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Journal of Health Economics 21 (2002) 497–513

Defining health inequality: why Rawls succeeds

where social welfare theory fails

Antoine Bommiera, Guy Stecklovb,∗

aInstitut National d’etudes Démographiques (INED), INRA-Jourdan, Paris, France

bDepartment of Population Studies, Department of Sociology and Anthropology, Mount Scopus Campus,

Hebrew University of Jerusalem, 91905 Jerusalem, Israel

Received 12 May 2000; received in revised form 7 November 2001; accepted 7 December 2001

Abstract

While there has been an important increase in methodological and empirical studies on health

inequality,notmuchhasbeenwrittenonthetheoreticalfoundationofhealthinequalitymeasurement

We discuss several reasons why the classic welfare approach, which is the foundation of income

inequality analysis, fails to provide a satisfactory foundation for health inequality analysis. We

propose an alternative approach which is more closely linked to the WHO concept of equity in

health and is also consistent with the ethical principles espoused by Rawls [A Theory of Justice.

Harvard University Press, Cambridge, MA, 1971]. This approach in its simplest form, is shown to

be closely related to the concentration curve when health and income are positively related. Thus,

thecriteriapresentedinourpaperprovideanimportanttheoreticalfoundationforempiricalanalysis

using the concentration curve. We explore the properties of these approaches by developing policy

scenarios and examining how various ethical criteria affect government strategies for targeting

health interventions. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Health inequality; Social welfare; Concentration curve

1. Introduction

Increasing interest in health inequality has intensified efforts to provide the tools to mea-

sure and analyze the distribution of health. Most of this research is empirically related

and is aimed at measurement and evaluation of changes in health inequality. In particu-

lar, economists have become increasingly involved in research on health inequality and—

borrowingheavilyfromrelatedresearchonincomeinequality—haveproducedanumberof

∗Corresponding author. Tel.: +972-2-588-3320; fax: +972-2-532-4339.

E-mail address: stecklov@mscc.huji.ac.il (G. Stecklov).

0167-6296/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0167-6296(01)00138-2

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A. Bommier, G. Stecklov/Journal of Health Economics 21 (2002) 497–513

important methodological and substantive contributions to this research (recent review by

WagstaffandvanDoorslaer,2000).Thesecontributionsincludestatisticaltestsofhealthin-

equalityindicators(Kakwanietal.,1997)andempiricalanalysesofhealthinequalitytrends

in developed and developing countries (van Doorslaer et al., 1997; Wagstaff et al., 1991).

Despite these important empirical and methodological advances in the field of health

inequality,thereisnotgeneralagreementonthedefinitionandmeaningofhealthinequality.

Wagstaff and van Doorslaer (2000) discuss two contrasting approaches: pure inequalities

in health and socioeconomic inequalities in health. The pure inequalities in health approach

focuses entirely on the distribution of the health variable itself within the population (see

Murray et al., 1999, 2000; Le Grand and Rabin, 1986). The socioeconomic inequalities in

health approach focuses on the distribution of health across social and economic groups

(Wagstaff et al., 1991; Braverman et al., 2000). The former approach leads to usage of

standard income inequality measures such as the Lorenz curve and Gini coefficients but

where the unit of measure is in terms of health rather than income. The theoretical bases for

understandingmeasuresofpurehealthinequalitybasicallymirrortheapproachesdeveloped

foranalysisofincomeinequality.Widespreadinterestinthesecondapproach,analysisofthe

distributionofhealthacrosssocialandeconomicgroups,hasledtothedevelopmentofother

toolssuchastheconcentrationcurveandconcentrationindex.Yet,therehasbeenverylittle

research on the theoretical foundations of these health inequality measures and we believe

this partly explains the divergence in opinion on the measurement of health inequality.

In this article, we explore the different approaches for defining and measuring both pure

and socioeconomic health inequalities. We begin by recalling how the social welfare def-

inition of inequality, which is widely used to study income inequality, can be extended

to define a concept of multi-dimensional inequality. We show how this notion of multidi-

mensional inequality may be used to define health inequality. However, our discussion will

show why this approach, which is a natural extension of the theory of income inequality,

fails to provide an acceptable definition of health inequality. This leads us to consider a

new definition of socioeconomic health inequality that captures both the distribution of

health and the association between health and income and which is consistent with Rawls’

definition of equity. We show that this approach provides a theoretical basis for the use of

the concentration curve as well as a direction for continued theoretical research towards

defining health inequality. Finally, we develop a series of policy scenarios to highlight the

differences between the various ethical approaches to health inequality.

