Welfare, Justice, and Pareto Efficiency

SVEN OVE HANSSON

WELFARE, JUSTICE, AND PARETO EFFICIENCY

Accepted 15 July 2004

ABSTRACT. In economic analysis, it is usually assumed that each individual’s well-being(mental welfare) depends on her or his own resources (material welfare). A typology isprovided of the ways in which one person’s well-being may depend on the material resourcesof other persons. When such dependencies are taken into account, standard Paretian analysisof welfare needs to be modified. Pareto efficiency on the level of material resources need notcoincide with Pareto efficiency on the level of well-being. A change in economic conditionsthat is Pareto efficient in the standard sense, i.e., with respect to material resources, maynevertheless sacrifice one person’s well-being to that of another. It is shown that underplausible assumptions, Pareto efficiency on the level of well-being may require the reductionof inequality on the level of material resources.

KEY WORDS: envy, equality, Gini inequality, interdependence, justice, Pareto efficiency,positional goods, preference, welfare, well-being

1. INTRODUCTION

In economic analysis, it is usually assumed that each individual’s well-being depends on her or his own material resources. In actual life, eachindividual’s well-being depends also on other factors, among them mate-rial resources that belong to other persons. Such human interdependenceshave consequences for the traditional measure of efficiency in welfare eco-nomics, Pareto efficiency. If each person’s well-being depends only on herown material resources, then a distribution that is Pareto efficient on thelevel of material resources will also be Pareto efficient on the level of well-being. However, when each person’s well-being can also be influenced bymaterial resources belonging to other persons, Paretian efficiency on thetwo levels will no longer coincide. It is the purpose of the present con-tribution to investigate how this type of human interdependence shouldinfluence economic analysis and what consequences it may have for therelationship between welfare and justice.

After clarifications of the meanings of welfare and justice (Section 2)and the distinction between mental and material welfare (Section 3), a sys-tematic treatment is offered of the ways in which one person’s well-beingmay depend on the material resources of another person (Sections 4–5).It is shown how such dependencies create problems for traditional Pare-tian analysis of welfare (Section 6). A model for two-levelled Paretian

Ethical Theory and Moral Practice 7: 361–380, 2004.C© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

362 SVEN OVE HANSSON

analysis of material and mental welfare is proposed (Section 7). Finally, itis shown how the traditional conflict between equality and efficiency canunder certain conditions be resolved in the proposed model (Section 8).

2. THE RELATIONSHIP BETWEEN JUSTICE AND WELFARE

Welfare means to fare well. The basic meaning is summarized in the OxfordEnglish Dictionary as “[t]he state or condition of doing or being well;good fortune, happiness, or well-being (of a person, community, or thing);thriving or successful progress in life, prosperity.” Hence, in order to specifywhat we mean by welfare in economics we have to make clear what a goodlife consists in.

The term justice has many meanings (Hansson, 2001, 2004). The mostimportant of these for economics is justice as distributional goodness. Aparticular distribution satisfies the requirements of justice to the extent thatit is good with respect to how it apportions advantages and disadvantagesamong recipients. Sometimes, “justice” is used to denote social goodnessmore in general, including non-distributional considerations. (John Rawls’sA Theory of Justice is probably the most important contemporary exampleof this usage.) Here, the focus will be on justice as distributional good-ness, i.e., justice in the distribution of resources, goods, services, and otheradvantages among the members of a population. (Justice can also refer tothe distribution of burdens, such as taxes and participation in war, but thatusage will not be treated here.)

With this approach to justice, justice and welfare are closely relatedconcepts. As a first approximation, economic justice consists in the justdistribution of welfare, or the conditions of a good life. One consequenceof this is that whatever problems we encounter in the explication of theconcept of welfare will also tend to complicate the analysis of justice.

3. MENTAL AND MATERIAL WELFARE

According to one view, a person’s welfare is a matter of her material con-ditions, such as access to food, shelter, healthcare and, generally speaking,the necessities and perhaps luxuries of life. According to another view,welfare is a matter of the inner, or mental state of the person, such as hap-piness or satisfaction. (In what follows, I will use the term “well-being” asan approximate synonym of mental welfare.)

