an equivalizing procedureand the corresponding equivalizing transformations.

Therefore we now consider the problem of welfare maximization. For an arbitraryβ ∈ Dwe solve

PROBLEM β

maxX1,...,Xn

W(X) =n∑

i=1

Vi (Xi )

such that1

n

n∑i=1

Xi = β.

β is the average income. The optimal income distributionX(β,W) is characterized by thefirst-order conditions9

V ′i (Xi (β,W)) = V ′j(Xj (β,W)

)for i, j = 1, . . . ,n. (9)

Not surprisingly, the social marginal evaluation of income has to be the same for all house-holds (or types). The approach proposed here is entirely based on an exploitation of theoptimality conditions.10 Investigating the solutionX(β,W) we obtain

PROPOSITION5 For every W ∈ W the solution to problemβ defines a unique incomeexpansion path, or X(·,W) ∈ IEP for W ∈ W . If W ∈ W andW ∈ W are cardinally fullycomparable then X(β,W) = X(β, W) for all β ∈ D.

It demonstrates that the solution to the above maximization problem condenses the in-formation contained in the social welfare functionW. Informationally equivalent socialwelfare functions generate the same income expansion path; but an income expansion pathcan be yielded by social welfare functions which are not informationally equivalent (seeproposition 8 below).

Proof: SinceW is once continuously differentiable,Xi (β,W) is continuous onD, i =1, . . . ,n. By construction: 1

n

∑ni=1 Xi (β,W) = β. The functionsXi (β,W) are strictly

increasing becauseW is strictly concave. Finally, the image ofXi (β,W) is equal toD sinceV1, . . . ,Vn satisfy the regularity condition(R) implying V ′i (D) = R++ for i = 1, . . . ,n.Therefore for anyt ∈ D there always existsβ ∈ D such thatt = Xi (β,W). If W, W ∈ Ware cardinally fully comparableX(β,W) = X(β, W) is implied by (8).

Furthermore, every income expansion path is equivalent to an equivalizing procedure.Thus, applying Lemma 1, we conclude from proposition 5

PROPOSITION6 For every W∈ W the solution to problemβ determines a unique equival-izing procedure T= T(W, r ) for each reference type r,1 ≤ r ≤ n, or T(W, r ) ∈ EP(r ),where T is defined by

Ti (t) = V ′r−1 (V ′i (t)) for all t ∈ D and i = 1, . . . ,n.


The definition ofT is based on an appropriate interpretation of the income expansionpathX(β,W) and the optimality conditions:X(β,W) always represents levels of incomewhich lead to the same standard of living of the households involved, or, for fixedβ, theincome vectorX(β,W) implies that all households are equally well-off according to thesocial welfare functionW. If this would not be the case, the income distributionX(β,W)

would not be optimal. Therefore the income vector defines the corresponding equivalizingprocedure.

Proof of Proposition 6: Lemma 1 yieldsTi (t) = Xr (β,W) if t = Xi (β,W).On the other handV ′i (Xi (β,W)) = V ′r (Xr (β,W))or Xr (β,W) = V ′r

−1◦V ′i (Xi (β,W)).This impliesTi (t) = V ′r

−1 ◦ V ′i (t)

Summing up, the second normative approach of deriving equivalizing procedures is basedon the following idea: If a social decision maker is able to specify a social welfare functionreflecting her value judgements, this welfare function also embodies information aboutoptimal income distributions. They describe a distribution of a fixed total income amongdifferent households which cannot be improved socially. As long as the total income is notchanged, no household can be made better-off without worsening another one. From a social(welfare) pont of view the situations of all households are equal: The corresponding incomescan be interpreted as leading to the same standard of living. Therefore an equivalizingprocedure is implicitly defined.

Next we discuss the same questions as above (cf. proposition 3–4). The second approachis also able to generate all equivalizing procedures. We have

PROPOSITION7 For every S∈ EP(r ) there exists a social welfare function W∈ W suchthat S= T(W, r ).