2. From single to multi-dimensional inequality

The theoretical foundations of inequality analysis have primarily focused on developing

criteria and empirical tools which allow us to consistently rank distributions with respect

to their level of inequality. For the most part, this line of research has proven highly suc-

cessful and has generated wide acceptance of empirical tools such as the Lorenz curve and

Gini coefficient. At the same time, the emphasis has been almost entirely on inequality

in a single dimensional factor. As economists have become increasingly concerned with

evaluating inequality in more than one dimension—for example, income and health—they

have naturally sought to extend the classical theory.

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499

In fact, the social welfare (SW) approach for income inequality analysis turns out to

be quite adaptable to multi-dimensional inequality analysis. Basically, the SW approach

developed in Dalton’s seminal article of 1920 suggests that one society can be said to be

less unequal than another if, for the same mean income level, it provides a larger social

welfare level. Obviously, a precise notion of welfare is central to this definition. Since

any definition of SW can be disputed, ranking in terms of inequality is only considered

conclusive if the SW criterion gives the same ranking for a wide class of SW functions.

Dalton focused on additive SW functions:

W =

1

N

N

?

i=1

U(yi)

(1)

where the social welfare function is the average utility level in society and where the utility

of person i is an increasing and concave function of income, yi. It has been shown that

this additive SW function can be extended to the more general case where W is symmetric,

continuous, monotonic and quasi-concave (Dasgupta et al., 1973; Rotshchild and Stiglitz,

1973).

Typically, inequality is measured with respect to a single dimension such as income.

However, when another dimension is introduced, such as health, education, or another

outcome, Eq. (1) can be easily modified. Instead of having only one argument in the utility

function, such as income, one might include two arguments, xi and yi, so that the SW

function becomes

W =

1

N

N

?

i=1

U(xi,yi)

(2)

This is the approach developed by Atkinson and Bourguignon (1982) and which they

illustrate with an analysis of inequality in income and life expectancy across countries.

It is worth noting that in addition to generalizing from one to two dimensions, Atkinson

and Bourguignon’s approach also differs from the pure utilitarian approach of Dalton in

that they assume that the function U can be derived from the preferences of a social planner

or decision maker and not necessarily from individual preferences. As in the single dimen-

sional case, the shape of the U function is important, and economists look for inequality

rankings that are valid for the widest possible set of SW functions. Obviously, restrictions

on the utility function make it easier to conclude whether one distribution is more or less

unequal than another distribution. However, imposing restrictions also reduces the gener-

ality of the results. The most common assumptions are that the U function is increasing

and concave in both arguments. What is left is deciding on the association between the two

variables.AtkinsonandBourguignonproposevariousassumptionsonthecrossderivatives.

We present four cases:

• Case A: No assumption is made on the cross derivatives.

• Case B: The cross derivatives are assumed to equal zero, U??

• Case C: The two factors are assumed to be substitutes, U??

• Case D: The two factors are assumed to be complements, U??

xy= 0.

xy< 0.

xy> 0.

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Determining the appropriate restrictions will depend on the variables that are to be an-

alyzed. In the following section, we focus on the analysis of inequality in income and

health, and we examine the appropriateness of each of these cases for the definition of

health inequality. We will ignore measurement issues for now, particularly with respect to

the health variable, and simply assume that the income (y) and health (h) variables are both

continuously measurable. We consider the problem of measurement of the health variable

in Section 4.

3. A social welfare approach to health inequality

Atkinson and Bourguignon’s approach allows us to define a bi-dimensional notion of

inequality that depends on the distribution of both health and income. However, our interest

is not in a fully two-dimensional measure of inequality, but rather in providing a definition

andmeasureofhealthinequalitythatisresponsivetothedegreeoffairnessinthedistribution

of health but is relatively insensitive to the degree of inequality in the existing income

distribution. One solution involves using the bi-dimensional SW definition of inequality

to compare societies with identical income distributions. Where income distributions are

the same, differences in bi-dimensional inequality reflect differences in the distribution of

health or in its association with income. In practice, societies have very different income

distributions.However,incomevariationscanberescaledinawaysuchthatthenewincome

distribution fits some common reference income distribution and bi-dimensional inequality

measures can then be used to define health inequality.1Health inequality in country A

will be said to be greater than in country B with the same income distribution, if the

bi-dimensional health inequality as defined by Atkinson and Bourguignon is greater in A

than in B.

Healthinequalitymeasuresbasedonsuchadefinitionencompasstwodistinctcomponents

of health inequality. The first is simply a function of the distribution of health itself within

the population and is taken into account by the second derivative of U with respect to health

(U??

on SW through the cross derivative U??

different ways by specifying additional assumptions on U??

Bourguignon.

