Measurement of income inequality*

Journal of Public Economics 47 (1992) 3-26. North-Holland

Measurement of income inequality* Experimental test by questionnaire

Yoram Amiel

Ruppin Institute, 60960. Isrcrel

Frank A. Cowell

London School of Economics, Houghton Street, London WCZA ZAE, UK

Received March 1990, revised version received March 1991

We reconsider some of the basic assumptions of the literature on inequality analysis. Students’ views on the meaning of inequality comparisons were elicited by means of a questionnaire involving both numerical and verbal questions. The responses suggest that two important axioms - the principle of transfers and decomposability - are not universally accepted.

1. Introduction

What do we mean by inequality comparisons? Applied economists - and others - are apt to make clear, unqualified statements to the effect that income inequality in one situation, place or period is higher or lower than in another. Statements of this sort are commonly taken to be a yardstick of economic performance; they sometimes provide a basis for social comment; and it has even been known for policy-makers to act upon them. It seems, therefore, important to give some thought to the information that they are supposed to convey.

In fact, of course, such statements are usually founded upon the implicit assumption that there are agreed criteria about the meaning of ‘inequality’,

*Our thanks go to the Hebrew University, the Ruppin Institute and ST/ICERD for making our collaborative work possible. We are grateful to Anja Green and Ann Harding who toiled over the completed quesiionnaire forms aid produced the tabulations for us, and to Dieter B6s, Sore1 Cahan. Wolfgang Eichhorn, Jim Gordon, David Levhari, Avishai Margalit, Wilhelm Pfiihler, Dan Slottje and Yossi Yahav who helped to organise the question&es for us in various institutions. We want to thank Tony Atkinson, Serge Kolm, the participants at the Symposium on Distributive Justice, Bonn-Bad Godesberg, a co-editor and two referees of this Journal who commented on earlier versions of the paper; Janet Stockdale for helping us with the questionnaire design; and also the colleagues who commented on preliminary drafts of the questionnaire. Above all we are indebted to the respondents who entered into the spirit of the exercise.

0047-2727/92/$05.00 6 1992-Elsevier Science Publishers B.V. All rights reserved

4 Y Amiel and F.A. Cowell, Measurement of income inequality

and hence about the basis on which comparisons of income distributions may usefully be made. It is the purpose of this paper to subject these criteria to critical examination from an unconventional standpoint.

A cursory glance at the literature on the measurement of income inequality reveals a variety of attitudes and approaches to the choice of appropriate criteria. For example, one commonly-held view is that the nature of income distribution is no different from the nature of any other distribution, so that one may, without further ado, use tools of other disciplines - dispersion measures, distance functions and the like - and apply them to economic inequality. On the other hand, some economists prefer to derive cardinal or ordinal indices by explicitly defining properties or axioms that inequality indices or social welfare functions ought to fultil.’ Although each approach has its own distinctive characteristics, it is possible to identify several criteria that almost all types of approaches have in common. The fact that these criteria are common to many diverse methodologies does not, however, mean that they should pass unchallenged.

In fact this paper addresses some of the most important and fundamental criteria for income inequality comparisons. Our aim is to investigate whether some of these supposed ‘consensus criteria’ really do form a consensus, or whether there are grounds for seriously questioning some of the standard assumptions made in the literature. A question that arises immediately is: A consensus amongst whom? The general public? Economists? Economists who are specialists in this particular area?

It is a tricky question. In our opinion it would be presumptuous to take for granted that only the views of this last - prestigious but narrow - group should be heeded. In principle one should take into account the views of people who are able to think about the problem analytically, but who are not specialists in the area of inequality analysis. They need not even be economists. In so doing one might avoid becoming hostage to the conven- tions that accompany any academic specialism.

On the other hand, although such a broad-minded approach sounds attractive, there are problems with an attempt to elicit the views of a very broad range of people. Members of the general public may have an intuitive idea about what economic inequality means; they may also have strong feelings about the matter. However, the public’s ideas about inequality are often tied to specific situations or examples rather than based on precisely defined abstract principles of distributive justice. Moreover, intuition alone can prove to be a poor guide to criteria of general applicability.

In this paper we suggest an appropriate ‘target’ sub-population whose

‘See, for example, Atkinson (1970). Champernowne (1974), Cowell (1977), Dalton (1920), Ebert (1988a. b), Eichhorn and Gehrig (1982), Foster (1983), Kakwani (1980), Kolm (1976a, b), Kuga (1973). NygBrd and Sandstriim (1981), Pyatt (1985), Sen (1973), Shorrocks (1988) and Thon (1982).

Y. Amiel and F.A. Cowell, Measurement of income inequality 5

views might reasonably be sought; we outline a method of eliciting their views; and report the results of some fieldwork trials.

The plan of the paper is as follows. Section 2 sets out our general approach. Section 3 introduces the specific assumptions required for our underlying model and the types of questions that are to be posed. Sections 4 and 5 report the empirical results and consider their implications for the standard approach to inequality measurement. Section 6 concludes.

2. The approach

The analysis of economic inequality suffers from an obvious disadvantage. Unlike the analysis of household expenditure patterns or of asset purchases, which can be used to provide evidence on behaviour and hence about preference structures, there is little in the way of ready-made concrete evidence about people’s views - or preferences - about distributional issues. Yet it is clear that such views may be relevant to the way in which one formulates an approach to inequality measurement. How does one go about discovering what they are?

As we have indicated in the Introduction, if one wanted to seek out views and attitudes to inequality, it is not clear what sort of people ought to be targeted. Experts may be prejudiced and hidebound; sensible and well- motivated lay people may just not formulate their ideas coherently. We were acutely aware of this point in designing the study reported in this paper: if you are going to elicit numerical responses from people, then it is important to use as subjects those who are likely to avoid arithmetical mistakes and logical slips. As a compromise between the narrow specialist and the well- meaning but innumerate layman, we decided to try using students.

Students are not, of course, mistake-free. Nor are they innocent of prejudice. However, they are usually accustomed to working through simple numerical problems and reflecting upon logical propositions. They can bring to bear the discipline of reasoned thought upon propositions of general interest. In addition, if one approaches the right students at the right time, it may be possible to avoid the problem of a bias in favour of ‘received wisdom’. It is possible that they may have a freshness of view from which their teachers may beneficially learn. Moreover, there are other advantages in using students, as we discuss below.

