MULTIDIMENSIONAL INEQUALITY MEASUREMENT WITHIN THE HUMAN DEVELOPMENT FRAMEWORK

KOEN DECANCQ

THIS VERSION: NOVEMBER 2004 WORK IN PROGRESS

Abstract: In this paper I investigate how the two step procedure proposed by Maasoumi (1986, 1989 and 1999) can be used to measure multidimensional inequality. Therefore an application, based on 2003 data, is provided with a sensitivity analysis for different parameters and standardization methods. Furthermore, I look into the mutual enrichment of Maasoumi’s approach and the Human Development paradigm, developed under the auspices of the UNDP. Notwithstanding some unresolved theoretical issues, Maasoumi’s two step approach seems to be an intuitive appealing and applicable procedure to multidimensional inequality measurement.

1

1. Introduction Traditionally, the literature on inequality measurement has concerned itself with the comparison of one-dimensional indicators of economic well-being, in particular with the measurement of income inequality. Seminal contributions of Kolm (1977), Atkinson and Bourguignon (1982) and Sen (1985, 1992, 1997) have brought under attention that income as sole indicator of well-being is not necessarily a good one. Problems arise especially when populations are socially heterogeneous, i.e. when people differ in dimensions other than income or wealth. Sen (1992) says: “An important and frequently encountered problem arises from concentrating on inequality of incomes as the primary focus of attention in the analysis of inequality. The extent of real inequality of opportunities that people face cannot be readily deduced from the magnitude of inequality of incomes, since what we can or cannot do, can or cannot achieve, do not depend just on our incomes but also on the variety of physical and social characteristics that affects our lives and make us what we are.” (Sen, 1992, pp. 28).

A popular solution to the problem of social heterogeneity has been to use equivalence scales. Deflating the raw money income by its equivalence scale rate converts the income distribution of each heterogeneous population into a homogenous distribution of equivalent income. A great range of inequality measures can be applied to the equivalent incomes. This approach makes several implicit assumptions about the relationship between income and the physical and social characteristics. There may not be a wide agreement about what these appropriate assumptions should be (the continuing co-existence of many different equivalence scales illustrates this view). (Jenkins and Lambert, 1993). In this paper another approach is followed to settle the social heterogeneity problem. We take the relevant characteristics explicitly into account and calculate a multidimensional inequality measure, which summarizes the inequalities with respect to different attributes of well-being. The design of a multidimensional inequality measure seems to be a logical extension of the research on multidimensional indices of well-being and development. Under the auspices of the United Nations Development Programme (hereafter UNDP), the

2 human development concept is proposed as framework of thought to develop such a broad index of well-being. In his review article Maasoumi (1999) describes three possible roads towards multidimensional inequality measurement. The two first methods generate a multivariate inequality index and provide a total ordering, whereas the last approach leads to a partial ordering. The first method is proposed by Maasoumi (1986, 1989) and invokes a “two-step” method. In the first step insights from information theory are used to aggregate the different attributes into a summary measure of well-being. The second step applies a univariate inequality index (such as the Generalized Entropy index) to the aggregates to obtain an inequality measure. Second, Tsui (1995, 1999) and Ebert (1995) have formulated some “one-step” measures of multidimensional inequality based on a set of axioms suited to the multivariate setting. The identification of a set of commonly acceptable axioms has proven to be more elusive than in the univariate case, though. The third approach, which is based on sequential generalized Lorenz dominance, is advocated by Atkinson and Bourguignon (1982, 1987, and 1989). Conditions are derived for the generalized Lorenz dominance of one distribution over another on the basis of welfare functions which are separable for different population groups. This separation allows for different welfare evaluations for different groups which are identified by their non-income characteristics. This paper is an application of Maasoumi’ s two step approach to global inequality and investigates how Maasoumi’ s approach can enrich the human development concept and vice versa. I opted for the two step approach for its intuitive appeal and the existence of some empirical applications in the literature1. In section two the human development paradigm is presented as the underlying theoretical framework of this application. Section three goes deeper into the particular formulation of Maasoumi’ s approach and its relation

1

See, for example, Maasoumi and Jeong (1985); Maasoumi and Zandvakili (1986 and 1989); Maasoumi and Nickelsburg (1988); Hirschberg, Maasoumi and Slottje (1991).

3 with information theory. In section four the results about global inequality are provided. Section five rounds off with a conclusion.

4

2. The human development paradigm The concept of human development has organically been built up in the annual Human Development Reports (HDR) and background papers published by the UNDP from 1990 onwards. Theoretically, the concept of human development derives from two distinct strands in development studies (Desai, 1991). First, it traces back from the literature on inequality and poverty, pioneered by Sens contributions on the measurement on poverty (1976), his later work on entitlements and his approach based on functionings and capabilities2. Illustrative for the dependence of the concept of human development on Sens capability approach is the opening sentence of 2002 HDR: “ Human development is about people, about expanding their choices to lead lives they value […] Fundamental to enlarging human choices is building human capabilities: the range of things that people can do or be.” (UNDP, 2002)

The other strong root of the paradigm is the concern that income and growth are not the sole criteria of development. Growth in income is considered as being necessary but not sufficient for human development. Early contributions to noneconomic indicators of development are from UNICEF (child related measures) and David Morris (1979) who developed the Physical Quality of Life Index (PQLI). Central in the concept of human development is the rejection of the traditional emphasis on consumption of commodities3. As alternative an individual wellbeing function is proposed, which relies on basic capabilities or functionings directly rather than translating them back to utilities first. The human development paradigm can be considered non-welfarist as defined by Sen (1977).

2

See, for example, Sen on entitlements (1981) and on functionings and capabilities (1985, 1987 and 1992).

3

This concentration on commodities is called commodity-fetishism by Sen (1985).