Again one must not be worried about this result. It means that every equivalizing pro-cedure can be justified and therefore characterized in this way. Thus the underlying valuejudgements can be revealed.

Proof: Suppose thatS = (S1, . . . , Sn) ∈ EP(r ) is given. Choose any differentiablefunction11 g: D→ R such thatg(t) > 0 andg′(t) < 0.

g can be used to define a household welfare functionVi by

Vi (t) :=∫ t

dg(Si (t)

)dt for i = 1, . . . ,n.

It is once continuously differentiable, strictly increasing, and strictly concave:

V ′i (t) = g (Si (t)) > 0

andV ′i (t) is strictly decreasing int . Furthermore the regularity condition(R) is satisfied.ThereforeVi ∈ V for i = 1, . . . ,n andW(X) :=∑n

i=1 Vi (Xi ) ∈ W.The corresponding solution to problemβ is implicitly given by

V ′i (Xi (β,W)) = V ′r (Xr (β,W))

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i.e. g (Si (Xi (β,W))) = g (Sr (Xr (β,W))) = g (Xr (β,W)) sinceSr (t) = t .Monotonicity ofg impliesSi (Xi (β,W)) = Xr (β,W)

or S= T(W, r ) by definition.

The proof does not rest on the choice of the functiong(t). It determines the form of thehousehold welfare functions and of the social welfare functions. Its role is to equivalizethe marginal social welfare levels in such a way that the given equivalizing procedureS isimplied. g(t) can be chosen arbitrarily as long as it is positive and strictly decreasing andfulfills the regularity condition. Therefore every equivalizing procedureS∈ EP(r ) can befounded on several social welfare functions. The next result characterizes the respectiveset of functions.

PROPOSITION8 An equivalizing procedure S∈ EP(r ) is generated by W= ∑Vi ∈ W

and W = ∑Vi ∈ W , i.e. S= T(W, r ) = T(W, r ) if and only if there is a continuous,

strictly increasing function h: R→ R such that

V ′i (t) = h(V ′i (t)

)for all t ∈ D and i = 1, . . . ,n.

Two social welfare functions yield the same equivalizing procedureS only if there is afunctional relationship between the respectivemarginalsocial welfare functions which dueto additivity coincide with the marginal household welfare functions. The transformationh(t) must be the same for all types. One marginal social welfare function has to be atransform of the other one. This result is surprising at first sight: One might have expectedthat the welfare functionsW andW had to be ordinally equivalent. But this conjectureis wrong. Themarginalsocial welfare functions are essential! They determine a uniqueincome expansion path which in turn defines the equivalizing procedure. The path dependson the first-order conditions (9). They are not violated if all marginal household welfarefunctions are transformed by the sameh(t). The same is true for the marginal rate ofsubstitution of householdi ’s income for householdj ’s income for anoptimal incomedistribution

M RSi j = V ′i (Xi (β,W))/

V ′j(Xj (β,W)

). (10)

It equals one. This property is retained if the marginal household welfare levels are trans-formed byh.

Therefore proposition 8 can be restated in another way: An equivalizing procedureS ∈EP(r ) is generated byW andW if and only if the social indifference curves ofW andWare parallel atX(β,W) = X(β, W) or on the income expansion path corresponding toS.

Proof of Proposition 8: Suppose thatS= T(W, r ) = T(W, r ). Then by definition

Si (t) = V ′r−1 (V ′i (t)) = V

′r

−1 (Vi (t)

)for t ∈ D

and thereforeV ′i (t) = V ′r ◦ V ′r−1 (V ′i (t)) for t ∈ D, i = 1, . . . ,n.

Thus the transformationh(t) is given byV ′r ◦ V ′r−1(t) which is continuous and strictly

increasing int ∈ D.


Conversely, the conditionV ′r (t) = h(V ′i (t)

)yieldsV

′i

−1 = V ′i−1◦h−1(t) for i = 1, . . . ,n.