It seemed to us that the best way of seeking out students’ views was to use a questionnaire - a form of investigation that has several precedents in the literature.* In doing this we were acutely aware of many of the well-known

‘The use of questionnaires in economics, and specifically in the area of welfare and inequality, is not new. See, for example, Cowell (1985), Gevers, Glejser and Rouyer (1979), Morawetz (1977) Ptingsten (1988) Tversky and Kahneman (1974) van Praag (1977, 1978), and Yaari and Bar Hillel (1984). For an interesting inquiry about existence of consensus by economists and politicians on questions of public economics, see B&tan (1973).


drawbacks of such an approach, and attempted to take measures that would mitigate them. As will be evident, our particular scheme was designed to reduce the problems of non-response and of misunderstanding of the nature of the questions involved. We also attempted to allow for the ‘learning-by- doing’ that inevitably takes place in the course of completing such a questionnaire. Moreover, we tried to keep the exercise as short as was practicable: one of the greatest enemies of this type of approach is simple fatigue.

Our questionnaire is reprinted in full in the appendix. It is designed with two, interrelated sections. The first section is numerical: it contains a set of simple choices between successive pairs of specific income distributions, A and B. The distributions are presented as vectors, without explicit currency units, and no hints were provided to the students as to what sort of living standards or welfare levels might correspond to those numbers.3 In each case the respondent is merely invited to state which of A and B appears to be the more unequally distributed. The second section is mainly verbal: respondents are confronted with various general propositions about income distribution; they have to choose with which view among several about income inequality in hypothetical changes of income distribution they agree. The issues raised in the second section are closely linked to the numerical questions of the first. Respondents have the opportunity to change their answers to the questions in the first section; in fact we invite them to do so. They are also invited to put forward possibilities other than those given in the list, and to record any comments they might want to make.

In the first section of the questionnaire we only asked about one particular case for each of the criteria. If the answers of most respondents happen to be in accordance with the relevant criterion this does not of course mean that they agree with the criterion, as a generul principle. However, if most answers are at variance with the specific criterion, then we evidently have to think again. The asymmetry between answers which are and which are not in accordance with theory is well known and we solve it partly by questions about opinions in the second section of the questionnaire. There is a place of course for checking accordance by additional examples of comparisons of income distribution.

Among the possible responses presented in the questions in the second section there is usually one that could be characterised as the ‘traditional’ view found in the literature of measurement of income inequality. Some of the other responses printed in the questionnaire correspond to views that have also appeared in the literature, some of them were suggested by

‘In Amiel and Cowell (1991a) we discuss the relationship between social we!fare and income inequality, and introduce currency units to the questions about income distribution. In Amiel and Cowell (1991b) we report on another questionnaire study in which respondents were given specific guidance on the purchasing power of the income which is being notionally redistributed.

Y Amiel and F.A. Cowell, Measurement of income inequality 7

Table 1

Institution

North Texas University Southern Methodist University London School of Economics University of Bonn University of Karlsruhe Hochschule fur Unternehmungsfiihrung, Koblenz Ruppin Institute Hebrew University, Jerusalem (economists) Hebrew University, Jerusalem (non-economists)

Country

U.S.A. U.S.A. U.K. F.R.G. F.R.G. F.R.G. Israel Israel Israel

Number of respondents

41P 108 106 356

53 50

174 170

54

All 1,108

respondents in a previous pilot questionnaire, and some of them just seemed to us like a good idea at the time. Although many questions offered a ‘none of the above’ response, and although we invited free expression by asking respondents to write down their own comments, it is clear that the structure of the alternative responses may have influenced the broad direction of the students’ replies; but it is not clear in which direction. Perhaps if the choices had been more circumscribed - if only the traditional economists’ doctrine and a single alternative had been given - the answers would have been more in accordance with standard doctrine. Perhaps also other arguments, which simply did not occur to us, might have shifted respondents’ answers in the opposite direction, away from orthodoxy. This is evidently a problem that is not confined to our questionnaire alone.

The questionnaires were filled in during late 1988 and early 1989 by 1,108 students in eight colleges and universities spread amongst four countries, as shown in table 1. Having as respondents students at roughly similar stages in their education, but from different countries, permitted us to examine the important question of whether there appear to be substantial cross-cultural differences in perceptions of the fundamentals of inequality analysis.

In most cases we arranged for the questionnaire to be completed by upper- level undergraduates who had some training in economics4 but who had not previously taken courses that included studying the measurement of income inequality.5 We exploited the advantage of having students as respondents to ensure a high level of participation: we arranged that completion of the questionnaire (about 20 to 30 minutes) shoud be incorpor- ated within a lecture or class. Note that the purpose of the questionnaire was

4We also carried out the questionnaire with non-economist students at the Hebrew University. The results appear in the last row of each of the relevant tables.

slsraeli economics students usually study the Lorenz curve as part of their standard first year course. This is not the practice in other institutions.

8 I: Amiel and F.A. Cowell, Measurement of income inequality

explained to the respondents on the first page, and that it was completed anonymously.

3. Assumptions and propositions

In this section we are not going to introduce any revolutionary new propositions. However, it is useful to set out briefly some of the standard assumptions in the literature, and to consider their implications for propositions on inequality analysis. In so doing one may be able to appreciate better the results of our questionnaire investigation. For the sake of brevity and precision we shall state the assumptions in formal terms: readers who are unhappy with this kind of presentation may care to skip a page and go straight on to the intuitive discussion that follows.

First, some notation. We assume that someone has already defined what the concepts ‘income’ and ‘income receiver’ mean. We index the members of the population by i = 1,2,. . . , n, and assume that person i’s income is a non- negative scalar xi. The symbol n denotes a vector of such incomes

(x1,x2,..., x,,), and I denotes a vector of ones. For convenience we write n(x) and p(x), respectively, for the number of components and the arithmetic mean of the components of vector x. Also, we let x[m]: =(x,x, . . . ,x), a mn- vector that represents a concatenation of m identical n-vectors x.

We write I for the set of integers { 1,2,. . , , n} and X for the set of all possible income vectors, which we will take to be the set of all finite- dimensioned non-negative vectors excluding the zero vector. We also write X’ for the subset of X that excludes all vectors of the form al, a>O. By an inequality comparison we mean a binary relation t on the members of X. Since our questionnaire is phrased in terms of inequality, the statement ‘x>x” is to be read as ‘x represents an income distribution that is at least as unequal as distribution x”, It is possible, of course, that it does not represent a complete ordering on X. We define the strict inequality comparison > and inequality equivalence - in the usual way.