Of the four cases presented above, Case A is obviously the most appealing and least

restrictive since no assumption is made on the cross derivative of the utility function. At the

same time, due to its greater generality, we are unable to rank most societies under Case A.

ClassBmakesthestrongestassumptionabouttherelationshipbetweenincomeandhealth

since it assumes that the change in utility from income is independent of health and vice

versa. The independence between health and income means that the general utility function

can be rewritten as, U(y,h) = U(h) + U(y). The independence assumption leads to a

situation where bi-dimensional measures of health inequality correspond to the definition

hh). The second is due to the association between income and health and has an impact

hy. The second component may be accounted for in

hy, as proposed by Atkinson and

1Rescaling in this case involves fitting the actual income distribution with an income distribution of reference.

In practice, this can be done by using individual ranks in the real income distribution to assign fictive incomes

whose distribution will correspond to the reference distribution.

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501

of health inequality in a single dimension. Thus, the same classical tools developed for the

analysisofincomeinequality,suchastheLorenzcurveorGinicoefficient,canbeappliedto

thehealthvariabletocomparehealthinequalityacrosssocieties.However,suchanapproach

leaves us unable to differentiate between a situation where all health is distributed to the

poor, to the rich, or randomly throughout the population.

Class C assumes that health and income are substitutes—an assumption consistent with

themainstreampoliticsofhealthinequality.Itisconsistentwithstatedhealthinequalitypoli-

cies of international organizations that focus on averting undue concentration of ill-health

among the poor. It suggests that concentrating ill-health among the poor is less equitable

thannotconcentratingill-healthamongthepoor.Inequalityinincomemaybecompensated

through inverse inequality in health. Of course, the extent to which poor health should be

distributedtowardstherichwilldependonthedistributionofincomeandontheexactshape

of the utility function.

Class D assumes that income and health are complements. This is the opposite of Case

C. Under this scenario, the social planner should prefer to concentrate ill-health among

the poor. We ignore this case because it leads to social preference of health inequality

distributions where the poorest are the sickest—a situation which conflicts with most social

goals for health.

The above discussion suggests that Class B and Class C are the main focuses of interest

with respect to health inequality. Among others, Class A does not allow any ranking in

practice while Class D leads us to prefer a society where the rich also enjoy better health

thanthepoor.Thenextsectionconsidersseveralconsequencesofdefininghealthinequality

using this bi-dimensional SW approach.

4. Weaknesses in the SW definition of health inequality

We have shown that the SW approach appears to provide a useful framework for defining

andanalyzinghealthinequality,butthatitsbenefitsdependonourabilitytoplacerestrictions

on the evaluation of welfare derived by the combination of given income and health levels.

Furthermore, we should consider a series of methodological issues that arise when we try

to apply the above concept of health inequality.

The first of these issues concerns the measurement of the health variable. The theory of

income inequality has been developed under the assumption that income can be measured

on a linear scale and that the measure of income inequality should be scale-independent

(Atkinson,1970).TheSWcriteriondoesnotformallyallowincomeinequalitytoberanked

between societies with different mean income levels. When researchers are faced with this

situation, the problem is typically resolved by rescaling income so both societies have

identical mean income levels.

Health, however, is more difficult to measure. Health is sometimes measured on a

linear scale using anthropometric measures on child height-for-age or weight-for-height

z-scores, blood pressure measures on adults, or life expectancy or related measures such

as quality-adjusted or disability-adjusted life years (QALYs and DALYs). There is no dif-

ficulty in such cases since the health variable can be easily rescaled. However, health data

are often also based on ordinal categories, such as self-assessed health status, which are

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impossible to rescale. Health inequality should then only be compared between societies

having the same marginal distributions of health, although not necessarily the same joint

distributionofhealthandincome.Whenhealthismeasuredbyadichotomousvariable,such

as child stunting or living status (dead/alive), the mean level is obviously defined. However,

equal means imply identical distributions for dichotomous variables. Thus, the problem is

essentially the same and, de facto, comparison is only appropriate between societies with

identical health distributions.2

Thislimitationalsohighlightstheimportanceoffurtherresearchonpossiblehealthmea-

sures. Wagstaff and van Doorslaer (1994) propose a method to translate categorical health

indicators to continuous measures and their approach has been applied in van Doorslaer

et al. (1997) to analyze health inequality based on self-reported health data from several

industrializedcountries.Furthermore,theirmethodissupportedbyresultsfromavalidation

study conducted on Swedish data (Gerdtham et al., 1999). These are important efforts and

continued research in these directions will provide us with more tools to analyze health

inequality.