Of course, in many cases measures aimed at achieving mental and mate-rial welfare will coincide. Proposals to improve material welfare are typi-cally put forward with the conviction that the mental welfare of the persons

WELFARE, JUSTICE, AND PARETO EFFICIENCY 363

concerned will also improve. But there are also situations in which materialand mental welfare do not coincide.

An obvious standpoint on the relation between the two is to regardmaterial welfare as having only derived value, so that mental welfare is allthat counts. Why arrange, for instance, for better housing if it does not makethe people who will live in these houses happier? Can material welfare beof any value at all if it does not affect the feelings and experiences of thoseconcerned? Is it not the only reasonable standpoint to consider only mentalwelfare to have intrinsic value?

This view, that subsumes the material under the mental, has a longtradition. All the major religions have branches that emphasize the futilityof material possessions and claim that various religiously defined aspectsof a person’s mental state are all that is important for her welfare. We findthe same basic standpoint in classical utilitarianism, but here the mentalstate that is counted as the sole final end is a secular one, namely happiness.

From the viewpoint of widely shared moral intuitions, it is highly prob-lematic to treat welfare as an exclusively mental phenomenon. We all knowthat people can live under deplorable material conditions but yet be happyand contented. As one example of this, it has long been known that Afro-Americans who live in black neighbourhoods tend to be more satisfied withtheir conditions of life than Afro-Americans who live under much bettermaterial conditions in white neighbourhoods. A plausible explanation ofthe difference is that contrary to the ghetto inhabitants, the latter group tendto compare themselves with the white middle class (Wilson, 1971). Shouldwe then accept the ghetto as more conducive to welfare than an ethnicallyintegrated society? There is also evidence that contentedness is a commonway for oppressed persons, including the poor in the third world and op-pressed women in all countries, to cope with their situation (Sen, 1985,p. 11). If mental states are all that count, then we will have to concludethat the social states in which this coping strategy has been successful areacceptable from the point of view of welfare.

This is a serious problem for religious ethics. Religious leaders who havesided with oppressors have told the poor to care only about their mentaland spiritual welfare, and to accept material conditions as they are. Interest-ingly enough, hedonistic utilitarianism – arguably the paradigmatic secularethical theory – has the same problem. According to classical hedonism,everything is fine if people are happy and oppressed.

Let us then turn to the other interpretation of welfare, the material conceptof welfare. It saves us from the problems just mentioned, but unfortunatelyit is problematic in other ways. Consider a person who lives under goodmaterial conditions, but is still unhappy and perhaps even contemplatessuicide. Most of us would hesitate to say that such a person’s welfare isgood.


The sensible conclusion to draw from this is that our intuitive conceptof welfare contains both a material and a mental component. In most dis-cussions on welfare, this ambiguity in the concept will not be noticed sincewe act under the assumption that the two components go together. How-ever, we seem to have reliable intuitions about welfare only in cases whenmaterial and mental welfare coincide. Therefore, cases when they do notcoincide have to be treated with the utmost care.

It is common in the literature to reserve the term “welfare” (and simi-larly “welfarist”, etc.) either only for mental or only for material welfare.1

Neither of these attempts to restrict the meaning of the term is supportedby common usage. I propose that we use the term “welfare” for the generalconcept, and specified terms for the more restricted conceptions of welfare.

This is not only a terminological issue, but also a substantial one. Anexclusive focus on mental welfare may lead to a neglect of material livingconditions. Similary, an exclusive focus on material welfare may lead to aneglect of the experiences and feelings of those concerned. Social policiesneed to take both into account.

4. POSITIONAL GOODS AND INTERPERSONAL DEPENDENCE

The standard approach in welfare economics is to treat each individual’swell-being as dependent on her own material conditions, see Figure 1.However, it is easy to show that this is a too far-reaching simplificationof the relationship between material and mental welfare. A person’s well-being depends not only on her own material conditions, but also on thoseof others who live in the same society.

This dependence can take many forms. The variant that has been mostdiscussed is connected with the role that our material conditions, and inparticular the objects that we own, have as indicators of social status. Usingthe terminology introduced by Hirsch (1976) we can talk about “positionalgoods”, goods that give their owner a place in the social hierarchy. Havinga colour TV at the time when this was a new and exciting technologycontributed positively to the owner’s social status. This effect decreasedin importance as colour TVs became common. Access to a particular typeof positional goods typically increases with economic growth, and it canthen lose its positional value and be replaced by other, newer objects, asmarkers of social status.