To give economic meaning to the relation 2, a number of assumptions are conventionally imposed. The principal assumptions which are commonly used as building blocks in an axiomatic approach to inequality measurement can be expressed in the following way:

0. Anonymity. For all x E X and any permutation matrix P, x - Px.

la. Scale-invariance. For all x E X and positive scalars a, ax -x.

lb. Translation-invariance. For any XEX and scalar b such that x + bl EX,

x+bl-x.

2. Principle of Population. For all x E X and positive integers m, x[m] -x.

I! Amid and F.A. Cowell, Measurement of income inequality 9

3. Transfer Principle. For any x E X’ for any i, jE I, and for any scalar 6 > 0 such that xi>Xj and Xi--6>Xj+6, X~(X1,X2,.,.,Xi_6,...,Xj+6,...,X,).

4. Decomposability. Let X~E X and X$E X, g = 1,2,. . . , G, be income vectors such that n(x,) = n(x& p(xJ =p(x$) and x&x; for all g, then (x,, x2,. . . ,x,)2 (x:,x; )...) xb).

The anonymity axiom says that if people are identical in all relevant characteristics other than income, then inequality comparisons treat them equally. This axiom is of fundamental importance in other related fields as well as in inequality analysis, and - as long as incomes and income receivers have been appropriately defined - some form of the anonymity axiom may be taken as essential to a rational discussion of the meaning of inequality comparisons.6

The rest of the assumptions are presented in an order that is convenient for comparison with the format of our questionnaire rather than that suggested by a logical sequence.

Except in trivial cases scale-invariance and translation-invariance (assumptions la and lb) should be treated as mutually exclusive alternatives: the choice between them was discussed by Dalton (1920) and by Kolm (1969, 1976b). However, other, similar assumptions may also be reasonable. For example, it has been argued’ that some compromise position between the two principles la and lb may be appropriate, e.g. one might require that inequality should remain unchanged under some transformation such as ax+ bl, where a and b are specific constants with a>O.

The population principle has an important implication for inequality indices that one might construct on the basis of one’s axiom system. Given this assumption such indices would be invariant under replications of the population. Note that this is considerably stronger than the requirement that inequality comparisons be invariant under replication, i.e. x’[m]>x[m] if and only if x’>x.

The transfer principle (assumption 3) states simply that if a small income transfer is made between two persons of unequal income, inequality rises (falls) according as the recipient is richer (poorer) than the donor. Observe that this property is required to hold independently of the incomes possessed by any other members of the community.

The decomposability assumption (4) can be viewed as a type of ‘indepen-

6Cf. the remark by Nygard and Sandstrom (1981): ‘a rejection of SYM [the anonymity axiom] would deprive us of all means of judging inequality’. Of course, if there are subgroups in the population between which income comparisons are not possible - for example groups where income receivers have different personal characteristics such as family size - then the anonymity axiom cannot apply universally: see Cowell (1980).

‘See Bossert and Ptingsten (1990) and Kolm (1976b).

10 Y. Amiel and F.A. Cowell, Measurement of income inequality

dence of irrelevant alternatives’ axiom. In the present case we investigated a weaker form of this property where G =2 and xZ =n;.

Assumptions 3 and 4 have tremendous consequences for the selection of appropriate tools with which to carry out inequality analysis. As is well known, there is a simple proposition that connects the transfer principle with the ordering on income distributions induced by S-concave welfare functions and the ordering of distributions using the Lorenz curves,’ and the greater part of the received wisdom on the positive and normative approaches to economic inequality is founded upon this principle and its corollaries. Furthermore, the acceptance of assumption 4 significantly affects the structure of the class of indices that are admissible as inequality measures: for example, it rules out popular indices such as the Gini coefficient, the relative mean deviation and the logarithmic variance.’ Accordingly we shall find it of particular interest to see how these two conventional assumptions about the structure of inequality comparisons correspond with the views revealed by our questionnaire analysis.

4. Results

Within the framework of a simple questionnaire there are evidently some issues about which it is extremely difficult to elicit useful information. For example, we made no attempt to investigate the anonymity axiom (assumption 0) since in the present context it was not clear to us what meaning might be attached to the responses on any simple question that we might put on this topic.” Therefore for the sake of clarity in each numerical question we asked the respondents to compare the distributions represented as ordered vectors. However, we can use the questionnaire responses to throw some light on four fundamental issues involving people’s approach to inequality comparisons relating to assumptions 1 to 4 in section 3 above: the effect of income transformations involving translation and scale, the effect of population replication, the principle of transfers, and decomposition by population subgroups.

4.1. Income transformations

We begin with an examination of the alternative principles cited as assumptions la and lb above. This could be regarded as a purely formal point: should one normalise so that inequality rankings are independent of a simple change in the scale of incomes, or of the origin from which one

‘See Atkinson (1970) and Dasgupta et al. (1973). ‘See Shorrocks (1984) and Cowell (1988). “‘However we have investigated this in a later questionnaire study involving issues of social

welfare - see ‘Amiel and Cowell (1991a).

Y Amiel and F.A. Cowell, Measurement of income inequality

Table 2

11

Double income (ql) Add 5 units (q2)

N Down Up Same Down Up Same

(%) (%) (%I) (%) (%) (%)

N. Texas SMU LSE Bonn Karlsruhe Koblenz Ruppin Hebrew econ. Hebrew non-econ.

All

(37) (108) (106) (356)

(53) (50)

(174) (170)

(54)

(1.108)

27 32 41 38 19 38 42 32 8 32 59 63

11 31 58 63 7 40 53 76

26 36 38 72 13 32 55 70 21 38 41 51 11 39 50 57

14 35 51 59

11 46 19 46 2 35 8 28 9 15 8 20 4 26

19 30 2 41

9 31

chooses to measure incomes ? However, the adoption of either of these principles - or of some alternative - is value-laden, and is of particular concern to those who take a welfare-theoretic approach to the analysis of inequality. For example, proportionate increases in all real incomes may change perceptions of inequality for a number of reasons.’ ’

Questions 1 and 2 attempt to elicit respondents’ views about the effects of each of these two transformations by using two simple numerical examples. In question 1 distribution B is formed by doubling the incomes in distribution A; in question 2 distribution B is formed by adding 5 units to each of the incomes in distribution A. Loosely translating this pair of questions we might say: Under the transformation A +B does inequality go down, go up, or stay the same? More than one ‘standard’ answer to this has been offered in the literature: it is common for authors to assume scale-invariance, although Kolm (1976a) has argued for translation-invariance.