The difficulties of measuring health variables are a key concern but they are common

to both the SW approach and the Rawlsian approach to be presented below. However, we

now turn to the fundamental weakness of the SW approach. This weakness is due to the

inconsistency of the SW approach with the basic notion of a just or equitable distribution

of health. The literature on health inequality suggests that a fair health distribution does

not imply equal health status for all individuals since individuals may differ in their health

endowments (Culyer and Wagstaff, 1993). Instead, this ideal rests on the notion that avoid-

able differences should be reduced or eliminated. This is implicit in the Global Strategy for

Health for All resolution (WHA32.30) adopted by the World Health Assembly in 1979 and

WHO Health for All. As Whitehead notes (1990), “Equity is ... concerned with creating

equalopportunitiesforhealthandwithbringinghealthdifferentialsdowntothelowestlevel

possible.” More generally, these aims are consistent with a social justice approach to health

(Wagstaff and van Doorslaer, 2000).

Webelieveitisusefultodefinethehealthdistributionintheidealequitablesocietyasone

where access to health has not been determined by socioeconomic status or income. Why

then is the SW approach inconsistent with a social justice approach to health inequality?

This is because, regardless of the assumptions that are made on the cross derivatives of

the utility function, the sum of utility functions depending on two arguments, health and

income, cannot provide an indication of the level of unfairness in the health distribution.

Whatever restrictions we place on the cross derivatives of the utility function, the SW

approach is unable to reject income-based discrimination in access to health. In Case B

income-based discrimination does not matter. On the other hand, in Case C, society will

choose discrimination in favor of the poor, even if health is not related to income. In Case

D, society will choose discrimination in favor of the rich. Only Case A avoids this pitfall,

but we already know Case A is not helpful since it is unable to rank most societies. In fact,

the failure of the SW approach is due to the fact that it gives symmetric roles to income

2We should also note that another measurement problem arises from the fact that we only observe the health of

survivors. Nonetheless, the expected utility should be measured by the average destiny of all people and not only

from the average destiny of survivors.

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503

and health. Thus, the SW approach will always opt for conditioning health access on the

distribution of income, even if income is not a source of discrimination in access to health.

Thus, the SW approach cannot provide a definition of the ideal equitable health distribution

orwiththebasisforameasureofthegapbetweentheidealandtheactualhealthdistribution.

Instead, we turn to a different approach which is related to Rawls’ Theory of Justice and

helps us provide a definition of both the ideal health distribution in the equitable society

and a measure of the distance from this ideal.

5. A Rawlsian approach to health inequality measurement

Our goal is to offer a definition of health inequality that is consistent with prevailing

egalitarian principles with respect to health inequality. Our approach follows from Rawls’

First Principle of Justice (1971). According to Rawls’ First Principle of Justice, basic free-

doms must be distributed equally throughout society. However, we must address the fact

that the First Principle cannot be directly applied to health. Rawls himself labels health a

natural good and explicitly rules out health as a basic freedom (Rawls, 1982). We argue

nonetheless that Rawl’s first principle can be used as a basis for defining health inequality

oncewerecognizethattheactualhealthofindividualsdependsonbothindividualhealthen-

dowments as well as on how health endowments are transformed into actual health through

access to health resources. We agree with Rawls’ perspective that it makes little sense to

include actual health status itself as a basic freedom since individuals differ in their health

endowments and it would be absurd to define the ideal society as the one where all individ-

ualsweregeneticallyidentical.Instead,weproposethataccesstohealthresourcesisabasic

freedom and therefore that health access should be distributed equally in the ideal society.

Thus, all individuals should have the same opportunity to achieve their potential health

levels. This means we should expect two individuals with equivalent health endowments

to reach the same health level, regardless of their socioeconomic status. A society where

some individuals have limited access to health, relative to others, is to be considered unfair,

regardless of the source of the discrimination.

Once we recognize health access (a) as a basic freedom, Rawls’ First Principle of Justice

provides a natural step towards defining the concept of health inequality. Rawls’ First

Principle then indicates that health access should be equally distributed. In this case, the

ideal measure of health inequality might appear to be a simple unidimensional measure

such as the Gini coefficient applied to a health access measure. The problem is that access

to health is not observed. We can measure the actual health status (h) of individuals but

we have no information on their health endowments (e) and therefore the gap between the

two is unmeasurable. However, if we assume that e is distributed independently of one

variable, y, and a is identical for the entire population, actual health, h, should then also be

independent of y. Any existing relationship between h and y is proof of health inequality

and stronger association between y and h indicates higher levels of health inequality. Thus,

byexaminingtheassociationbetweenyandhwecanassessonecomponentofinequalityin

access to health, even if access to health is not directly measurable. This is why we propose

to measure health inequality by measuring the association between observed health, h, and

another variable, y, which is unrelated to the health endowment and is discussed below.