Therefore

Si (t) = Ti (W, r ) = V′r

−1 ◦ V ′i (t) =(

V ′r−1 ◦ h−1

)◦ (h ◦ V ′i (t)

)= V ′r

−1 (V ′i (t)) = Ti (W, r ) = Si (t),

i.e. the same equivalizing procedure is generated.

5. Ranking Needs

Up to now no needs ranking has been assumed. The results in section 3 and 4 demonstratehow far one can go without ranking. (A referee pointed out that in principle one may not beable to achieve a complete ranking.) In this section it will be supposed that the household(types) can be ranked and are numbered bydecreasing needs. The way of numberingfacilitates the following discussion.

We would expect that the living standard reached by a household decreases with needswhenever the household income is kept fixed. Since households are numbered by decreasingneeds equivalizing transformations then should satisfy

Si (t) ≤ Si+1(t) for t ∈ D, i = 1, . . . ,n− 1. (11)

These conditions are not satisfied automatically. The requirements for corresponding utilityprofiles are described by

PROPOSITION9 Assume that S= S(U, r ).Then Si (t) ≤ Si+1(t)⇔ Ui (t) ≤ Ui+1(t) for all t ∈ D, i = 1, . . . ,n− 1.

Considering the definition ofS(U, r ) the proof of proposition 9 is immediate and theresult is not surprising. The proceeding in the proof of proposition 3 suggests an alterna-tive interpretation of an equivalizing procedure. It can be interpreted as a specific utilityprofile, namely as one which measures utility in income (for the reference type), or put inanother way: IfS is generated by a utility profileU then the ‘utility profile’ S andU areinformationally equivalent.

Next we want to examine the relationship betweenSi (t) andSi+1(t) for i = 1, . . . ,n−1when a social welfare function is the fundamental concept. Here it proves to be useful toconsider the redistribution of income between households. We introduce (cf. Ebert (1997a,2000)).

Definition. A transfer of incomeδ > 0, changingXi to Xi + δ and Xi+1 to Xi+1 − δfor any i , 1 ≤ i ≤ n − 1, is called a progressive transfer between households as long asXi + δ < Xi+1− δ.

It defines a transfer from a less needy household to a needier one. In our frameworkhouseholds of different types have different needs. According to the above assumptionsit is not possible to quantify them; but households can be and are ranked by decreasing

632 EBERT

neediness. A progressive transfer is restricted to the case that the recipient’s income (afterthe transfer) is still lower than the corresponding income of the other (less needy) house-hold.

We are able to establish

PROPOSITION10 Assume that S= T(W, r ).Then Si (t) ≤ Si+1(t) for all t ∈ D and i = 1, . . . ,n− 1 if and only if every progressive

transfer between households increases the level of social welfare W.

There is a relationship between the way social welfare is changed by a progressive transferand the living standard of different household types—measured by equivalent income.Equivalent income is decreasing with needs (if the level of income is kept fixed) if and onlyif the social decision maker is (weakly) inequality averse: Progressive transfers (as definedabove) increase the level of social welfare.

Proof of Proposition 10: In the following we can use a result which is contained in theproof of the Theorem in Ebert (1997a). Every progressive transfer increasesW(X) =∑

Vi (Xi ) ∈ W if and only if

V ′i (t) ≥ V ′i+1(t) for all t ∈ D andi = 1, . . . ,n− 1.

The last condition is equivalent toV ′r−1 (V ′i (t)) ≤ V ′r

−1 (V ′i+1(t))

sinceV ′r is decreasing,or to Si (t) ≤ Si+1(t) for t ∈ D, i = 1, . . . ,n− 1 whereS= T(W, r ).

A comparison of proposition 2–4,9 with proposition 6–8,10 demonstrates the similaritiesand differences between both normative approaches. The next section makes an attempt atcombining them.