The responses to questions 1 and 2 may then be summarised as in table 2.i2 The responses to the corresponding verbal questions (10 and 11) generally correspond to the pattern indicated by the numerical questions: they are given in table 3.

It is perhaps easier to see what is going on from an examination of a cross-tabulation of the responses to questions 1 and 2, as in table 4: the position of a person who believes in scale-invariance is marked with a single asterisk, that of a person who believes in translation-invariance is marked

“On this point see Miller (1970): ‘... it is conceivable that a proportionate income increase means more to the poor than to the rich’.

‘*The percentage given in each case refer to the overall sample numbers and not just to those who responded on a particular question. As a consequence of this (and also because of rounding) some of the percentages in this and subsequent tables will not add up to 100.

12 Y Amiel and F.A. Cowell, Measurement oJ income inequality

Table 3

Double income Add fixed sum Deduct fixed sum

(q10) (qlt) (qll)

N Down Up Same Down Up Same Down Up Same

(%) (%) (%) (%) (%) (%) (%) (%) (%)

N. Texas (37) 1:

24 70 38 11 51 11 43 46 SMU (108) 32 48 25 8 62 5 41 51 LSE (106) 13 38 46 61 7 32 5 68 25 Bonn (356) 12 30 58 61 35 6 64 27 Karlsruhe (53) 17 55 28 62

2:

Koblenz (50) 26 36 36 48 8 4;

4 79 17

8 72 Ruppin (174) 10 49 40 67 6 27 7 70 :‘: Hebrew econ. (170) 52 40 75 6 12 73 Hebrew non-econ. (54) 14 52 35 39 4 :; 11 41 :8”

________ All (1,108) 12 40 47 58 6 35 7 64 28

Table 4

Add 5 units (q2)

Down Up Same

(%) (%) (%)

Double income (ql) Down 8 (2) (5) UP 15 17** Same 37* (:) (9)

with a double asterisk. The situations corresponding to entries in parentheses are illogical under the assumptions normally made concerning inequality measures.

Scale-invariance (37%) is a clear winner, with translation-invariance com- ing a distant second (17%). Perhaps one should register mild surprise that support for the conventional view - scale-invariance - was not exactly overwhelming. However, the pattern of responses is more reassuring if one adopts a broader interpretation of ‘the conventional view’. Notice that 15% of the respondents adopted a position intermediate between the scale- invariance and translation-invariance principles, but only 11% adopted positions more extreme than either principle; the implication of this is that there is clear support (69%) for the following composite view on the effect of income transformations ‘inequality judgements should respect either the scale-invariance principle, or the translation-invariance principle, or should conform to some intermediate position between the two.“j Finally, note that

13Cf Bossert and Ptingsten (1990). We have investigated further the issue of the relationship between income transformations and income inequality in Amiel and Cowell (1991b).


Table 5

Add fixed sum (911)

Down Up Same (%) (%) (%)

Double income (q10) Down 7

UP 21 Same 30*

(1) (4)

(:, ,::;*

table 4 suggests that there is a bias in favour of saying ‘the same’ even when such a response would appear illogical: perhaps this reflects an innate ‘safety- first’ response on the part of student respondents.

Once again the broad tenor of these conclusions is borne out by the cross- tabulation of the response to the verbal questions, which is given in table 5. The entries have been annotated as in table 4. It seems that, on this issue at least, respondents were not likely to have been influenced by whether the questions were presented in numerical or verbal form.

However, apart from the numerical versus verbal issue, when one contrasts the pair of questions 1 and 2 with the pair 10 and 11, one is not exactly comparing like with like. On reflection it is clear that if the style of question is changed from ‘compare A and B’ to something of the form ‘consider what happens if you go from A to B’, an extra factor is being introduced, namely the order in which the cases are presented. It is possible that this order may be perceived as significant; for example, A+B may be a process that occurs in real time. If this is so, then judgements about inequality comparisons may be sensitive to the particular initial distribution that has been specified: history matters. There are perfectly respectable arguments for allowing such considerations to affect assessments of distributional fairness, although the standard theory of inequality measurement rules them out.

It is clearly worth investigating whether such considerations might have had a major influence in the responses to our questionnaire. To do this we might look for ‘reversal consistency’ in the respondents’ answers. Without such consistency inequality comparisons are bound to depend on information other than the pair of income vectors with which one is confronted. We can investigate such consistency by looking at the first two parts of question 11: loosely paraphrased, will the removal of one unit from everybody have exactly the opposite effect on inequality from that brought about by the addition of one unit?

The analysis of the relevant responses is as shown in table 6. Those entries marked with an asterisk indicate reversal consistency. Once again, of those responses that appear to violate the principle of reversability, a substantial proportion may have been influenced by ‘safety first’ considerations - see the 9% in the row labelled ‘the same’.


Table 6

Deduct fixed sum

Down Up Same

(%) (%) (%)

Add fixed sum Down 4 53* 1 UP 2* 3 1 Same 1 9 2s

Table I

N

N. Texas (37) 41 19 41 SMU (108) 44 18 35 LSE (106) 29 9 61 Bonn (356) 31 6 62 Karlsruhe (53) 28 13 58 Koblenz (50) 40 10 50 Ruppin (174) 32 5 63 Hebrew econ. (170) 18 15 66 Hebrew non-econ. (54) 39 13 48

Numerical (q3) Verbal (q12)

Down Up Same

(%) (%) (%)

Down Up Same

(%) (%) (%) _-, ~_~ 38 14 49 32 12 51 25 9 65 19 6 73 25 8 68 22 2 76 22 9 66 18 12 65 22 11 63

All (1,108) 31 10 58 22 9 66

4.2. Population replication

It is common in both empirical and theoretical work to find inequality measures normalised so that the principle of population is automatically satisfied. However, acceptance of the principle is not quite universal. If we extend the ambit of inequality measures to include also measures of concentration, it is not unusual to find instances where the value of the index varies under replication of the population. For example, the Herfindahl index can be defined as

n xi 2 c [-I9 i=l w

which clearly decreases under population replication.’ 4 We investigated respondents’ views on this principle with one numerical

question (q3) and one verbal question (q12). In each case respondents were invited to consider the effect on inequality of doubling the population. The summary of responses is given in table 7. Those whose responses were in line