Access to positions in the social hierarchy is limited. Hirsch (1976,p. 52) claimed that the competition for positional goods is a zero-sum game.

1One example is Dworkin’s (1981, p. 188) proposal to use the term “welfare” only formental properties.


Figure 1. The traditional view, according to which each person’s well-being depends onher own material resources.

However, a zero-sum situation holds only for purely positional goods, i.e.,goods that have no other value for the owner than their effects on theowner’s social position. In most practical cases, there is both a positionalcomponent and a use value. The colour TV did not only provide a highersocial status; it also enhanced the joys of TV viewing.

We cannot adequately understand the social role of new consumer prod-ucts, if we treat them as purely positional goods with no other functionthan that of conferring status to their owners. On the other hand, we cannoteither understand them if we treat them merely as utility objects. Both thesefunctions need to be included in economic analysis.

Decisive support for the importance of social positions for welfare hasemerged from studies of public health. It is known since long that there aregreat differences in health between rich and poor countries. A clear sign ofthis is the large difference in average length of life. It turns out, however,that among the rich industrialized countries, there is no clear correlationbetween wealth and length of life. In other words, it makes a big differencefor a person’s expected length of life if she lives in a poor third world coun-try or in an industrialized country. However, it makes no difference if shelives in one of the richer or one of the less rich among the industrializedcountries. Once a country has reached the level of the poorest countries inWestern Europe, additional wealth and income does not increase longevity.Instead, however, among the industrialized countries a surprisingly strongcorrelation has been found between average length of life and equality inthe distribution of wealth and income. The more economic equality there isin a country, the longer do its inhabitants live (Wilkinson, 1994). We do notknow what causes this connection. (It has been shown that differences inresources spent on healthcare cannot explain the correlation.) To the extent


that we may take length of life as an indicator of well-being, these results in-dicate that relative positions in the social hierarchy have a substantial impacton well-being, and in particular that the types of relative positions createdby large social differences have a strong negative effect on well-being.

5. A TYPOLOGY OF DEPENDENCE EFFECTS

In addition to position effects, there are several other ways in which oneperson’s material resources can influence another person’s well-being. Inorder to show how diversified such dependencies are, and how importantthey are in our economies, I will introduce a simple typology that includesthree main types of dependencies, each of which can be divided into apositive and a negative subvariant. For simplicity, I will refer to the two-person case. The first of the three types is based on the effects on oneperson’s well-being that are mediated by effects on another person’s well-being. See Figure 2.

Type I, The well-being that B achieves through owning X has an influ-ence on A’s degree of well-being.,

Type I+, The well-being that B achieves through owning X increasesA’s degree of well-being.,

Type I−, The well-being that B achieves through owning X decreasesA’s degree of well-being.Type I+ includes the dependence created by so-called “benevolent” pref-erences, the type of preferences that you have when you are pleased tofind that your friend has bought a better car or when you are relieved tofind that a destitute person has obtained the means necessary for a decentlivelyhood.

Type I− includes dependencies originating in “malevolent” preferences,such as when you are annoyed that someone you dislike could afford abetter house or when you react negatively to reports on the luxuries of thevery rich. One possible source of dependence of type I− is envy. This isalso one of the forms of dependence that is most frequently referred to in

Figure 2. Dependence of type I.


Figure 3. Dependence of type II.

the economic literature. It is commonly claimed that preferences referringto the welfare or well-being of others should be disregarded in economicanalysis, since envious preferences should not be allowed to influence socialprocesses (Cudd, 1996, p. 22). There is also a fairly extensive literature onhow preferences driven by envy can in practice be excluded from economicanalysis (Varian, 1974; Vohra, 1992; Suzumura, 1981; cf. Le Grand, 1990,1991). However, it must be emphasized that preferences that give rise todependencies of type I− need not be envious or in any other way improper.In fact, already Adam Smith emphasized the distinction between envyand legitimate indignation.2 In my view, it is sensible to follow AdamSmith in making distinctions between the different types of other-regardingpreferences, rather than excluding them all since some of them are drivenby envy.