6. Combining Both Approaches

Both approaches are based on a social evaluation of income. The first approach employs autility profileU = (U1, . . . ,Un)whereUi (t)denotes the utility level amemberof householdi attains when the household income is equal tot . The second one uses a household welfareprofile V = (V1, . . . ,Vn) whereVi (t) evaluates the level of total welfarehousehold ireaches. In the following we investigate the case in which both concepts are related. Thismakes sense only if the measurability and comparability of the evaluation functions areappropriate and if further regularity conditions are satisfied.

Suppose that a decision maker is able to choose a utility profile

U = (U1, . . . ,Un)

whereUi ∈ V , i.e. Ui is once continuously differentiable, strictly increasing and strictlyconcave and satisfies the condition(R). Furthermore, assume that the decision makeris in the position to measure and compare the evaluation functionsUi cardinally. Thentwo profilesU andU are informationally equivalent if there is a positive affine function


8(t) = a+ b · t , b > 0 such that

Ui (t) = a+ b ·Ui (t) for t ∈ D andi = 1, . . . ,n. (12)

In other words we postulate cardinal full comparability (cf. Roberts (1980) again).In order to combine both approaches we now assume that household welfare can be

described by

Vi (t) = ci ·Ui (t), ci > 0, (13)

The multiplicative form of the right hand side of (13) is justified by the following interpreta-tion: total household welfare is decomposed into the product of an indicator for the numberof members and of the utility level attained by a representative member. Here the weight(factor of proportionality)ci reflects the composition of householdi . ci could be equal e.g.to the number of persons, the number of equivalent adults, or to any other constant whichtakes into account the size and/or needs of householdi .

Given the weightsc1, . . . , cn and a profileU we obtain a welfare profile

V = (V1, . . . ,Vn) ∈WP.

Moreover, ifUi (t) = a+ b ·Ui (t) (U andU are equivalent) we get

Vi (t)=ci ·Ui (t)=ci (a+b·Ui (t)) =ci ·a+b·ci ·Ui (t)=ci a+b·Vi (t); (14)

i.e. V andV are equivalent according to cardinal unit comparability. ThereforeV definesa social welfare functionW(X) =∑Vi (Xi ) ∈ W.

In this case we are able to derive two equivalizing procedures:S(U, r ) andT(W, r ). Thequestion arises whether they differ or whether they are identical. It is answered by

PROPOSITION11 Assume that U∈UP, V∈WP, Vi (t) =ci·Ui (t), that W(X)=∑Vi (Xi )=∑ci ·Ui (Xi ) and assume cardinal full comparability of U.

Then the following statements are equivalent

a) S= S(U, r ) = T(W, r )

b) Ui (t) = Ur

(crci

t +(1− cr

ci

)d)

for i = 1, . . . ,n if D = (d,∞)or there exist b1, . . . ,bn ∈ R,br = 0 such that Ui (t) = Ur

(crci

t + bi

)for i = 1, . . . ,n

if D = R

c) W(X) =∑ni=1 ci ·Ur

(crci

t +(1− cr

ci

)d)

if D = (d,∞)or there exist b1, . . . ,bn ∈ R,br = 0 such that W(X) = ∑n

i=1 ci · Ur

(crci

t + bi

)if

D = R

d) The equivalizing transformations are given by

Si (t) = cr

cit +

(1− cr

ci

)d for i = 1, . . . ,n if D = (d,∞)

634 EBERT

or there exist b1, . . . ,bn ∈ R,br = 0 such that Si (t) = crci

t + bi for i = 1, . . . ,n ifD = R.

Both approaches generate the same equivalizing procedure in a specific case: The utilityfunctionsUi are interdependent and the way they are related to one another depends on theweights ci (cf. b). In other words, the implications of both approaches are the same only ifthe utility profileU also depends on the weights introduced to evaluate household welfare.Furthermore the domain of feasible incomesD is relevant. It has to be equal to an intervalhaving no upper bound. In every case we obtain linear equivalizing transformations, i.e. acombination of equivalence scales and equivalence differences.