14See also the argument of Cowell (1977, p. 63) where this assumption is questioned using an example of extreme inequality

Y Amiel and F.A. Cowvll, Measurement of income inequality 15

with the principle of population on the numerical question also answered in agreement with the principle on question 12: less than 10% of those who had answered ‘Same’ to question 3 then wanted to respond differently on question 12. Of those whose responses violated the principle of population on question 3, a large proportion subsequently agreed with the principle when expressed verbally in question 13: 34% of the ‘q3 - Downs’ and 41% of the ‘q3 - Ups’ responded in this way, and 43% and 41% of each of these subgroups, respectively, explicitly indicated that they would then want to change their response on question 3 to the ‘right’ answer. So it seems as though verbal argument persuaded some who had not immediately seen the issue when confronted with the numerical example.

4.3. Transfer principle

Do people accept the principle of transfers? This is a case where one has to be extremely careful what one means. Clearly if we hypothesise a situation in which there are only two persons - rich Tweedledum and poor Tweedledee - and if we further suppose that there is a small transfer Tweedledee-rTweedledum, then we would expect any one who understands basic English to agree that such a transfer would increase inequality. However, if we consider a society that has a more complex structure than just a pair of twins, the answer may not be so blindingly obvious.15

To investigate this issue we used questions 4 and 13. Notice that in question 4 distribution B clearly Lorenz-dominates distribution A. However, casual inspection of the two income vectors might not immediately suggest the superiority of B to all observers. Notice that there is no special trick of presentation here: the incomes appear in ascending order in both cases; only one pair of incomes is affected; and the pair involves adjacent incomes. However, the pair of affected incomes does not include the richest or the poorest members of society. If people assert that in fact distribution B is more unequal than A, then we shall (provisionally) take this as strong disagreement with the transfer principle; if they regard A and B as equivalent, then we shall take this as disagreement with the principle. Question 13 invites people to respond to a specific verbal statement of the transfer principle; if they specifically reject this by picking answer (b), we shall take this as strong disagreement with the principle; answer (c) can be taken as disagreement with the principle. On this basis the results are as in table 8.

Taken overall, 38% of the sample do not agree with the transfer principle if the issue is presented in numerical or verbal form. Less than half the

‘5Pigou - who can lay claim to being the discoverer of this principle - was doubtful about its validity in this case [Pigou (1912, pp. 24-25)].


Table 8

Numerical (q4) Verbal (q13)

Strongly Strongly N Agree disagree Disagree Agree disagree Disagree

(%) (%) (%) (%) (%) (%)

N. Texas SMU LSE Bonn Karlsruhe Koblenz Ruppin Hebrew econ. Hebrew non-ccc

All

(37) (108) (106) (356)

(53) (50)

(174) (170)

In. (54)

(1,108)

22 51 27 41 38 19 32 51 14 44 29 20 45 39 16 66 23 11 31 45 24 54 28 15 45 34 19 57 30 13 46 46 8 62 26 12 33 38 29 62 24 14 42 38 19 83 7 6 24 43 33 50 28 20

35 42 22 60 24 14

Americans and half the non-economist Israeli students agree with it and - with one interesting exception to which we shall return - in each of the other subsamples less than two-thirds agree with the principle. Moreover, if the issue is presented in numerical form, then nearly two-thirds of the sample fail to agree with the transfer principle.

It could be argued that we are placing too demanding an interpretation on the responses to question 13: perhaps the 14% or so who checked (c) were not disagreeing with anything; perhaps they just had no idea what the question was about. However, even if we simply disregarded the (c) responses to question 13, the picture is not exactly encouraging to the principle of transfers. Of those who took a specific position in responding to question 13, 29% voted against the principle.

Were people persuaded by verbal argument where the numerical example had been misunderstood or misinterpreted? Of those who disagreed strongly with the transfer principle on question 4, 54% then agreed with the transfer principle when it was presented verbally; 25% continued to disagree strongly, and 18% just registered disagreement. The story was the same for those who originally just disagreed with the transfer principle (ranking A and B as equivalent): when faced with the verbal question this group then subdivided into the categories ‘Agree’, ‘Strongly disagree’ and ‘Disagree’ in the propor- tions 47%, 32% and 19%. However, of those who had originally agreed with the principle (on question 4), only 75% continued to agree when confronted with question 13.

The questionnaire gave people the opportunity to change their minds over their responses to the numerical questions. On an issue as fundamental to the study of inequality as the transfer principle, it would be particularly interesting to see whether respondents felt that, having reasoned the matter


through verbally, they might want to reconsider their answers. Only 4% of those who agreed with the principle on both questions first time around subsequently wanted to change their answer to question 4; and of those who registered disagreement or strong disagreement with the principle on both questions first time around, only 8% wanted to change their answer to question 4. Of those who ‘saw the light’ - those who did not agree with the transfer principle on their first pass through question 4 and then agreed with the principle when put in verbal form - 30% wanted to change their answer on question 4 so that it would now register agreement. That this proportion is not much higher may once again be evidence of respondents ‘playing safe’: of those whose responses differed as between question 4 and question 13, the majority did not indicate any desire to go back and change their response to question 4.

One other interesting point to emerge from the possibility of switching one’s response is as follows: only 36% of the sample were prepared to agree with the transfer principle both when expressed in the example and when stated verbally, even after allowing people to record any change of mind. In the light of this it may be that people experience a considerable difficulty translating an apparently appealing general principle into concrete examples; it may thus also be that they have difficulty translating them into actual situations.

Finally, let us come back to the obvious exception in table 8. Why was there a markedly different response of economics students at the Hebrew University on the verbal question about the transfer principle? One possible answer is that - unlike the majority of our student respondents - those at the Hebrew University are taught about the Lorenz curve as part of their introductory economics course. Perhaps the received wisdom of the pro- fession is making its mark on perceptions of inequality comparisons at an early stage in students’ intellectual development.

4.4. Decomposability

In order to investigate whether there is a consensus view on decomposability by population subgroups we need to use three questions. Questions 5 and 6 are clearly very similar problems: if respondents gave the same rankings in each of these two questions we may take this as (weak) agreement with the principle of decomposability (table 9).