5 The first Human Development Report in 1990 introduced the Human Development Index (HDI). This index became the flagship and signboard of the HDR’ s. The HDI summarizes the achievements of a country in three basic dimensions of human development: •

The longevity of the human life, as measured by the life expectancy at birth. This indicator measures the life expectancy of the newly born cohort and is meaningful for populations, but the life expectancy figures cannot be applied directly to the other cohorts of a country or to individuals. (Pradhan, Sahn and Younger, 2001).

•

The education, as measured by the weighted average of the adult literacy rate (with two-thirds weight) and the combined primary, secondary and tertiary gross enrolment ratio (with one-third weight). These weights are obtained in an arbitrary way.

•

The standard of living, as measured by the logarithm of the GDP per capita (in PPP US$)4. The logarithmic transformation of income is taken in order to reflect diminishing returns of transforming income into human capabilities (Anand and Sen, 2000 p. 87). This concave transformation captures the viewpoint that income is essentially seen as means to decent living and not as an end.

Performance in each dimension is mapped on a real number between 0 and 1 by applying formula (2.1). Arrow (1963, p.32) refers to this normalization as the Kaplan normalization. Maximum and minimum values for each indicator are given in table 1. Dimension index =

actual value - minimum value maximum value - minimum value

(2.1)

The HDI is then calculated as a simple average of the dimension indices. Taking the average implicitly assumes the dimensions to be perfect substitutes, since a bad performance on one dimension can be compensated by a good performance on another dimension.

4

In the calculation of the HDI a correction for PPP (purchasing power parity) has been made, contrary to the

income inequality calculations in the HDR’ s. The choice whether to make PPP corrections or not can have important influence on the final results as illustrated by Melchior, Telle and Wiig (2000).

6 Table 1 Goalposts for calculating the HDI (source: Human Development Report, 2003)

Indicator

Maximum value

Minimum value

Life expectancy at birth (years)

85

25

Adult literacy rate (%)

100

0

Combined gross enrolment ratio (%)

100

0

Logarithm of GDP per capita (PPP US$)

LOG (40 000)

LOG (100)

An intuitive appealing way of analyzing multidimensional inequality would be to look at the inequality in the HDI for the different countries. This approach suffers unfortunately from two important drawbacks. First, Appendix 1 makes clear that only the membership function of the knowledge dimension is a straight line through the origin or a “ pure” rescaling of the underlying indicator. The other two dimensions have a more complex membership function which influences the inequality of the index compared to the underlying indicator. This muddling makes the HDI a rather opaque tool for inequality measurement. Another flaw of the HDI for multidimensional inequality measurement is its crudity (Sakiko Fukuda-Parr, 2001). The addition of education and life expectancy has broadened undeniable the scope compared to GDP alone. The sole concentration of the HDI on these three dimensions, however, has led to a narrow interpretation of the human development concept, if it were exclusively about education and survival. The ideal multidimensional inequality measure should incorporate other dimensions of human development as well. Sakiko Fukuda-Parr suggests adding information about political and civil rights. Concluding, we can say that the human development concept offers an attractive theoretical framework for a broad analysis of global inequality. Unfortunately, the workhorse of the human development concept, the HDI, is not very useful for inequality analysis. In the following section we investigate an alternative based on the two-step approach of Maasoumi. This approach will provide us with a tool to use the human development framework for inequality measurement that goes beyond income and beyond the HDI as well.

7

3. Maasoumi’s two step approach In this section we describe in detail the two step approach to multidimensional inequality measurement as proposed by Maasoumi (1986, 1989 and 1999). First we define the distribution matrix X, introduced by Fisher (1956). Let the NxM distribution matrix Xif be an element of X, denoting the amount of attribute f = 1,…, M received by individual (household, country, etc) i = 1,…, N. Furthermore, let Xi = (Xi1, … XiM)’ be the ith row of the matrix X and Xf = (X1f, X2f, …, XNf)’ the fth column of the distribution matrix. The summary or aggregate attribute function is denoted by Si = h(Xi).

3.1 First step: Aggregation procedure In the first step the different relevant attributes are aggregated into a summary measure Si for every country i. Crucial for Maasoumi’ s approach is the following implicit assumption about aggregate Si: Assumption 3.1: It is reasonable to require the aggregate Si to have a distribution that is as “ close” as possible to the multivariate distribution of the Xf’ s.

This assumption seems reasonable as long as we limit our interpretation of Si to “ the aggregate function approaching the multivariate distribution of the Xf’ s” . Maasoumi however, goes further and interprets the Si as the standard of living or the utility function of the i-th unit5. In this paper I stick to the first interpretation, thereby staying away from interpreting Si as a well-being or utility function. Information theory provides us with a measure for the closeness of two distributions. The generalized Kullback-Leibler criterion measures the pairwise similarity or distance

5

For welfare economists, this interpretation may look like a bridge too far since it is not clear on which

welfare axioms Maasoumi’ s claim is based. Explicit clarification of the underlying welfare axioms seems a possible extension and deepening of Maasoumi’ s theory. This axiomatization goes beyond the scope of this paper.