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Our approach bears resemblance to a similar definition Roemer (1998) has offered for

equality of opportunity. However, Roemer’s approach and ours are focused on rather dif-

ferent philosophical concerns. Roemer is primarily concerned by the fact that individuals

may not invest equal levels of effort in the production of their own health. He argues that

unequal effort should not be confused with inequality per se. His definition for the perfectly

equal society is the one that provides the same level of outcome (health in this case) for

all those who invest the same degree of effort. Our work is less concerned by variations in

the effort levels of individuals in the production of health, and more concerned by hetero-

geneity in health resulting from unobservable natural factors (something we label “natural

endowments” but may simply be “luck”, etc.). There is definitely a difference between

Roemer’s concern and ours, or in other words between “degree of effort” and “luck”, but

since both of these factors are unobservable, the two definitions lead to similar definitions

ofhealthinequality.Infact,theunobservabilityofdegreeofeffortinRoemer’swork,andof

natural endowment in our work, makes it necessary to add similar additional independence

assumptions (Roemer, 1998, p.15).

Obviously, the association between h and y may not account for the entire variation in

a. Indeed, it may be possible that a is not equally distributed but that the variation in a is

orthogonal with the variation in y. Our approach is therefore conditional on the appropriate

choice of a variable y. Such a variable should be: (i) independent of health endowments

and (ii) likely to reflect a major cause of discrimination in access to health. There is no

obvious choice for such a variable. Discrimination in access to health is likely to be based

on socioeconomic-status and income seems a good choice with respect to the second point.

However, it does not fully satisfy condition (i) since a poor health endowment may be a

cause of low income. Better candidates may be parental socioeconomic status which may

be more consistent with point (i) or an instrumented socioeconomic status measure. For

simplicity, we will thereafter refer to income as the y variable even though we recognize

that such a variable does not fulfill requirement (i). In practical applications, more thought

should be given to the choice of this variable according to the characteristics of the data

that are used and to the socioeconomic environment of the population under study. Once

we choose a measure of y, such as income, our concept of health inequality simply focuses

on the association between the health and income distributions.

Unfortunately, there is no universal method for measurement of the association between

two distributions. Obvious and simple candidates include the covariance, the correlation,

and the Spearman correlation. However, each of these measures relies on an implicit value

judgement. For example, the correlation is appropriate for detecting a monotonic trend

between two variables and weights each observation according to its distance from the

mean. The Spearman correlation is based solely on the rank of the observations and thus is

only affected by relative positions in both the health and income distributions. This means

that measures of health inequality based on the covariance implicitly assume that only the

relativeincomeandhealthlevelsmatter,whilemeasuresbasedontheSpearmancorrelation

dependonlyonpositioningalongtheincomeandhealthscales.Thetwocriteriaclearlyrely

on different conceptions of health inequality. The Relative Index of Inequality is another

common indicator which is closely related to the Concentration Index and is obtained by

regressingtherelativehealthlevelontherankingintheincomedistribution(Kakwanietal.,

1997). In fact, the relative index of Inequality reflects properties of both the covariance and

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505

the Spearman correlation since it focuses on how positioning on the economic scale affects

the relative level of health.

It is impossible to choose among the many potential measures of bivariate association

without first introducing some social value judgement. This problem is identical to trying

to choose among the many measures of income inequality such as the Gini coefficient,

Theil coefficient, or the variance of the logarithm. For this reason, economists prefer to use

inequality criteria that are based on a class of inequality measures, as illustrated with the

SW approach to income inequality. Here, we propose a parallel approach and offer criteria

that can be used with a wide class of health inequality measures.

Following the theoretical approach of Rawls and including our assumption that health

access is a basic freedom, we argue that the expectation of actual health given income,

E(hi|yi), should be equal to the average health level in the society, E(hi), and therefore

independent of income. Analysis of the variation of E(hi|yi) within societies provides a

usefulcriterionforassessinghealthinequality.Yet,wehavenotspecifiedhowthisvariation

shouldbemeasured-aproblemthatdependsonhowoneweightsminorvariationscompared

tolargevariationsbutourchallengeisnowreducedtothemoretraditionaloneofmeasuring

inequality in one dimension: inequality in E(hi|yi). Thus, we suggest one simple indicator

of health

W1=

1

N

N

?

i=1

U(E(hi|yi))