Amongst the European respondents and in the Ruppin Institute the pattern is quite remarkable: on the numerical test they all split roughly 60:40 in favour of a position consistent with decomposability. But on the verbal question the position appears to be reversed. More than 60% of those registering agreement with the principle on the verbal question (question 14) had already provided answers to the numerical questions that were consis-

18 Y. Amiel and F.A. Cowell, Measurement of income inequality

Table 9

Numerical (qq 5

& 6)

N Same Different (%) K)

N. Texas (37) 43 51 SMU (108) 55 41 LSE (106) 63 36 Bonn (356) 60 39 Karlsruhe (53) 64 36 Koblenz (50) 62 38 Ruppin (174) 62 38 Hebrew econ. (170) 48 51 Hebrew non-econ. (54) 48 48

All (1,108) 57 41

Verbal (q14)

Strongly Agree disagree Disagree

(%) (%) (%)

40 40 11 40 40 11 42 50 8 36 53 10 43 49 4 46 48 6 38 42 18 44 31 16 37 35 9

40 45 11

tent with this. Even if one disregards response (c) on question 14 one still finds a substantial falling-off of support for the principle of decomposability once the issue is stated explicitly: of those who responded (a) or (b), only 47% indicated agreement with the decomposability principle.

5. Relations among axioms

Up to now we have discussed the separate axioms one by one. But what we usually consider in measurement of income inequality are systems of axioms and not isolated axioms. What can we say about the interrelationship of support for different axioms ? Do respondents who agree with one axiom agree also with others? Are there well-defined patterns of thinking?

Let us start with the commonly-used Lorenz partial ordering. Agreement with the ordinary Lorenz partial order is indicated by accepting simulta- neously scale-invariance, the principle of population, and the transfer principle, in addition to anonymity. The combination of responses to scale- invariance, the principle of population, and the transfer principle shows a very low agreement with the Lorenz ordering, both numerically (questions 1, 3, and 4) and verbally (questions 10, 12, and 13). The results are given in table 10.

It might be objected that the low acceptance of the Lorenz ordering results from the stringent requirement of simultaneous agreement on all of the issues posed by these particular questions, but this is exactly what was needed for acceptance of the Lorenz criterion. If a person rejects even one of these axioms, then he cannot accept Lorenz dominance as a general criterion.

If we replace scale-invariance by translation-invariance we get the concept

I: Amiel and F.A. Cowell, Measurement of income inequality 19

Table 10

Numerical (qq 1, 3, 4) Verbal (qq 10, 12, 13)

N Agree Disagree Agree Disagree

(%) (%) (%) (%)

N. Texas (371 8 92 16 81 SMU (iosj LSE (106) 2; Bonn (356) 13 Karlsruhe (53) 23 Koblenz (50) 12 Ruppin Hebrew econ.

(174) :: (170)

Hebrew non-econ. (54) 6

All (1,108) 14

87 9 89 75 22 73 86 24 72 75 11 89 88 16 82 88 18 79 84 26 67 94 13 83

85 20 76

Table 11

N. Texas SMU LSE Bonn Karlsruhe Koblenz Ruppin Hebrew econ. Hebrew non-econ.

All

Numerical (qq 2, 3, 4) Verbal (qq 11, 12, 13)


(%) (%) (%) (%)

(37) 3 92 5 92 (108) 5 92 15 82 (106) 9 91 14 84 (356) 5 94 13 83

(53) 6 94 9 91 (50) 10 90 22 76

(174) 3 97 12 85 (170) 7 92 10 85

(54) 7 93 19 78

(1,108) 6 93 13 83

of Lorenz domination proposed by Moyes (1987). The support for Moyes’ idea is even lower than that for the ordinary Lorenz criterion, as we can see in table 11. The addition of the decomposition axiom to the list of axioms which are equivalent to the ordinary Lorenz criterion leads to the Genera- lised Entropy family of inequality measures [see Cowell (1980)]. In this case the agreement with the axiom system is reduced by half, as we see in table 12. We can conclude that there is no wide agreement about using Lorenz curve, but once we accept the Lorenz curve, then the family of General Entropy measures is preferred.

Finally, let us consider a broader issue: Do people view inequality comparisons in terms of a well-defined ordering? For this to be so all distributions must be comparable (in terms of the notation of section 4, for all x, ~‘EX, either x2x’ or x’tx or both), and transitive. Although respondents were encouraged to write comments on the questionnaire, none


Table 12

Questions Questions 10, 12, 13” 10. 12, 13, 14”


(4;;) (Xl) 0”) (%)

N. Texas (37) 16 81 8 84 SMU (108) 9 89 5 86 LSE (106) 22 73 12 83 Bonn (356) 24 72 10 85 Karlsruhe (53) I1 89 8 89 Koblenz (50) 16 82 10 88 Ruppin (174) 18 79 6 88 Hebrew econ. (170) 26 67 11 77 Hebrew non-econ. (54) 13 83 7 76

All (1,108) 20 76 9 84

“We use the terms ‘agree’ and ‘disagree’ only for those respondents who had clear answers to all the relevant questions, so in some cases the percentage of disagreement with four axioms is less than disagreement with three of them.

of them raised the possibility of non-comparability;16 no one argued that the distributions presented in questions 1 to 9 could not be ranked. And only a very few failed to answer all of the questions. As far as transitivity is concerned, the cross-tabulations of responses to questions 7, 8 and 9 reveal that about 11% of the sample violated this property.” Of course all this does not mean that everyone has a complete transitive ordering of inequality comparisons, but nevertheless it is interesting to see that so few of our respondents provided evidence that they do not have such an ordering.

6. Conclusion

In the course of a radio interview Peter Ustinov once explained why he had obtained such bad marks in music at his English preparatory school. One had, apparently, to understand the nature of the question paper. The first question ran: ‘Who was the world’s greatest composer? The right answer to which was ‘Beethoven’. Ustinov got no marks for ‘Mozart’. To the second question - ‘Name one Russian composer’ - the right answer was ‘Tchaikowsky’: the hapless Ustinov was penalised for showing off and writing down ‘Shostakovitch’ .