Therefore proposition 11 demonstrates that both approaches are really independent andgenerally yield different procedures. This might not be surprising since their focus is differ-ent and since the assumptions concerning measurability and comparability differ as well.

Proof of Proposition 11: Suppose thatS= S(U, r ) = T(W, r )andW(X) =∑ ci Ui (Xi ).ThenSi (t) = U−1

r (Ui (t)) = V ′r−1 (V ′i (t)).

By definitionV ′i (t) = ci ·U ′i (t).ThereforeV ′r

−1 (V ′r (t)) = t = V ′r−1 (cr ·U ′r (t)

).

Replacet by U ′r−1(t/cr ) and getU ′r

−1(t/cr ) = V ′r

−1(cr ·U ′r (U ′r−1

(t/cr ))) = V ′r−1(t).

We obtainSi (t) = V ′r−1 (V ′i (t)) = U ′r

−1(

ci

crU ′i (t)

)(∗)

SinceUi andUr are differentiable we get

S′i (t) =(U−1

r (Ui (t)))′ = U−1

r′(Ui (t)) ·U ′i (t) =

U ′i (t)U ′r(U−1

r (Ui (t))) = U ′i (t)

U ′r (Si (t)).

Insertion of(∗) yields

S′i (t) =U ′i (t)

U ′r(U ′r−1(

cicr

U ′i (t))) = cr

ci.

ThusSi (t) has to be linear: there existb1, . . . ,bn ∈ R such thatSi (t) = crci

t + bi and ofcoursebr = 0 sinceSr (t) = t .

If D = (d,∞), Si (t)→ d for t → d.

This impliescrci

d + bi = d or bi =(

1− cr

ci

)d.

Moreover, by definitionUi (t) = Ur (Si (t)) = Ur

(cr

cit + bi

)Therefore a) yields b), c), and d).The converse and the caseD = R are obvious.

It is instructive to relate the recipes of this proposition to existing equivalizing procedures:in practice equivalence scales are mostly used. Proposition 11 demonstrates that they can


simultaneously be justified by both approaches. Moreover a corresponding social welfarefunction has a particular form. Assuming that the reference household type is a single adult,i.e.cr = 1, we can interpret this welfare function as one for a fictive population consistingof equivalent adults (cf. Ebert (1997a, 1999)). Consider any householdi with incomeXi .Its living standard (measured by equivalent income) is given by(cr /ci ) · Xi . Therefore itsrepresentative equivalent adult contributesUr ((cr /ci )Xi ) to social welfare. Householdicorresponds toci adults. Thus one has to attach the weightci to Ur ((cr /ci )Xi ). Summingover all households we obtain total social welfare. Similarly equivalence differences (i.e.equivalizing transformationsSi (t) = t + bi ) are based on welfare functions which are‘utilitarian’” W(X) = ∑

Ur (t + bi ) (sinceci = cr for i = 1, . . . ,n w.l.o.g.). In thiscase all weights are equal. The equivalence difference acts like a deduction (or transfer) ofincome in the utility of income functionUr . Otherwise all households are treated equally. Inthe general case, if the equivalizing transformations employ a combination of equivalencescales and differences the underlying social welfare functions employ the scales as weightsand use deductions (or transfers).

The findings of proposition 11 will be illustrated by some examples in the next section.

7. Applications

This section presents a number of examples demonstrating the application of the methodol-ogy proposed. Several equivalizing procedures are derived from a utility profile and a socialwelfare function, respectively. Moreover, the role of equivalizing procedures is illuminatedby a discussion of income tax systems when the principle of horizontal equity is imposed.

7.1. Examples12

It is almost obvious how to generate equivalence scales and equivalence differences bymeans of the first approach. Given some constantsmi ∈ R++ andei ∈ R (i = 1, . . . ,n)and a utility of income functiong ∈ U we can define a utility function for a representativemember of householdi in the one case byUi (t) := g(t/mi ) for t ∈ R++ and in theother byUi (t) := g(t − ei ) for t ∈ R (i = 1, . . . ,n). The corresponding equivalizingtransformations are given bySi (t) = mr

mit andSi (t) = t − (ei − er ) for i = 1, . . . ,n; i.e.

the utility profileU = (U1, . . . ,Un) implies an equivalizing procedureS(U, r ) based oneither equivalence scales or differences, respectively.