(3)

whereUisanincreasingandstrictlyconcavefunction.Suchindicatorssharecertainimpor-

tant properties with the SW function defined by Eq. (2). In particular, they have an additive

structure which makes it relatively easy to compute society’s “willingness” to improve one

person’s health relative to another or, in other words, to define a health policy. However, W1

type indicators are also quite different since they do not assume separability between the

health and income of different individuals. This means it is generally impossible to define

interventionprioritiesbetweenanytwoindividualswithouthavingmoreinformationonthe

healthandincomedistributionswithinthepopulation.Clearly,foranyconcavefunction,U,

for a given average health level, the W1indicator will lead us to prefer the situation where

there is no association between hiand yi. The shape of the function U will determine the

tradeoff between inequality and the average health level. Strict adherence to the priority

for health based on the First Principle would require us to assume an infinitely concave

U function that would imply the equalization of expected health across society. Indicators

defined by Eq. (3) lead us to our first general criterion of health inequality, the W1criterion,

which states that health inequality in society A can be said to be greater than in society B, if

W1is always larger in B than in A, for all functions U which are increasing and concave.3

Thishealthinequalitycriterionisbasedonaclassofunidimensionalinequalitymeasures.

Thus, it can be easily interpreted using a graphical representation by drawing the Lorenz

curve for the variable E(hi|yi). One particular case is when E(hi|yi) is an increasing

3Like in the Social Welfare case, such a criterion can not rank all societies. As illustrated in the literature

on income inequality, additional restrictions may be imposed on the function U, and this makes it possible to

increase the number of societies that may be ranked. However, this possibility must be balanced by our interest

and willingness to make specific ethical assumptions on inequality at different health levels.

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function of income. Then ranking in terms of E(hi|yi) and ranking in terms of income

are identical and the Lorenz Curve exactly corresponds to the concentration curve. In fact,

our proposed W1criterion is closely related to the concentration curve, which is probably

the most widely used measure of health inequality. However, there has been very little ef-

fort geared towards developing the theoretical foundations of the concentration curve. Our

results suggest that the extensive use of concentration curves for comparisons of health in-

equality,evidentinboththeoretical(ContoyannisandForster,1999)andempiricalresearch

(e.g. van Doorslaer et al., 1997) is consistent with the Rawlsian approach.

One drawback to the above function, W1, is that it does not depend on the distribution

of health within income groups. However, according to a Rawlsian approach to health

inequality, the full distribution of health conditional on income, rather than simply its aver-

age,shouldbeindependentofincome.Therefore,animprovedmeasureofhealthinequality

might also depend on the distribution of health within income groups. This can be easily

resolved by assuming decreasing returns to health in the objective function. That is,

W2=

1

N

N

?

i=1

U(E(v(hi)|yi))

(4)

with U and v being increasing and concave. In Eq. (4), concavity in the function U still

reflects aversion to inequality across income groups while concavity in the function v

represents aversion to inequality within income groups. Measures like W2appear to be

simple extensions of W1measures. Such inequality measures will obviously depend on the

particular shapes of the functions U and ν, but again a general inequality criterion can be

defined considering the classes of increasing and concave functions. Such a criterion is

closer to the Rawlsian concept of health inequality, but it no longer has a direct relationship

to the concentration curve. While this clearly reduces its immediate relevance in relation

to the current literature, an indicator such as W2has the advantage of encompassing both a

unidimensional and bi-dimensional approach to defining and measuring health inequality.

In the next section, we further examine the properties of both measures using policy

scenariosandwealsocontrasttheirimplicationswiththoseobtainedfromalternativeethical

criteria to health inequality.

6. Implications for the design of health intervention policies

We illustrate here the implications of various ethical approaches to health care interven-

tion.Weassumethatgovernmentsacceptaparticulartradeoffbetweenequityandefficiency

and we consider a series of simple scenarios where the health technology remains constant

while the ethical goals of the government are varied. We assume the public sector has the

opportunitytodistributeanewhealthtreatmenttothepopulationbutthatthecostofthetreat-

mentmakesuniversalcoverageimpossible.Thus,thegovernmentmustdecideonastrategy

for targeting the new treatment. Whatever strategy is adopted, there will be a trade-off be-

tweenequityandefficiencyandthefinaldecisionwilldependonthegovernment’saversion

to health inequality. Our goal is to highlight how the equity component will be valued dif-

ferently according to the inequality measure that is used.

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507

Fig. 1. Equity-efficiency tradeoff: targeting when only health equality matters.