The second type of dependence refers to the direct effects that oneperson’s possessions can have on another person’s living conditions. SeeFigure 3. Again, there is both a positive and a negative variant.

Type II, That B owns X has a direct influence on A’s material conditionsof life, and therefore indirectly on A’s well-being.,

Type II+, That B owns X has a direct positive influence on A’s materialconditions of life, and therefore indirectly on A’s well-being.,

Type II−, That B owns X has a direct negative influence on A’s materialconditions of life, and therefore indirectly on A’s well-being.Type II+ is exemplified by catalytic exhaust emission control devices andABS brakes. A road-user’s welfare is improved when other car-ownershave these devices, since they make the air she breathes cleaner respec-tively decrease the risk of accidents. Note that another car-owner’s emis-sion control devices and ABS brakes constitute an improvement both forsomeone who owns and for someone who does not own the same types ofdevices.

2Smith 1808, 2:129–130 (Part VI, Section III). Aristotle made the same distinction inhis Rhetoric, 1387a (Book II; chapter 9). See also Rawls, 1972, p. 533 (§80).


Figure 4. Dependence of type III.

Type II− dependencies can be divided into two major classes. One ofthese refers to property that makes life more uncertain or unpleasant for oth-ers than their owners. Handguns are probably the best example in this cate-gory. The other class of examples are the positional goods referred to above.Suppose that owning a satellite phone confers social status. Then my statuswill increase if I acquire such a device, but it will also decrease when some-one else acquires one (irrespective of whether I own one myself or not).

The final type refers to how the value of one person’s possessions canbe influenced by property owned by another person. See Figure 4.

Type III, That B owns X has an influence on the value for A of A’s ownproperty, and therefore indirectly on A’s well-being.,

Type III+, That B owns X has a positive influence on the value for A ofA’s own property, and therefore indirectly on A’s well-being.,

Type III−, That B owns X has a negative influence on the value for A ofA’s own property, and therefore indirectly on A’s well-being.

There are several ways in which dependencies of type III+ can arise. Thevalue for me of having a telephone or internet access will increase if othershave the same means of communication. Since it is a greater pleasure toattend to a play or a concert in a full than an empty house, a correspondingeffect holds for theatre and concert tickets. It is also an advantage for acar-owner if others own the same make of car, since they create a marketfor spare parts and specialized garages.

Dependencies of type III− arise from what is called “congestion” ineconomic theory (Cornes and Sandler, 1996, chapter 8). One importantmechanism is our wish to avoid crowding, in a concrete physical sense. Asummer house on a small island will be worth more if no one else ownsa house there.3 Another mechanism is connected with limited access tocertain goods, in particular collector’s items such as art, coins, and stamps.

3Hirsch (1976, e.g. p. 3) made no distinction between types II− and III−, but includedexamples of both types among his examples of positional goods.


Figure 5. In actual fact, each person’s well-being depends both on her own material re-sources and those of other persons.

6. A PROBLEM FOR CLASSICAL PARETIAN ANALYSIS

Given all these types of dependencies, the traditional picture shown inFigure 1 is misleading. The true picture is more like Figure 5. The well-being of each person depends not only on that person’s material conditions,but also on the material conditions of other persons.

There are two standard ways to subject a distribution of resources andwelfare to an economic analysis. We can evaluate it from the viewpoint ofdistributive justice or from that of efficiency.

The concept of distributive justice can, without much difficulty, be ap-plied to material resources, i.e., to the left column of boxes in Figures 1and 5. We can determine how material advantages are distributed amongdifferent persons, and based on this we can assess how this distributioncompares with our criteria for a just distribution.

However, if we want to make an assessment in terms of distributive jus-tice on the level of well-being (the righthand columns in the same figures),then we run into problems. It is difficult – many would say impossible– to compare well-being or other mental states in different persons. Wehave no means to determine if one person is happier or more satisfied thananother, and the same applies to other criteria that we may have of mentalwell-being. (It is however feasible to compare one person’s happiness atdifferent points in time, for instance by comparing her own reports of herfeelings.) In summary, critera of justice are in practice applicable to materialresources (the left-hand boxes) but not to well-being (the right-hand boxes).