These equivalizing transformations or—more generally—linear transformations can alsobe derived by using the second approach. Introducing a generalization of Atkinson-typesocial welfare functions (cf. Bossert and Pfingsten (1990), Ebert (1995))

Wε,s(X) =

n∑i=1

ai (Xi + s)ε for ε < 1, ε 6= 0

n∑i=1

ai ln(Xi + s) for ε = 0

(15)

636 EBERT

whereai ∈ R++,∑

ai = 1, s ∈ R, and X ∈ Dn = (−s,∞)n we obtainsε,si (t) =mrmi

t − bi for i = 1, . . . ,n (Sε,s = T(Wε,s, r )). Of course there must be a relationship

between the coefficientsai and the equivalizing transformations:mi = (1/ai )1/(ε−1) and

bi = (1−mr /mi ) · s.The coefficientsai reflect the needs of householdi . For s = 0, the welfare functions

are homothetic and are identical to (an ordinal transform of) the usual Atkinson welfarefunctions with the exception of symmetry. Then only equivalence scales are employed. Fors 6= 0 the welfare functions also possess a homogeneity property. In this casemi andbi playthe role of an equivalence scale and equivalence difference. It is easy to see that differentsocial welfare functions can imply the same equivalizing procedure (cf. proposition 8). Theentire family of welfare functions is also relevant for the derivation of inequality measures(see Ebert (1995, 1996)).

In this example the equivalizing procedures can be generated by both approaches: define

U ε,si (t) := 1

ε

(t+smi

)and V ε,s

i (t) := mi Uε,si (t). Then S(U ε,s, r ) = T(Wε,s, r ) for ε <

1, ε 6= 0 (cf. proposition 11). An analogous result holds forε = 0.Similarly a generalization of Kolm-Pollak social welfare functions (the cases = −∞)

can be introduced

Wγ (X) = − 1

γ

n∑i=1

ai e−γ Xi (16)

for γ > 0, ai ∈ R++ and X ∈ Dn = Rn. They are translatable (i.e. there is an ordinaltransformWγ such thatWγ (X + α1) = Wγ (X) + α for α ∈ R and1 = (1, . . . ,1))and generate absolute inequality measures (cf. Ebert (1997b)). The corresponding equiv-alizing transformations employ equivalence differencesSγi (t) = t − (ei − er ) whereei = (1/γ ) ln ai . They are also linear and can be simultaneously generated by both ap-proaches ifVγ

i (t) := U γ

i (t) := − aiγ

e−γ t is used.

7.2. Income-Level-Dependent Scales

Up to now the equivalent scales and differences considered have been income-independent.Employing the second approach and an appropriate social welfare function we are able topresent an example in which a more complicated form of equivalizing transformations isfounded normatively.

Consider the family

Wδ(X) =n∑

i=1

ai ln(1− e−δXi ) (17)

for δ > 0, a1 > · · · > an > 0, X ∈ Dn = Rn++ and assume thatr = n. These welfare

functions belong to the setW and generate the equivalizing transformations

Si (t) = 1

δln

(1+ an

ai(eδt − 1)

), for i = 1, . . . ,n. (18)


They are nonlinear fori < n. The properties ofSi (t) can be investigated in terms of itsderivative. We obtain

S′i (t) = 1

/(1−

(1− ai

an

)e−δt

)(19)

and

S′i (t)→an

aifor t → 0

and S′i (t)→ 1 for t →∞. (20)

The derivative has, of course, to be positive. The equivalent incomeSi (t) increases int andthe (marginal) increase is the greater the highert .