8 between distributions, defined in formula (3.1). The Kullback-Leibler distance is weakly positive, approaching 0 in the case of perfect similarity. β  N    S i  ∑ Si   − 1   i =1  X if   M    Dβ ( S , X ; α ) = ∑ α f   β (β + 1) f =1      

(3.1)

,IZHVHW  ZHREWDLQ M N  D0 ( S , X ; α ) = ∑ α f ∑ Si log( Si / X if )  f =1  i =1 

(3.2)

M N  D−1 ( S , X ; α ) = ∑ α f ∑ X if log( X if / Si )  f =1  i =1 

(3.3)

$QGLI  -1, this gives

Shortly, the divergence criteria are the

f

weighted averages of pairwise Generalized

Entropy-divergences between the “ distributions” of S and Xf. Since we want these divergences as small as possible, we minimize D 6 ;  ZLWK UHVSHFW WR 6 After a derivation this gives us the following distribution of Si: M  Si ∝  ∑ δ f X if− β   f =1 

−1/ β

whereby δ f =

αf

(3.4)

M

∑α f =1

f

When available, Maasoumi (1989, p. 140) suggests deriving weights

f

based on market

prices. Market prices generally do not exist for the attributes of well-being, so other approaches are needed. Maasoumi advocates the method of principal components proposed by Ram (1982). The vector of weights

f

is then the normalized form of the first

eigenvector of the attributes’ correlation matrix, which accounts for the largest fraction of the variability in the data. This method, however, seems not to have any welfare-theoretic justification. These weights can only provide a benchmark for other determinations of the weights.

9 For positive Xif’ s the index Si is dHILQHGIRUDOO YDOXHVZLWKWKHOLPLWLQJFDVHVGHVFULEHG in Table 2. If the Xif’ s are allowed to be negative, as some standardized values typically are6, SiLVPDWKHPDWLFDOO\RQO\GHILQHGIRU YDOXHVIRUZKLFK β 2 ∈ Z hold, where Z is the set of integers. This limiting restriction can be by-passed by requiring the Xif’ s to be positive. For empirical applications7 alternative standardization procedures have to be developed. 3DUDPHWHU DFFRXQWVIRUWKHGHJUHHRIVXEVWLWXWLRQEHWZHHQWKe attributes. We can equalize

β = −1 + 1 σ which illustratesWKHFORVHUHODWLRQEHWZHHQ DQGWKHHODVWLFLW\RIVXEVWLWXWLRQ

Figure 1 depicts the iso-index curves8IRUGLIIHUHQWYDOXHVRI LQDWZRDWWULEXWHFDVH,I

HTXDOV-RU JRes to +’ DWWULEXWHVDUHVHHQDVsubstitutes. A bad performance on one attribute can be compensated by a good performance on another attribute. Large positive

YDOXHVRI RU DSSURDFKLQJIURPWKHULJKW FDQEHLQWHUSUHWHGDVVHHLQJWKHDWWULEXWHVDV complementary. When the attributes are perfect complements, only the worst performance is taken into account when calculating Si&RQYHUVHO\ FDQWDNHDODUJHQHJDWLYHYDOXH 

approaching 0 from the left), which leads to a concave iso-LQGH[FXUYH,I DSproaches -’ only the best performing attribute is relevant for the calculation of the index. In this case the marginal rate of substitution between the attributes is increasing for increasing attribute values, which is an unfamiliar result in economics9.

6

7

Following the standardization procedures Z =

X −µ

σ

, for example.

Most empirical applications are about income mobility, where every individual or country is characterized

by an income vector, build up by the incomes at different moments in time. The income data are all positive and are not standardized first. See, for example, Maasoumi and Jeong (1985), Maasoumi and Zandvakili (1986 and 1989). An application based on attributes of well-being is of Hirschberg, Maasoumi and Slottje (1991), they make a sensitivity analysis for different standardization procedures. An analogous sensitivity analysis is carried out in the application in section 4. 8

An iso-index curve can be defined as a curve that connects all combinations of attributes which lead to the

same index value Si. 9

In consumption theory, concave indifference curves are rare, but in some extreme addiction cases they can

be noted, as well.

Attribute 2

10

’

-’

-1 Attribute 1

Figure 1 Iso-index curves of different parameter values of 

A simple example can clarify: Little Thomas, 6 years old, returns home with his first school report. Since Thomas is a first year schoolboy, he has only two marks on his report: mathematics and reading. Thomas has a very good mark for mathematics and a very bad for reading. Thomas first shows his report to his mother (whose HTXDOV-1). His mother calculates the average of the marks and finds the report middling. His severe father (who’ s  LV ’  RQ WKH RWKHU KDQG KDV RQO\ H\H IRU WKH EDG PDUN IRU UHDGLQJ DQG LQIOLFWV D

punishment on Thomas. Finally Thomas shows his report to his grandmother (whose LV’ *UDQGPDLVYHU\SOHDVHGZLWKWKHJRRGPDUNIRUPDWKHPDWLFVDnd buys Thomas a sack

RIH[WUDVZHHWV7KLVOLWWOHH[DPSOHVKRZVWKHLPSRUWDQFHRIWKH SDUDPHWHUZKHQPDNLQJ evaluations, no matter whether it is about a first year school report or complicated wellbeing indices. Equation (3.4) is the harmonic mean of the considered attributes. The weighted geometric mean and the weighted mean, popular indices in economics can be obtained by choosing SDUDPHWHU . Furthermore, by choosing particular values for the SDUDPHWHU the formula for Si equals many popular utility functions in economics, such as the CES, Cobb-Douglas and the linear utility function. This makes it only more tempting to interpret Si as a utility function. An overview of some special cases is provided in Table 2.

11 Table 2 Overview of Maasoumi’s aggregation procedure

Divergence criterion D (.)

‘ideal’ aggregator

0

1

N  α ∑ f ∑ S i log( Si / X if )  f =1  j =1 

Si = ∏ X if f

-1

’

N  α ∑ f ∑ X if log( X if / S i )  f =1  j =1 

Si = ∑ δ f X if

 N  S  α f ∑ Si  i ∑  f =1  j =1  X if

M  Si ∝  ∑ δ f X if− β   f =1 

M

  -1

’

M

M

β       − 1 / β (β + 1)    

M

Utility function

δ

Cobb – Douglas

f =1

M

Linear

f =1

−1/ β

CES

3.2 Second Step: Inequality Measurement of Si

(

)

In the second step the relative inequality of the aggregate S = S 1 ,..., S N is measured. We measure it by an index of the Generalized Entropy (GE) class (see formula (3.5) - (3.7)).