For comparisons in term of efficiency, or more precisely Pareto effi-ciency, the situation is different. (A distribution is Pareto superior to another


distribution if and only if every person has at least as much of the distributeditem, and at least one of them has more. A distribution is Pareto efficient ifand only if there is no other feasible distribution that is Pareto superior toit.) The determination of Pareto superiority does not require interpersonalcomparisons. Furthermore, under traditional assumptions it does not mat-ter if we consider Pareto efficiency on the level of material resources oron the level of well-being. To see that, consider again the traditional pic-ture of the relationship between resources and well-being, Figure 1. Thestandard assumption is that each person’s well-being is an increasing func-tion of his or her own material resources. Under this assumption, Paretoefficiency on the resource level and on the level of well-being will coin-cide. In other words, a distribution of material resources is Pareto efficientif and only if the distribution of well-being that it gives rise to is Paretoefficient.

However, this does not hold with the more realistic assumptions aboutthe relationship between resources and well-being that are indicated inFigure 5. When each person’s well-being depends not only on her ownmaterial resources but also on those of others, then Pareto efficiency in ma-terial resources and in well-being need not coincide. A change in economicconditions that is Pareto efficient in the standard sense, i.e., with respectto material resources, may nevertheless sacrifice one person’s well-beingto that of another. This is unsatisfactory, and gives us reason to performParetian analysis on the level of well-being as well. In other words we needto investigate what social changes will lead to improved well-being forsome, but not to decreased well-being for any, of the individuals includedin the comparison.

7. A PARETIAN ANALYSIS OF WELL-BEING

To simplify the Paretian analysis of well-being, I will focus on a simpletwo-person society.4 Furthermore, I will make the simplifying assumptionthat we can measure not only material resources but also each person’swell-being in numerical terms. (Strictly speaking, this is not necessary,but a numerical approach has expository advantages over a relational one.)Interpersonal comparability will not be assumed, i.e., we will compare eachperson’s well-being under different conditions, but we will never have tocompare the well-being of different persons.

The resource vector 〈x, y〉 denotes that the first of our two individualshas the total amount x of useful things and that the second individual has

4Alternatively, the example can be interpreted as representing a society with equalnumbers of rich and poor.


Figure 6. Classical Pareto efficiency for two individuals with equal initial assets.

the amount y. We may (but need not) think of x and y as representing sumsof money.

The well-being of the two persons depends in part on their materialresources (and in part on other factors that we will have to abstract fromhere). As already mentioned, the traditional assumption is that a persons’swell-being is an increasing function of the value of her material possessions.From a mathematical point of view, this means that there is a function u1

that, given the resources of the first individual, takes us to her well-being,and a similar function u2 for the second individual. Hence, if the distributionof resources is represented by the vector 〈x, y〉, then the first individual hasthe level of well-being u1(x) and the second individual the level u2(y). Thismeans that from the resource vector 〈x, y〉 we can derive a well-beingvector 〈u1(x), u2(y)〉.

In Figure 6 this classical approach is shown in a case where both indi-viduals have equal initial assets, and in Figure 7 in a case where one ofthem has three times more material resources to begin with. The shadedarea represents those states that would, from the respective starting-point,be Pareto improvements.

But as we have already seen this is too simplified. The first individual’swell-being depends not only on her own resources but also on those ofthe other person. Hence her well-being is a function of 〈x, y〉 rather thanof x. For the same reason, the second person’s well-being is a functionof 〈x, y〉 rather than of y. Hence, from the resource vector 〈x, y〉 we canderive the well-being vector 〈u1(〈x, y〉), u2(〈x, y〉)〉.5 This framework can

5Cf. Bronfenbrenner, 1973, p. 18.