A number of simplifying assumptions are introduced. We assume that the health (h) and

income(y)variablesarecontinuousandmeasurable.Weassumethatthehealthintervention

produces only a marginal effect on individual health levels and thus an infinitesimally small

impact on the health distribution. We assume that this health impact depends on the health

and income levels of those treated. The health impact of the treatment is assumed to be

larger for persons at lower initial health levels or higher income levels.4This means that

program efficiency may be increased by targeting interventions towards less healthy or

more wealthy persons. This is implicit in the positive slope of the iso-technology lines (see

dashed lines) shown in Figs. 1–4. Each iso-technology line represents persons along the

joint health and income continuum for whom the treatment will produce the same change

in health for a given investment in the health sector. Efficiency is increased as we move

the treatment towards lines that lie in the southeasterly direction. Each subsequent line

4These assumptions are not essential but serve to simplify our example. They are also reasonable. Treating

less healthy individuals is often more efficient than treating the more healthy which explains the first part of our

assumption. For a given level of health, treating the richer may also be more efficient since they are typically more

educated and more efficient in the use of medical care. Furthermore, the wealthier are also more likely to pay for

treatment. This may further raise the efficiency of per capita public health expenditures on the rich.

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Fig. 2. Equity-efficiency tradeoff: targeting using the social welfare criteria.

representsinterventionpossibilitiesthatareα timesmoreefficientthaninterventionsonthe

previous line.

The government policy will depend on technological constraints represented by the

iso-technology lines and their intersection with welfare and equity principles. We con-

sider several different ethical criteria: health egalitarianism, SW as well as the Rawlsian

measures we have proposed and we show how each of these leads to a different government

targeting strategy. Ethical criteria may be represented by government indifference curves

(see dotted lines) as shown in Figs. 1–4. The indifference curves in each graph represent an

equal gain in the government’s objective for a given improvement in health. The scenario

shown in Fig. 1 represents the case where the government is solely concerned with reduc-

ing health inequality by raising health levels of the least healthy, regardless of their income

levels. This leads to the horizontal indifference curves in Fig. 1 since the government’s

objective is assumed to depend solely on the health levels of those that are treated. The gain

is larger when improving health of individuals at lower health levels and each consecutive

curve represents gains in government’s objective that are α times larger. Thus, the distance

between the curves measures the level of the government’s aversion to health inequality:

the smaller the distance the higher the aversion to inequality.

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509

Fig. 3. Equity-efficiency tradeoff: targeting based on Rawlsian criteria W1.

Combining the technological constraints and government preferences on the same figure

enables us to see their tradeoff and provides a tool to target health interventions. Turning

again to Fig. 1, position b and c represent equivalent equity levels since they lie on the

same indifference curve while a and b represent similar efficiency levels because they lie

on the same iso-technology line. Positions c and d lie on the line that represents α percent

greater health returns than positions a and b for every dollar invested while a lies on the

indifference curve representing α percent greater return in terms of equity for every unit

of health improvement than positions b and c. Furthermore, positions a and c are equally

desirable in terms of targeting. Although, a is α times preferable relative to c in terms of

equity, c is also α times preferable in terms of efficiency. A series of iso-targeting lines may

bedrawn(seesolidlines)toreflectinterventionsthatprovideequalgaininthegovernment’s

objective for a given dollar investment. Positions a and b lie on different iso-targeting lines

and it is easy to see why a is a preferable investment from the government’s perspective

since it is on the same iso-technology line but offers a higher return in terms of equity.

The simplified geometry underlying these scenarios allows them to help us identify how

a government with limited resources might use the iso-targeting lines as boundaries for

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Fig. 4. Equity-efficiency tradeoff: targeting based on Rawlsian criteria W1.

health interventions. Of course, all the above conclusions depend on the assumption that

is made on the equity-efficiency tradeoff of the government. A government more averse

to inequality would have indifferences curves much closer together resulting in almost

horizontal iso-targeting lines and meaning that intervention a would be preferred to almost

any other intervention. In contrast, a government little concerned by inequality will have its

policy mainly driven by efficiency goals, which would result in iso-targeting lines having

almost the same slope as the iso-efficiency lines. In any case, the slope of the iso-targeting

lines will lie somewhere between the slope of the indifference curves (horizontal in this

case) and the slope of the iso-technology curves—depending on the extent to which the

government is ready to trade efficiency for equity.

Incontrasttothepurehealthinequalitycase,wenowconsidercaseswherethegovernment

priority includes both the distribution of health as well as the association between health

and income. Fig. 2 presents the first such case where the technological relationship is the

same as Fig. 1 but the government’s aim is to maximize SW (see Eq. (2)) and views health

and income as substitutes (U??

healthandincomeinSW,theindifferencecurvesarenegativelysloped.Fig.2showsthatthe

hy< 0). Because of the acceptability of the tradeoff between

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511

tradeoff between efficiency and welfare will lead the government to choose to compensate

the poor with additional health, as compared to the health egalitarianism case illustrated in

Fig. 1. If the government were to view health and income as complements (Case D), the

slope of the indifference curves would reverse and become positive (not shown). We would

then expect the slope of the iso-targeting lines to be steeper than in Fig. 1 to reflect a greater

likelihoodoffocusingtreatmenttowardsbetteroffindividuals(bothintermsofincomeand

health).