3DUDPHWHU  UHSUHVHQWV WKH ZHLJKWV JLYHQ WR WKH GLVWDQFHV EHWZHHQ LQFRPHV DW GLIIHUHQW parts of the incRPHGLVWULEXWLRQ)RUORZHUYDOXHVRI *(LVPRUHVHQVLWLYHWRFKDQJHVLQ

the lower tail of the distribution, and for higher values GE is more sensitive to changes that affect the upper tail. It is instructive to analyze this measure in the discrete case.  S ∗ 1+γ  pi  i  − 1 ∑  pi   i =1 I γ (S ) = LI -1 γ (γ + 1) N

(3.5)

N  S∗  I 0 ( S ) = ∑ Si∗ log  i  Theil’ s first index i =1  pi 

(3.6)

N p  I −1 ( S ) = ∑ pi log  ∗i  Theil’ s second index i =1  Si 

(3.7)

Thereby Si∗ = Si K with K = ∑ Si and pi is the i-th unit’ s population share (typically pi = 1 N )10. Under the assumption of equal population shares formula (3.5) can be written as: 10

The assumption of equal population shares is considered more deeply in section 4.4.

12

N

I γ (S ) =

1 ∑ i =1 N

 S 1+γ   i  − 1  Si   γ (γ + 1)

(3.8)

Parallels can be drawn between this approach and the traditional solution to the social heterogeneity problem based on equivalence scales. Whereas the equivalence scale approach takes the non-income characteristics implicitly into account by the choice of the equivalence scale, Maasoumi’ s approach explicitly looks at the non monetary characteristics to obtain an aggregate. In the final step of both approaches a univariate inequality measure is used to analyze the equivalized incomes and aggregates Si respectively.

3.3 Properties of the two step approach In this paragraph we investigate whether the two step method satisfies the axiomatic principles which are thought to be desirable for a good index of inequality. Cowell (1995 p. 54) highlights four principles in the univariate case: Principle of transfers In a one-dimensional setting the transfer principle states that a good inequality measure should decline in response to a mean preserving redistribution from rich to poor. Although straightforward in a one-dimensional setting, the multidimensional generalization of this principle is more complicated. Kolm (1977) suggests making use of a bistochastic matrix11, as mean preserving equalizing device. Kolm defines X to be strongly more equal than X if there exists a bistochastic matrix B such that X = BX , whereby B is not the permutation matrix P12. Smoothing a multivariate distribution with a bistochastic matrix should lead to a

11

A nonnegative square matrix is bistochastic if all of its row and column sums are equal to 1.

12

A Permutation Matrix, P, is obtained by permuting the ith and jth rows of the identity matrix. Every row

and column therefore contains precisely a single 1.

13 decrease of the inequality. Weymark (2003) calls this result the Uniform Matrix Majorization. In his 1986 article Maasoumi introduces the following proposition stating that the principle of transfers holds for his two step procedure: Proposition 3.1 If X = BX, where B is a bistochastic matrix, the Iγ ( S ) ≤ Iγ ( S ) for all γ ∈ R and any h(.) ∈ H , such that S i = h( X ) and Si = h( X ) . Hereby the set H contains all concave, non-decreasing aggregate functions and is R the set of real numbers. Following this proposition the inequality I (S) would decrease after a premultiplication with a bistochastic matrix. Maasoumi (1986, p. 995) sketches a proof using Kolm (1977, theorem 6).

In a short paper Dardanoni (1995) provides a counterexample to proposition 3.1. In his example the smoothing process with a bistochastic matrix leads to an increase in inequality. Take:

0 10 10 10 1 0 10 10 10      x = 10 90 10 , B = 0 0.5 0.5 so that x = Bx = 50 50 10  90 10 10 0 0.5 0.5 50 50 10  Choose any symmetrical concave function S i ( x j ) that denotes the utility function S for individual i based on attribute vector j. Then S 2 ( x2 ) = S 3 ( x3 ) ≤ S 2 ( x 2 ) = S 3 ( x 3 ) whereby any Schur-convex inequality index will increase after the transformation. In fact, after the smoothing, the two richer individuals are made better off so that one should argue that inequality is actually increased after a bistochastic transformation. This counterexample shows that proposition 3.1 is not correct and that therefore the transfer principle not holds generally for the results of the two-step approach. There exist, more accurately, an incompatibility between the generalization of the transfer principle using a bistochastic matrix and Maasoumi’ s proposition 3.1. In the literature two different answers to this incompatibility are given.

14 First, according to Dardanoni (1995) the bistochastic transformation as proposed by Kolm (1977) can not be used to impose restrictions on the class of allowed summary utility functions. He concludes that the bistochastic transformation is “ uninformative for evaluating the amount of inequality in society” Dardanoni (1995, p. 202). As alternative to the bistochastic transformation, Dardanoni proposed the unfair rearrangement. He defines the unfair rearrangement as a rearrangement so that the first individual gets the lowest amount of all attributes, the second individual the second lowest amount and so on. An unfair rearrangement should not increase social welfare. Dardanoni proves that this requirement implies the strong restriction of additive separability on the class of summary functions. Second, in a recent article Weymark (2003) takes the bistochastic transformation for granted and uses the counterexample of Dardanoni to question the suitability of Maasoumi’ s two-step approach to multidimensional inequality measurement. Weymark (2003, p. 20) states: “ A more natural conclusion [than the one of Dardanoni] would be to question the appropriateness of Maasoumi’ s two-stage aggregation procedure for constructing multidimensional inequality indices.” Until now Maasoumi has only reacted on Dardanoni’ s article. His reaction (1999) is twofold. First he admits that the strong result he derived in his 1986 paper, only holds for a limited range of S functions since I (S) does not contain all Schur-concave social welfare functions and is not everywhere increasing. He nuances proposition 3.1 in a less strong proposition. Proposition 3.2: Let X = BX , where B is a bistochastic matrix. Then W ( S ) ≥ W ( S ) for all Schur-concave W(.), and all positive real valued concave h(.) ∈ H , such that S i = h( X ) and Si = h( X ) .