For an interpretation we assumen = 2 and that household 2 consists of a single adult andthe type 1 household is a couple (comprises two adults). We seta2 = 1 anda1 = 2.

If the couple’s incomeX1 is low the increase of the equivalent income is approximatelyequal to the ratioa2/a1 (in our case 1/2); i.e. the equivalent income ofX1 equals the fractiona2/a1 of X1. This reflects the respective number of persons. The impact of the equivalizingtransformationsS1 can be compared to the effect of an equivalence scale. For very highincomes the equivalent income increases by the same amount asX1. Then X1 and itsequivalent incomeS1(X1) grow equally, but certainly they will be different:X1 − S1(X1)

corresponds approximately to an equivalence difference. Thus the equivalizing proceduremight be interpreted as a (variable) combination of equivalence scales and differences: thenumber of persons belonging to a household is relevant for low incomes (equivalent incomeis approximately determined by equivalence scales); but it is no longer important for veryhigh incomes. That seems to be an attractive property of the equivalizing procedure.

7.3. Application to Taxation

A typical field of application for equivalizing proceduresS is the design of tax systems forheterogeneous populations. Equivalizing transformations allow us to compare gross andnet incomes, and the respective standard of living of different household types. Then it ispossible to check whether a tax system satisfies the principle of horizontal equity or not.The latter can be defined by

HE Horizontal Equity

A tax system t1(X1), . . . , tn(Xn) is equitable if Xr = Si (Xi ) implies Xr − tr (Xr ) = Si (Xi −ti (Xi )) for i = 1, . . . ,n.

The implications of HE for a tax systemt1(X1), . . . , tn(Xn) can easily be described: NetincomesXi − ti (Xi ) and Xj − tj (Xj ) are equivalent (imply the same standard of living)whenever the corresponding gross incomesXi andXj are equivalent, for all(X1, . . . , Xn) ∈Dn andi, j = 1, . . . ,n.

638 EBERT

Using the fact thatXr = Si (Xi ) and rearranging terms we obtain

Xi − ti (Xi ) = S−1i (Si (Xi )− tr (Si (Xi )))

or ti (Xi ) = Xi − S−1i (Si (Xi )− tr (Si (Xi ))) for i = 1, . . . ,n.

(21)

The net income of householdi can be calculated by computing the gross equivalent incomeSi (Xi ), the corresponding net income of a typer household having that gross income, andby transforming that equivalent income back to the corresponding net income of householdtype i . Similarly, the tax scheduleti for 1 ≤ i ≤ n is already completely determined bythe scheduletr and the equivalizing transformationSi (Xi ). This relationship betweentiandtr is always implied by horizontal equity. Conversely, a tax system satisfying (21) fori = 1, . . . ,n is equitable, i.e. fulfills the principle of horizontal equity.

If one of the linear equivalizing procedures derived in section 7.1 is employed an equitabletax system possesses a particularly simple form (cf. Ebert and Moyes (2000)):

Suppose thatmr = 1 andSsi (Xi ) = Xi /mi − bi , wherebi = (mi − 1)s/mi then we

obtain

ti (Xi ) = mi tr

(Xi

mi− bi

). (22)

For s = 0 (homotheticity of the social welfare function) the equivalizing procedure andhorizontal equity imply ‘family splitting’ using the equivalence scalemi (herebi = 0).Income has to be equivalized by means of equivalence scales, the tax schedule of thereference type has to be applied and, afterwards, the tax liability has to be multiplied by theequivalence scale again. For a translatable social welfare function (s = −∞) the result iscompletely different:S−∞i (Xi ) = Xi − ei if er = 0. We get

ti (Xi ) = tr (Xi − ei ). (23)

Then income has to be equivalized by subtracting a type dependent equivalence differenceei and the common tax scheduletr has to be employed.ei corresponds to an allowancewhich depends on needs. Ifs 6= 0 ands 6= −∞ a combination of both methods has to beused. Whenever the equivalizing procedure is nonlinear, tax liability is also determined byequation (21), but its calculation is more complex.