Something like this seems to apply to the results from our questionnaire. It is interesting to notice which of the questions induce the unconventional

160n the other hand, we did not actually suggest it to them. “Questions 7, 8, 9 and 15 also raise several other issues connected with the work of Temkin

(1986) which are fully explored in Amiel and Cowell (1990).


responses. To summarise roughly, on the more pedestrian axioms of inequality analysis (scale- or translation-irrelevance, population replication) one seems to get the sort of responses that would correspond to conventional views. However, on the two axioms that play such an important part in the logic and structure of inequality measures, there do appear to be some difficulties. Respondents are split almost down the middle on whether it is possible to decompose inequality. More controversially, many do not appear to agree with the transfer principle, as it is usually expressed; and since this principle is at the heart of inequality analysis, such a finding could have disturbing implications.

What is one to make of respondents who consistently give the ‘wrong’ answers in questionnaires about attitudes to economic inequality? For economists who have long been trained in the received wisdom of inequality analysis it would perhaps be easy to react like Ustinov’s music teacher: ‘no marks for Mozart’. But the very persistence of the ‘Mozart’ and ‘Shostakovich’ answers suggest that we take them more seriously. Were the respondents perhaps drawing our attention to important aspects of inequality which conventional wisdom has neglected?‘* Or were they just being capricious, careless or deliberately stupid? We have all come across vapid and irreflective students, and those who take delight in being contrary, so it would be disturbing if there were evidence that our questionnaire responses had been contaminated by such traits. As it happens there are some pairs of questions that can afford us insights as to whether this was a significant problem in the present case.

Consider again the answers to questions 3 and 12 about the effects of population replication. This is one instance where the wording of the verbal question matches extremely closely the precise structure of the numerical problem. As we have seen, there was a substantial majority that appears to concur with the principle, but on both the numerical and the verbal questions there was a significant dissenting minority. Did the members of this minority just fail to understand the questions? We cannot answer that directly, but we can deal with a related problem that has a bearing on the issue: Were people’s answers self-contradictory? On the whole they were not: after they had been allowed to have second thoughts, just under 4% of the sample indicated both ‘Up’ and q3 and ‘Down’ on q12 or vice versa (on the first time around the figure had been just under 5%); this would seem to be an acceptably low proportion of silly responses.

So, if the unconventional replies are not to be dismissed as merely

“The (few) written comments we received suggested that people were thinking carefully about alternative views; as an example, take the student who said that he would have concurred with the transfer principle had the richest or the poorest members of the society been involved in the transfer.


unperceptive or unreasoning replies, what is to be learned from them? Consider the role played by cultural differences in all this. If ‘culture’ is interpreted as ‘country’, then such differences did not have a great impact, although the Americans in our sample clearly responded in a less conventional way than the rest. However, there may be another type of ‘cultural’ factor at work: subject specialism does appear to be important in influencing respondents’ thinking. The extreme example of this phenomenon is found in the evidence from the Hebrew University on the transfer principle, where the non-economists were much less inclined than the rest of our sample to support the principle, and the economics students (brought up on the Lorenz curve) responded ultra-conventionally.

Appendix: Income inequality questionnaire

This questionnaire concerns people’s attitudes to income inequality. We would be interested in your views, based on some hypothetical situations. Because it is about attitudes there are no ‘right’ answers. Some of the suggested answers correspond to assumptions commonly made by economists: but these assumptions may not be good ones. Your responses will help to shed some light on this, and we would like to thank you for your participation. The questionnaire is anonymous: please do not write your name on it.

In each of the first nine questions you are asked to compare two distributions of income. Please state which of them you consider to be the more unequally distributed by circling A or B. If you consider that both of the distributions have the same inequality, then circle both A and B.

Z Amiel and F.A. Cowell, Measurement of income inequality 23

1. A=(5, 8, 10) B=(lO, 16,20)

2. A=(5,8, 10) B=(lO, 13, 15)

3. A=(5, 8, 10) B=(5, 5, 8, 8, 10, 10)

4. A=(l, 4, 7, 10, 13) B=(l, 5, 6, 10, 13)

5. A=(4,8,9) B=(5, 6, 10)

6. A=(4, 7, 7, 8, 9) B=(5, 6, 7, 7, 10)

7. A=(5, 5, 5, 10) B=(5, 5, 10, 10)

8. A=(5, 5, 10, 10) B=(5, 10, 10, 10)

9. A=(5, 5, 5, 10) B=(5, 10, 10, 10)

In each of questions 10 to 14 you are presented with a hypothetical change and three possible views about that change, labelled (a), (b), (c). Please circle the letter alongside the view that corresponds most closely to your own. Feel free to add any comments which explain the reason for your choice.

10. Suppose we double the ‘real income’ of each person in a society, when not all the initial incomes are equal. (a) Each person’s share remains unchanged, so inequality remains

unchanged. (b) Those who had more also get more, so inequality has increased. (c) After doubling incomes more people have enough money for basic

needs, so inequality has fallen.

In the light of the above, would you want to change your answer to question l? If so, please write your new response - ‘A’ or ‘B’ or ‘A and B’ (if you now consider the two distributions to have the same inequality):

11. Suppose we add the same fixed amount to the incomes of each person in a society, when not all the initial income are equal. (a) Inequality has fallen because the share of those who had more has

fallen. (b) Inequality remains the same. (c) Inequality has increased.

Suppose instead of adding we deduct a fixed amount from each person’s income. Then inequality

24 E Amiel and F.A. Cowell, Measurement of income inequality

(a) is the same (b) increases (c) decreases.

In the light of both of the above, would you want to change your answer to question 2? If so, please write your new response (‘A’ or ‘B’ or ‘A and B’) here:

12. Suppose we replicate a three-person society by merging it with an exact copy of itself (so that we now have a society of six people consisting of three sets of identical twins). (a) The income inequality of the six-person community is the same as

that of the three-person community because the relative income shares remain unchanged.

(b) The income inequality of the six-person community is less than that of the three-person community because in the six-person community there are some people who have the same income.

(c) The income inequality of the six-person community is greater than that of the three-person community.

In the light of the above, would you want to change your answer to question 3? If so, please write your new response (‘A’ or ‘B’ or ‘A and B’) here:

13. Suppose we transfer income from a person who has more income to a person who has less, without changing anyone else’s income. After the transfer the person who formerly has more still has more. (a) Income inequality in this society has fallen. (b) The relative position of others has also changed as a consequence of this

transfer. Therefore we cannot say, a priori, how inequality has changed.