Secondly, Maasoumi reacts to Dardanoni’ s alternative based on unfair rearrangements. He points at the implicit assumption of Dardanoni that all attributes are valued equally by all individuals and that they are perfect substitutions (setting

 -1). Moreover, Maasoumi’ s

investigations have shown that additive separability is not required over a wide range of QHJDWLYHYDOXHVIRU DQGFRPPRQYDOXHVRI 

15 Concluding we can say that up till now there is no agreement in the literature on whether the two step approach satisfies the transfer principle or not. More fundamentally there seems to be disagreement on the proper multidimensional generalization of the transfer principle itself. Income scale independence - homogeneity In the one-dimensional context, the scale invariance principle states that a desirable measure should be homogenous of degree zero. That is, if we rescale all the incomes by the same number, the measure of inequality should not change and becomes invariant to the units in which incomes are measured. In the multivariate context, various invariance properties are proposed. Homotheticity. For all X, Y ∈ D DQGDOO !; ; Y if and only if ; ; 0

Strong homotheticity was proposed by Tsui (1995). If the distribution matrix Xif is defined as a matrix of shares13 of attributes, I (S) satishies the strong homotheticity propery and is made invariant to the scale of every attribute separately. This requires the rather unusual assertion that individual well-being depends on shares of attributes. Furthermore the appeal of this strong property can be questioned when applied to inequality measurement. See, for example, Bourguignon (1999, p. 479) . Principle of population. For all n ∈ N , X , Y ∈ D , Iγnk (Y ) = I γn ( X ) , where Y is the k-fold replication of X, k ≥ 2 This principle postulates that the ranking of two income distributions is unchanged if these distributions are replicated. (Dalton, 1920). This property allows us to compare inequality

13

A similar standardization procedure is defined in equation (4.4) as the Z2 procedure.

16 of distributions over different population sizes. Maasoumi’ s two step procedure satisfies this principle. Decomposability A decomposable inequality measure can be written down as the sum of the inequality within groups and inequality across groups. Maasoumi (1986 and 1999) proves a far reaching result about the decomposability of I (S). The result holds under the restriction    - , establishing a relation between the risk aversion and the degree of substitution. This restriction is unfortunately hard to interpret. The core result decomposes the multidimensional inequality in inequality within each attribute and the inequality across the attributes. Formally Maasoumi proves the following. LI  - the following result holds: M

( )

I γ ( S ) = ∑ δ f W f1+γ I γ X f =1

whereby W f =

Tf K

f

M



 f =1



+  ∑ δ f W f1+γ − 1 γ (γ + 1)

(3.9)

N

, K = ∑ Si and T f = ∑ X if i

i =1

This is clearly an interesting and powerful result, but its Achilles’ heel is the initial restriction, which is hard to interpret in a convincing way.

17

4. Application 4.1 Data and Attributes In this section I apply Maasoumi’ s approach to multidimensional global inequality. Therefore I make use of data from the World Bank (World Development Indicators, hereafter WDI 2003) and the UNDP (Human Development Report, 2003) supplemented with the Freedom House report 2003. Data have been collected for 18 attributes of wellbeing for 175 countries. The attributes are tabulated in Table 3. Strongly limited by the availability of data, I choose the attributes as close as possible to the Human Development Concept. The data have been put in a distribution matrix, A, with 175 rows and 18 columns. The results of this application depend strongly on the content and limitations of this distribution matrix (somewhat like only looking for the keys under the lamppost because that is where the light is). Table 3 attributes of well-being and their source.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Attributes

Source

Fertility rate Percentage of people living in urban areas Life expectancy at birth Under-5 mortality rate Malnutrition prevalence Access to essential drugs The number of physicians per 1000.000 Access to an improved water source Access to improved sanitation facilities Adult literacy rate Female to male literacy rate Combined enrolment ratio Number of internet users The number of mobile phones GDP per capita (PPP, international $) Annual percentage growth rate of GDP per capita Gastil’ s political rights index Gastil’ s civil liberties index

WDI (2003) WDI (2003) WDI (2003) WDI (2003) HDR (2003) HDR (2003) HDR (2003) WDI (2003) WDI (2003) HDR (2003) HDR (2003) HDR (2003) WDI (2003) WDI (2003) WDI (2003) WDI (2003) Freedom house (2003) Freedom house (2003)