Figure 7. Classical Pareto efficiency for two individuals with unequal initial assets.

be used to perform a Paretian analysis of well-being (in our simplified two-person case). Given the functions u1 and u2, we can determine whether ornot different distributions of resources are Pareto efficient on the level ofwell-being.6

In this framework, we can construct a simple mathematical model ofdependence of type II+. We can assume that a person with this type ofdependence will experience the same increase in well-being, when theother person receives a certain amount of rescources that she would havedone, had she herself received a certain fraction of those resources. This canbe expressed by assigning to the first person a factor (positive real number)k1 such that u1(〈x, y〉) = u1(〈x + k1y, 0〉) and to the second person a factork2 such that u2(〈x, y〉) = u2(〈0, y + k2x〉). Under the further assumptionthat u1(〈x, 0〉) and u2(〈0, y〉) are increasing functions this results in thefollowing condition for when a resource allocation 〈x′, y′〉 is at least as goodas a resource allocation 〈x, y〉 from the viewpoint of a Paretian analysis ofwell-being:7

〈x′, y′〉 is at least as good as 〈x, y〉 if and only if:

x ′ + k1 y′ ≥ x + k1 y and y′ + k2x ′ ≥ y + k2x .

6The classical case with no dependencies corresponds to functions u1 and u2 such that thevalue of u1(〈x, y〉) is independent of y (depends only on x) and that the value of u2(〈x, y〉)is independent of x. With the additional assumption that both functions are increasing, wewill then find that if x′ is greater than x, then u1(〈x′, y′〉) is greater than u1(〈x, y〉). Similarly,if y′ is greater than y, then u2(〈x′, y′〉) is greater than u2(〈x, y〉).

7When k1 �= 0 �= k2, the condition can be simplified to −k2 ≤ (y′ − y)/(x′ − x) ≤ −1/k1.


Figure 8. Pareto efficiency for two individuals with utilitarian preferences and equal initialassets.

Figure 9. Pareto efficiency for two individuals with utilitarian preferences and unequalinitial assets.

Clearly, if k1 = k2 = 0 then we are back in the classical case illustrated inFigures 6 and 7. If k1 = k2 = 1 then both individuals are strict “utilitarians”who value advantages for themselves just as much as they value advantagesfor others. The latter case is illustrated in Figures 8 and 9, with an equalrespectively unequal point of departure. Note that here and in what follows,though the axes of the diagrams represent the material possessions of thetwo individuals, the shaded areas denote Pareto improvements in terms ofwell-being, not material resources.


Figure 10. Pareto efficiency for two individuals with type II+ dependence.

Figure 11. Pareto efficiency for two individuals. The richer person has type II+ dependence,whereas the poorer person’s well-being is independent on the other’s resources.

In Figure 10 both individuals have type II+ dependence, with k1 = k2

= 1/3. In Figure 11, the rich person’s well-being is positively influencedby material improvements for the poor person, whereas the poor person’swell-being is uninfluenced by improvements for the rich (k1 = 0, k2 = 1/3).In Figure 12 the reverse relationship holds (k1 = 1/3 and k2 = 0).

Figure 13 represents the arguably somewhat extreme case in which therich person gains more in well-being from improvements for the poor


Figure 12. Pareto efficiency for two individuals. The poorer person has type II+ depen-dence, whereas the richer person’s well-being is independent on the other’s resources.

Figure 13. Pareto efficiency for two individuals. The richer person has type II+ dependenceto a very high degree, whereas the poorer person’s well-being is independent on the other’sresources.

person than from improvements for herself (k1 = 0 and k2 = 1.5). In thiscase some redistributions (i.e., changes with x + y constant) will be Paretoefficient. This situation was discussed by Hochman and Rogers (1969), whoclaimed that in practice the rich have so much to gain from improvementsfor the poor that redistributions from rich to poor people are Pareto efficient.


Figure 14. Pareto efficiency for two individuals with dependence of type II−.

Dependencies of type II− can be treated with the same type of math-ematical model that we used for type II+. This means that if one of theindividuals obtains an increase in resources, then the other person experi-ences a decrease in well-being that is the same as if she herself had obtaineda decrease in her resources of the same size as a specified portion of the firstperson’s increase. We can use the factors k1 and k2 as above. This meansthat the resource allocation 〈x′, y′〉 is at least as good as 〈x, y〉 from theviewpoint of a Paretian analysis of well-being if and only if:8

x ′ − k1 y′ ≥ x − k1 y and y′ − k2x ′ ≥ y − k2x (1)

Figure 14 shows what happens if both participants’ well-being dependsin this way on the material resources of the other (k1 = k2 = 1/3). In Figure15 only the rich person has this type of dependence, whereas the poorhas the classical (independent) pattern. Figure 16 represents the contrarysituation, and corresponds to the situation discussed by Ahn (1990), inwhich the luxury consumption of the rich decreases the well-being of thepoor.