The question arises whether there are equivalizing procedures rationalizing tax creditsfor an equitable tax system.13 Unfortunately the answer is negative: Lambert and Yitzhaki(1997) prove that tax systems with family size-related credits cannot be considered equitable,given their property of horizontal equity (their equation(3)). But this property is identicalwith equation (21) and therefore with our principle HE since an equivalizing procedureS= (S1, . . . , Sn) can be interpreted as a utility profileU = (U1, . . . ,Un) which forms thebasis of Lambert and Yitzhaki’s investigation.

8. Conclusion

The paper presents a systematic analysis of equivalizing procedures and of two normativemethods of founding them. Furthermore the requirements concerning the measurability


and comparability of individual and household utility-or-income functions are revealed.Different utility profiles and welfare orderings, respectively, can generate the same equiv-alizing procedure, i.e. a procedure can be compatible with different norms. The propertiesof a utility profile or an underlying welfare ordering can, at least in principle, be recoveredby means of an axiomatic characterization. Therefore the normative approaches are able toexamine the value judgements yielding an equivalizing procedure. The methods can deter-mine the implicit normative assumptions involved when using equivalizing transformationsor equivalence scales whether they be subjective, pragmatic, or derived from demand anal-ysis. This result is all the more important as every equivalizing procedure can be justifiedwithin this framework.

On the other hand the methods proposed can be employed to generate some new equival-izing transformations (not used up to now). Looking for them is attractive since equivalencescales are often criticized because of their simplicity: They are independent of the level ofincome. This property does not necessarily have to be satisfied in the framework discussed.An example shows us a new possibility: equivalence scales which depend on income levels.It is, of course, obvious that equivalence scales from demand theory could be income-level-dependent; but our example presents anormativefoundation for such scales.

The above analysis is based on the assumption that there is exactly one household of eachtype. If this restriction is dropped nothing changes for the first approach. In the secondone we only have to impose partial symmetry on the welfare ordering. It requires thathouseholds of the same type have to be treated equally. They then possess the same utility-of-income function. We obtain the same first-order conditions as above and further onesrequiring that the incomes of households of the same type are equal. Therefore the sameequivalizing procedure is generated, though the corresponding income expansion path hasto be defined (slightly) differently, taking into account the dimensionality of the problem.

Notes

1. I am grateful to Patrick Moyes, Oskar von dem Hagen, three anonymous referees, and an Editor of this journalfor most helpful suggestions and comments. The paper forms part of the research programme of the TMRnetworkLiving Standards, Inequality and Taxation [Contract No. ERBFMRXCT980248] of the EuropeanCommunities whose financial support is gratefully acknowledged.

2. The population is kept fixed. Household formation is not investigated in this paper. Sometimes there areeconomies of scale in living costs and other motivations as discussed by Becker (1991).

3. Further properties could be imposed. Cf. the discussion below.4. Clearly living standards can be negative in this case, and then it may require one or other income level to be

negative for comparability.5. The income expansion path is defined in income space and not—as usually—in commodity space.6. This assumption can be satisfied whether household members are identical or different, e.g. if some members

have certain non-discretionary expenditures (e.g. due to disability).7. Si (t) is an abbreviation ofSi (t;U, r ).8. Again the members of a household may have different needs which are reflected byVi (t).9. Assumption (5) is crucial for the form of the first-order conditions.10. Kaplow (1996) employed this approach to define tax rules for singles and couples.11. We have to distinguish between case (i) ifD = (d,∞) and case (2) ifD = R. In the first case we suggest

g(t) := 1/(t − d) and in case (ii)g(t) := e−t .Both functions are positive and strictly decreasing int on D.

640 EBERT

12. The details of the examples in section 7.1 and 7.2 have been delegated to an appendix which can be obtainedfrom the author on request.

13. This point was raised by a referee and the editor.

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Equivalizing Incomes: A Normative Approach