We now examine W1, which is the first of our proposed Rawlsian measures. Instead of

assuming a tradeoff between health and income, the government is assumed to be primarily

concerned with avoiding differences in expected health levels across income groups. Thus,

the indifference curves are drawn as vertical lines since there is no stated preference to help

individuals at different health levels within an income category. In the first case (Fig. 3),

we assume that health and income are strongly and positively correlated and this leads

us to draw the indifference curves relatively close together. In the next case (Fig. 4), we

considertheconsequencesofmaintainingasimilarlevelofinequalityaversionbutassuming

a weaker association between health and income. This leads to greater spacing between the

indifference curves. In both cases, indifference curves to the left imply greater gains in

term of equity since they concern income groups who have on average lower health levels.

The slope of the iso-targeting line is greater in Fig. 4 than in Fig. 3, although the only

change that has been introduced is the statistical correlation between health and income in

the population (aversion to inequality and government preferences are the same). Had we

introduced a second figure for the SW scenario to represent the difference between weak

and strong correlations with no other differences, the targeting line in Fig. 2 would have

remained the same, as a consequence of the assumption of additive separability underlying

the SW function (Eq. (2)).

This last point leads us to stress a fundamental difference between the SW and Rawlsian

approaches. In the case of the SW approach, the government targets health interventions

towards people with lower utility levels. This means it targets interventions towards the

poor if income and health are viewed as substitutes and this targeting is independent of

any existing association between income and health. According to the Rawlsian approach,

healthinterventionsshouldbetargetedtowardsthepoorforethicalreasonsonlywhenthere

is a positive correlation between health and income. Stronger correlations lead to greater

targeting. Thus, the SW approach targets health interventions towards the poor in order

to diminish welfare inequality that is due to inequality in both income and health levels.

This means that it may be advisable to apply discriminatory health policies even when all

individuals have the same health levels in order to compensate for inequality in income.

On the other hand, according to W1, targeting health interventions towards the poor is only

advised if low income is statistically found to negatively affect health, since this targeting

aims to correct for income-based discriminations in health access.

Restricting the indifference curves in Figs. 3 and 4 to be vertical may seem rather incon-

sistentwithcommonethicalcriteriasinceitmeansentirelyignoringanypreferencestotreat

the less healthy individuals within income groups. In fact, this is the fundamental advan-

tage of our second proposed measure of health inequality, W2, which provides the basis for

discriminatory health interventions within income groups as well as across income groups.

This measure considers both the full distribution as well as the mean level of health within

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each income group. Certain assumptions on the form of the function v would allow us to

draw the indifference curves for this case as well. In fact, the indifference curves would be

identical to those for W1except for a slight counterclockwise rotation of the curves leading

to a similar slight rotation in the iso-targeting lines (not shown). The extent of the coun-

terclockwise rotation would depend on both the government’s aversion to health inequality

within income groups and how within inequality varies between income groups.

7. Conclusion

There is widespread interest in reducing health inequality. Economists and other social

scientistshaveengagedinthiseffortbyprovidinganalyticaltoolsandempiricalassessments

aimed at facilitating the measurement and reduction of health inequality. However, there

has yet been very little research on the theoretical foundations underlying the analysis of

health inequality.

Our paper evaluates the appropriateness of the classic social welfare (SW) approach to

the development of a conceptual definition of health inequality. We discuss several mea-

surement and ethical criticisms related to the analysis of health inequality that reduce the

desirability of the classic approach. In particular, the SW approach treats income and health

as comparable variables—an important drawback in terms of ethical considerations. We

propose an alternative approach which appears to be more closely linked to the WHO con-

cept of equity in health and is also consistent with the ethical principles espoused by Rawls

(1971).Thisconcept,initssimplestform,isshowntobecloselyrelatedtotheconcentration

curve as soon as we assume that health and income are positively related. Thus, the criteria

presented in our paper provide an important theoretical foundation for empirical analysis

using the concentration curve. In addition, we have presented an additional measure which

is theoretically more appealing but further study is need in order to determine its usefulness

in empirical research on health inequality.

Acknowledgements

We thank Shlomo Yitzhaki for critiques and suggestions on an earlier version of this

paper. We also benefited from comments from two anonymous reviewers.

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