18 The first attribute is the fertility rate (FERT) defined as the number of births per woman. I presume the lower this number, the better. This presumption can be substantiated by Becker’ s (1981) theory of the family, where children are presumed to be a consumption good whereby quality is substituted for quantity. Attribute 2 is the percentage of people living in urban areas (URB). We assume the higher this ratio, the better14. Life expectancy at birth (LEXP) is the third attribute. It indicates the number of years a newborn infant would live if prevailing patterns of mortality at the time of its birth were to stay the same throughout its life. Attribute 4 is the under-5 mortality rate (CMORT). It measures the probability that a newborn baby will die before reaching age five. The probability is expressed as a rate per 1,000. The fifth attribute is the malnutrition prevalence (MNUT) for children under five. It is measured by the percentage of children under five whose weight for age is more than two standard deviations below the median reference standard for their age as established by the World Health Organization. Attribute 6 is the percentage of people with access to essential drugs (DRUG). The data are based on statistical estimates from the World Health Organization obtained in 1998-99. These estimates are grouped into four groupings: very low access (0-49%), low access (50-79%), medium access (80-94%) and good access (95-100%). For every grouping the average value is taken. The number of physicians per 1000.000 (PHY) is attribute 7. Attribute 8 is the access to an improved water source (WAT). It refers to the percentage of the population with reasonable access to an adequate amount of water from an improved source, such as a household connection, public standpipe, borehole, protected well or spring, and rainwater collection. The ninth attribute refers to the access to improved sanitation facilities (SAN). It is measured as the percentage of the population with at least adequate excreta disposal facilities (private or shared, but not public) that can effectively prevent human, animal, and insect contact with excreta. Attribute 10 is the adult literacy rate (LIT). Attribute 11 measures the ratio of female to male literacy rate (FMLIT), thereby referring to Sen’ s work on gender inequality15. We assume the higher this ratio, the better. The combined enrolment ratio

14

This assumption fits in an optimistic view on urbanization, which considers cities as engines of growth and

urbanization as a tool against poverty. See, for example, URBAN21, World Conference on the Urban Future (4-6 July 2000) documentation p. 3, 23. 15

See, for instance, Sen (1990) or Nussbaum and Glover (1995).

19 (ENROL) is the twelfth attribute. The data on school enrolment are from the UNESCO Institute for Statistics and refer to the 2000-01 school year. Attribute 13 measures number of internet users (INT) per 1000 people. The number of mobile phones (MOB) per 1000 people is measured by attribute 14. Attribute 13 and 14 are based on data from the International Telecommunication Union and measure the capability to communicate, with a bias towards high tech communication. The fifteenth attribute measures the GDP per capita (PPP, international $) (GDP). The gross domestic product is thereby converted to current international dollars using purchasing power parity rates. An international dollar has the same purchasing power over GDP as the U.S. dollar has in the United States. Calculations are made by the International Comparison Programme of the World Bank. Attribute 16 is the annual percentage growth rate of GDP per capita (GRO) based on a constant local currency. Combining stock and flow aspects of GDP/capita is done with some reluctance, yet in the work of Slottje et al (1991) the GDP growth plays an important role as attribute of well-being. Attribute 17 measures Gastil’s political rights index (PR). Gastil (1980) has constructed indexes annually since 1973 of political and civil rights. His political rights measure ranked from 1 (the highest degree of liberty) to 7 (the lowest). It is based on several criteria such as the meaningfulness of elections, voting power, multiple political parties and minority self-determination. The eighteenth attribute is Gastil’ civil liberties index (CL). The measure ranks from 1 to 7 and is based on criteria as freedom of press, freedom of speech, freedom from arbitrary imprisonment, free trade unions, freedom of religion and freedom of corruption. An important flaw of the choice of these attributes is the combination of output and input data and the mix of stock and flow aspects. These mixtures are due to data-limitations and further work based on larger and better datasets should try to avoid them.

20

4.2 Preliminary reduction of the attributes by a cluster analysis Regrettably a lot of data points in matrix A are missing. By dropping the countries with incomplete data coverage a new distribution matrix, B, is obtained. B counts 18 attributes (columns) and 116 countries (rows). This group of 116 countries (86% of the total world population) excludes some important developing countries16, in my eyes an unsatisfactory result when talking about global inequality. Therefore in a first preliminary step a cluster analysis is carried out, although this step does not really belong to Maasoumi’ s approach. This summarizing statistical technique is used to divide the attributes into clusters or groups that are similar in their distribution among the countries. Selecting a representative well-documented attribute for every cluster will allow us to broaden the group of covered countries (the number of rows) and therefore gain on information. A price has to be paid in terms of attributes (the number of columns). A final distribution matrix, C, is then obtained. The exact dimensions of C depend on arbitrary choices during the cluster analysis. Unfortunately there are no clear guidelines for the tradeoff between number of attributes and number of countries. In the literature on cluster analysis several methods are used. The most commonly used are the agglomerative hierarchical methods. Hierarchical methods arrange the clusters into a hierarchy so that the relationships between the different groups are apparent. An agglomerative method starts with all the attributes to be clustered separate. Then the most similar attributes and clusters are successively combined until all are in a single, hierarchical group. Generally speaking, an agglomerative clustering algorithm proceeds as follows. 1. The similarity between each pair of cases is calculated and placed in a matrix. There are numerous types of similarity and distance measures that can be used.

16

Large developing countries that could not be covered were Congo, Cuba, Ethiopia, Indonesia, Iraq, North

Korea, Myanmar, Pakistan, Sudan and Ukraine.

21 2. The obtained similarity matrix is then scanned to find the pair of cases with the highest similarity (or lowest distance). These will be the most similar cases and should be clustered most closely together. 3. The cluster formed by these two cases can now be considered a single object. The similarity matrix is recalculated so that all the other cases are compared with this new group, rather than the original two cases. 4. The modified matrix is then scanned (as in step 2) to find the pair of cases or clusters that now have the highest similarity. Steps 2 and 3 are repeated until all the objects have been combined into a single group. For this application I opted for the clustering method proposed by Ward (1963). Wards method (also called the minimum variance method) uses the error sum of squares (SSE) of the attributes as the distance measure. The distance between clusters k and L is defined as

SSE = DkL

∑ (x =

k

− xl

1 1 + Nk N L

)

2

(4.1)

Where x L and x k are vectors of M arithmetic means of the attributes in cluster L and k, NL and Nk are the numbers of attributes in each cluster, and M is the number of countries. DkL is the sum of the squared differences between the cluster means. This method focuses on determining how much variation is within each cluster. The clusters to be joined in the next round of clustering are then chosen by determining which two would give the least increase in within-cluster variation. In this way, the clusters will tend to be as distinct as possible, since the criterion for clustering is to have the least amount of variation. Figure 2 gives the dendrogram of the cluster analysis carried out on the distribution matrix for the 116 countries. Note that before computing the SSE’ s each attribute was scaled so that the mean is zero and the standard deviation is one across all countries17. 17

The standardization procedure, Z0, is defined in (4.2). Other procedures could be followed as well, but

these extensions go beyond the scope of this paper.