It is instructive to compare Figures 11 and 16. It turns out that type II− de-pendence of the poor person Pareto-forbids the most unequal distributions,whereas type II+ dependence of the rich person will not do so.

8When k1 �= 0 �= k2, this can be simplified to k2 ≤ (y′ − y)/(x′ − x) ≤ 1/k1.


Figure 15. Pareto efficiency for two individuals. The richer person has type II− dependence,whereas the poorer person’s well-being is independent on the other’s resources.

Figure 16. Pareto efficiency for two individuals. The poorer person has type II− depen-dence, whereas the richer person’s well-being is independent on the other’s resources.

8. POSITIONAL PREFERENCES AND EQUALITY

Positional preferences give rise to dependence of type II−, but it is never-theless of interest to treat them more directly. A simple model of positionalpreferences in a two-person society can be constructed in the followingway: If the first individual has x units of resources, and the second has


y units, then we can use x − y as a measure of the first individual’s positionin relation to the second individual. In the same way we can use y − x asa measure of the second individual’s position in relation to the first. Wecan furthermore introduce a factor p1 that denotes the importance for thefirst individual of her position, in relation to the importance of her own re-sources, and a similar factor p2 for the second individual. Their respectivewell-being can then be derived from the resource vector, as follows:

u1(〈x, y〉) = u1(〈x + p1(x − y), 0〉) and

u2(〈x, y〉) = u2(〈0, y + p2(y − x)〉).

This means that the resource allocation 〈x′, y′〉 is at least as good as 〈x, y〉from the viewpoint of a Paretian analysis of well-being if and only if:9

x ′+ p1(x ′ − y′) ≥ x+ p1(x − y) and y′ + p2(y′ − x ′) ≥ y + p2(y − x)

We do not need any new diagrams to illustrate Pareto efficiency in thismodel of positional preference, since some of the diagrams from above canbe reused for the purpose. Figure 14 illustrates a case where both individualshave positional preferences of this type (p1 = p2 = 1/2). Figure 15 illustratesa case in which only the rich person has such preferences whereas the poorhas classical preferences (p1 = 0 and p2 = 1/2). Figure 16 shows the reversecase (p1 = 1/2 and p2 = 0).

Finally we can compare the effects of positional preferences to those ofreducing inequality. The most common measure of inequality is the Ginicoefficient. The shaded area in Figure 17 consists of all social states that areless unequal than the starting-point 〈1, 3〉, given that we measure inequalitywith the Gini coefficient.10

By comparing Figure 17 to some of the previous diagrams, we findthat requiring a Gini improvement (i.e., increased equality) in materialresources has effects that are quite similar to those of requiring Paretoefficiency on the level of well-being, given that we have dependence oftype II− (k1 = 1/3) on the side of the poor. In the immediate surroundingsof the present state, Figure 16 (type II− dependence on the side of the poor)coincides with the intersection of Figures 7 (classical Pareto-efficiency)and 17 (Gini improvement).

9If p1 �= 0 �= p2, then this can be simplified to p2/(1 + p2) ≤ (y′ − y)/(x′ − x) ≤ (1 +p1)/p1.

10If half of the population has x units each and the other half y units each, and x ≤ y, thenthe Gini coefficient is (y − x)/(2x + 2y). Since the Gini coefficient for 〈1, 3〉 is 1/4, it followsthat Gini improvements from this starting-point can be characterized by x/3 ≤ y ≤ 3x .


Figure 17. Decreased inequality (Gini improvements) for two persons.

The general conclusion that can be drawn from this is that under plausibleassumptions about human interdependence, Pareto efficiency on the level ofwell-being may require the reduction of inequality on the level of materialresources. Equality and efficiency are not as incompatible as they haveoften been believed to be.

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Philosophy UnitRoyal Institute of TechnologyTeknikringen 78, 100 44, Stockholm, SwedenE-mail: [email protected]

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Welfare, Justice, and Pareto Efficiency