(c) Neither of the above.

In the light of the above, would you want to change your answer to question 4? If so, please write your new response (‘A’ or ‘B’ or ‘A and B’) here:

14. Suppose there are two societies, A and B, with the same number of people and with the same total income, but with different distributions of income. Society A is now merged with C, and society B is merged with C’ where C and C’ are identical. (a) The society which had the more unequal income distribution before

the merger still has the more unequal distribution after the merger. (b) We can’t say which society has the more unequal income distribution

unless we know the exact distributions. (c) Neither of the above.


In the light of the above (and your answer to question 5) would you want to change your answer to question 6? If so, please write your new response (‘A or ‘B’ or ‘A and B’) here:

15. Suppose there is a society consisting of n people. There is one rich person and n- 1 identical poor people. One by one, some of those who were poor acquire the same income as the rich person, so that eventually there are n- 1 (identical) rich people and just one poor person. Please circle the appropriate response: (a) Inequality increases continuously. (b) Inequality decreases continuously. (c) Inequality at first increases and then decreases, (d) Inequality at first decreases and then increases. (e) Inequality remains the same throughout. (f) None of the above.

In the light of the above would you want to change your answer to questions 7, 8 and 9? If so, please note your new responses here . . .

7: 8: 9:

Please write your special subject here:

Thanks once again for your help!

References

Amiel, Y. and F.A. Cowell, 1990, Inequality changes and income growth, TIDI discussion paper No. 150, London School of Economics.

Amiel, Y. and F.A. Cowell, 1991a, Income inequality and social welfare, London School of Economics, Mimeo.

Amiel, Y. and F.A. Cowell, 1991b, Income transformation and income inequality, London School of Economics, Mimeo.

Atkinson, A.B., 1970, On the measurement of inequality, Journal of Economic Theory 3, 244263.

Bossert, W. and A. Ptingsten, 1990, Intermediate inequality concepts, indices and welfare implications, Mathematical Social Sciences 19, 117-134.

Brittan, S.E., 1973, Is there an economic consensus? (Macmillan, London). Champernowne, D.G., 1974, A comparison of measures of income distribution, Economic

Journal 84, 787-816. Cowell, F.A., 1977, Measuring inequality (Philip Allan, Oxford). Cowell, F.A., 1980, On the structure of additive inequality measures, Review of Economic

Studies 47, 521-531. Cowell, F.A., 1985, A fair suck of the sauce bottle - or what do you mean by inequality?,

Economic Record 6, 567-579. Cowell, F.A., 1988, Inequality decomposition - three bad measures, Bulletin of Economic

Research 40, 309-312.


Dalton, H., 1920, Measurement of the inequality of incomes, The Economic Journal 30, 348361. Dasgupta, P.S., A.K. Sen and D.A. Starrett, 1973, Notes on the measurement of inequality,

Journal of Economic Theory 6, 18&187. Ebert, U., 1988a, A family of aggregative compromise inequality measure, International

Economic Review 29, no. 5, 363-376. Ebert, U., 1988b, Measurement of inequality: An attempt at unification and generalisation,

Social Choice and Welfare 5, 147-169. Eichhorn, W. and W. Gehrig, 1982, Measurement of inequality in economics, in: B. Korte, ed.,

Modern applied mathematics - optimization and operations research (North-Holland, Amsterdam) 657-693.

Foster, J.E., 1983, An axiomatic characterization of the Theil measure of income inequality, Journal of Economic Theory 31, 105121.

Gevers, L., H. Glejser and J. Rouyer, 1979, Professed inequality aversion and its error component, Scandinavian Journal of Economics 81, 238-243.

Kakwani, N.C., 1980, Income inequality and poverty (Oxford University Press, London). Kolm, S.Ch., 1969, The optimal production of social justice, in: J. Margolis and H. Guitton, eds.,

Public economics (Macmillan, London). Kolm, S.Ch., 1976a, Unequal inequalities I, Journal of Economic Theory 12,41&442. Kolm, S.Ch., 1976b, Unequal inequalities II, Journal of Economic Theory 13, 82-l 11. Kuga, K., 1973, Measures of income inequality: An axiomatic approach, Discussion Paper 76,

Institute of Social and Economic Research of Osaka University. Miller, H.P., 1970, Recent trends in family income, in: E.O. Laumann, P.M. Siegel and R.W.

Hodge, eds., The logic of social hierarchies (Markham Publishing Company, Chicago). Morawetz, D., 1977, Income distribution and self-rated happiness, some empirical evidence,

Economic Journal 87, 51 l-522. Moyes, P. 1987, A new concept of Lorenz domination, Economic Letters 23, 203-207. Nygard. F. and A Sandstrom, 1981, Measuring income inequality (Almqvist and Wiksell

International, Stockholm). Pfmgsten, A., 1988, Empirical investigation of inequality concepts: A method and first results,

Karlsruhe University, Mimeo. Pigou, A.C., 1912, Wealth and welfare (Macmillan, London). van Praag, B.M.S., 1977, The perception of welfare inequality, European Economic Review 10,

no. 11, 1899207. van Praag, B.M.S., 1978, The perception of income inequality, in: W. Krelle and A.F. Shorrocks,

eds., Personal income distribution (North-Holland, Amsterdam). Pyatt, G., 1985, An axiomatic approach to the Gini coefficient and the measurement of welfare,

in: R.L. Basmann and G.G. Rhodes, eds., Advances in econometrics, vol. 4 (JAI Press, London).

Sen, A.K., 1973, On economic inequality (Clarendon Press, Oxford). Shorrocks, A.F., 1984, Inequality decomposition by population subgroups, Econometrica 52,

136991385. Shorrocks, A.F., 1988, Aggregation issues in inequality measurement, in: W. Eichhorn, ed.,

Measurement in economics (Physica Verlag, Heidelberg). Temkin, LX, 1986, Inequality, Philosophy and Public Affairs 15, 999121. Thon, D., 1982, An axiomatization of the Gini coeflicient, Mathematical Social Science 2,

131-143. Tversky, A. and D. Kahneman, 1974, Judgement under uncertainty: Heuristics and biases,

Science 185, 1124-l 131. Yaari, M. and M. Bar Hillel, 1984, On dividing justly, Social Choice and Welfare 1, l-24.

Measurement of income inequality*

Documents

Transcript of Measurement of income inequality*