22 Figure 2 Dendrogram of the Cluster analysis

To decide the exact number of clusters, the researcher has to make an arbitrary choice about the variance allowed within each cluster. Hirschberg, Maasoumi and Slottje (1991) make a sensitivity analysis for the impact of the number of clusters to the measurement of multidimensional inequality. Based on their paper and a similar study of Slottje et al (1991) I divided the different attributes into five distinct clusters18. 1. Civil liberties, Political Rights 2. GDP per capita, Number of internet users, Number of mobile phone users 3. Adult literacy rate, Female to male literacy rate 4. People with access to improved water, People with access to sanitation, Number of physicians 5. Life expectancy at birth, Under five mortality rate, Fertility rate, Enrolment ratio, Children with underweight, Urban population

18

This result is fairly robust against changing the clustering method and distance concept. Only the attribute

measuring the urban population (URB) clusters sometimes in the second cluster with GDP.

23 The attributes GDP growth and People with access to essential drugs have been put into their own clusters. Building on the results of the cluster analysis, I choose five representative attributes to form distribution matrix C: one for every cluster19. The criterion of selection is the data availability. The selected attributes in matrix C are: civil liberties, GDP per capita, adult literacy rate, people with access to improved water and life expectancy at birth, each representing a cluster. 145 Countries can be covered, accounting for 96% of the total world population. These five attributes show some resemblance with the attributes of the HDI20 extended with the Gastil’ s civil liberties index and the number of people with access to improved water. The former extension is also suggested by UNDP director Sakiko Fukuda-Parr (2001) and the latter extension reminds us of the work of Streeten (1981) on basic needs in which the access to water and sanitation plays an important role. In Figure 3 until Figure 7 the density functions of the different attributes are shown after rescaling so that the mean equals 0 and the standard deviation 1. Remarkable are the different shapes of the density functions across the attributes. Most of the functions are left skewed normal-like distributions, some with a second mode to the left for the badperforming countries. Exception is the second cluster, containing GDP/capita, which looks more like a lognormal distribution with a small mode to the right for the rich countries. Gastil’ s indices seem to be most unequally distributed. In Figure 6 attribute 7, the number of physicians, seems to be an outsider in its cluster.

19

In the calculations hereafter I will concentrate on the attributes representing a cluster with more than one

attribute. The sole attributes get a smaller principal component weight (smaller than 50% of the other weights) in the next step and are therefore not very influential to the final results, according to my calculations. 20

Note that the enrolment ratio has not been selected. Furthermore I look at the GDP/capita whereas the HDI

focuses on the log GDP/capita. According to my calculations the attribute log GDP would cluster very close to the enrolment ratio in cluster 5.

24

Figure 3 Density functions of Political rights and Civil liberties index.

Figure 4 Density functions of GDP/capita, internet users and mobile phone users

25

Figure 5 Density functions of adult literacy rate and female to male literacy rate

Figure 6 Density functions of access to water and sanitation and the number of physicians

26

Figure 7 Density functions of life expectancy, child mortality and underweight, fertility rate and urban population

4.3 First step: Aggregation procedure )LUVW ZH GHULYH WKH ZHLJKWV

f

as defined in equation (3.4). Since no market prices are

available for the attributes, Maasoumi (1989 p. 140; 1999 p. 448) suggests using the method of principal components. Although there is no clear welfare justification for the use of principal components it can serve as a benchmark for other weighting schemes. The results of the principal component analysis can be found in Table 4. The data are normalized first so that their mean is zero and the standard deviation is 1. The results in the table are rescaled so that their sum equals 121. (In Maasoumi’ s notation they represent the f’ s).

The variance captured by this principal component is about 68% of the total variance.

Remarkable is the fact that each attribute gets a similar weight. So using a principal component analysis or an arbitrary equal weighting scheme has no large effect on the final results.

21

Footnote 17 applies here as well.

27 Table 4 Principal components of distribution matrix (145X5) First Principal Component Adult literacy rate

0,201

People with access to improved water

0,202

Life expectancy at birth

0,212

Gastil’ s civil liberties index

0,184

GDP capita PPP, US$

0,201

Slottje et al. (1991) give an overview of some other techniques to obtain the weights. If individuals can rank the different attributes lexicographically in a social welfare function, weights based on the relative rankings of the attributes can be employed to construct a weighting scheme. An alternative is for the researcher to impose his own ranking on the relative importance of each attribute, but this method seems to be very ad hoc. After deriving the weights, the different attributes, Xf’ s, can be aggregated to obtain the distribution of aggregate Si. Since the attributes are very different in nature, they have to be standardized first. Different standardization procedures are possible. See, for example, Hirschberg, Maasoumi and Slottje (1991 p. 144). In the following I discuss three different standardization procedures. There are two problems with the commonly used standardization procedure, defined as follows. Z0 =

X −µ σ

(4.2)

First, this procedure involves a subtraction oIWKHPHDQ DQGLVWKHUHIRUHPRUHGUDVWLFWKDQ a simple rescaling, thereby influencing the inequality, as measured by a scale invariant inequality measure. This is roughly the same argument as made in section 2 about the HDI transformations. Secondly, this standardization leads inevitably to some negative Xif’ s. These negative values cause mathematical problems as described in section 3.

28 Pragmatically, I would suggest staying away from negative Xif’ s by translating the standardized values of Xif’ s by adding a large enough positive number. The drawback of this procedure is the further muddling of the inequality of the attributes. Hirschberg et al. (1991), translate the standardized values by the arbitrary value 10. This results in a standardized distribution with mean 10 and standard deviation of 1, defined as Z1 hereafter. Z1 =

X −µ + 10 σ

(4.3)

Hirschberg et al. (1991) also suggest looking at attribute shares, which is up to a scale factor equal to dividing by the mean. Inequality of any attribute, summarized with a scale invariant measure, doesn’ t change after this standardization procedure. A drawback of this procedure is that it does not take the standard deviations of the underlying attribute distributions into account. Hereafter the procedure is called Z2. Z2 =

X µ

(4.4)

A third standardization procedure Z3 is more in line with the HDI transformations of equation (2.1) and can be defined as follows. Z3 =

X X max

(4.5)

This standardization can be interpreted as the performance of every country vis-à-vis the best performing country. Again no information about the standard deviation is taken into account. It can be seen as an advantage that every attribute value is a member of the interval between 0 and 1 as in the HDI case22. The two last standardization procedures are only useful, if the Xif’ s are inherently positive. In general this is not the case for every well-being indicator23. Since no standardization 22

De facto, low standardization values are only reached for attributes GDP per capita and civil liberties. For

the other attributes, such as life expectancy, the worst performing country reaches still 0,4. This unequal lower bound makes Si (with large pRVLWLYH  YDOXHV  HVSHFLDOO\ VHQVLWLYH WR *'3FDSLWD DQG FLYLO OLEHUWLHV when standardization procedure Z3 is used (the same argument applies to Z2 up to a lesser extent). 23

Indicators, which are measuring flow aspects of development such as GDP/capita growth can be negative

in general. In analyses where flow aspects are considered, only procedure Z1 will be useful.

29 procedure seems to be really satisfactory, a sensitivity analysis is carried out in the following steps. In future research more theoretical work has to be done on the influence of the standardization procedure on the aggregation procedure and the measure of multidimensional inequality. In appendix 2 values for the Si’ s are given for every country. Thereby I report the results

for the three standardization procedures and fRU VL[YDOXHVRIWKHSDUDPHWHU (-200, -1, -

0.5, 0, 1 and 50)24, which captures the degree of substitution between the attributes. Since the weighting scheme derived from the principal component analysis and the equal weighting do not diverge much, only the results weighted by the principal components are reported. In appendix 3 the rankings based on the Si values of all countries are provided. Furthermore the ranks based on GDP/capita and the HDI25 are given. The Spearman rank correlation matrix26 is provided in appendix 4. Smallest rank correlations (0.78) are reached between Si calculated with standardization procedure Z1 (  = 50) and GDP/capita or Si calculated with standardization procedure Z2    -200). By carrying out pairwise rank correlation tests I could reject the null hypothesis of different rankingsRQDVLJQLILFDQFHOHYHO RI. So the ranking of the countries is statistically

robust for the choice of the standardization proceduresIRUDJLYHQ YDOXH, and moreover the ranking does not differ significantly of the ranking based on the HDI or on GDP/capita. This is especially true for the high-performing countries. 24

The values -1, -0.5, 0, 1 are commonly used in the empirical literature on Maasoumi’ s approach. See, for

example, Maasoumi and Jeong (1985); Maasoumi and Zandvakili (1986 and 1989). I added the values -200 and 50 to capture some extreme viewpoints, where attributes are treated as complements or oppositely. These YDOXHVDUHDUELWUDULO\FKRVHQODUJHHQRXJKWRPDNHWKHGLIIHUHQFHZLWKPRGHUDWH YDOXHVEXWQRWWRRODUJHWR avoid numerical problems. (51.275 seems to be the largest feasible value that could be reached with my software package for standardization Z3) 25

Rankings based on the HDI show some difference with the tabulated ranks in the HDR (2003), since I

reranked the countries to compensate for the countries without data coverage. 26

Pairwise 6*

Rs = 1 −

Spearman

rank

∑ (difference in ranking )

correlations

are

calculated

by

applying

2

i

i 3

n −n

, whereby n equals the number of countries (here 145).

formula

30 Countries for which the choice of the standardization procedure makes a lot of difference in their ranking are Botswana, Namibia, Oman, Saudi Arabia and South Africa. As can be seen in appendix 3 these 5 countries all have a large divergence between their ranking based on GDP/capita and a broader measure like the HDI. A close investigation of correlation matrix in appendix 4 or a comparison of Figure 4 and Figure 10 makes clear that standardization procedure Z3 (and Z2WRDOHVVHUH[WHQW PLPLFV*'3FDSLWDIRUKLJK  values, whereas standardization procedure Z1 is not27. Countries with different performance on the attribute GDP/capita and the other attributes are likely to be more sensitive to the choice of standardization procedure. The same image arises when we fix the standardization procedure and study the robustness IRUWKHGLIIHUHQW YDOXHVLQWKHUDQNLQJRIWKHFRXQWULHV7KHUDQNLQJRIKLJK-performing

FRXQWULHV LV UREXVW IRU GLIIHUHQW  YDOXHV whereas Benin, Brazil, Djibouti, Oman, Saudi

Arabia and South Africa change a lot of rankingV IRU GLIIHUHQW  YDOXHV 7KHse countries seem to be unequally developed28, performing much better on some attributes than on

others. Table 5 reports the rankings for the different attributes of the six most unequally developed countries. Djibouti, for instance, ranks number 1 for access to water, but ranks at least number 97 for all the other attributes Table 5 Rankings of some unequal developed countries

27

Adult literacy

Access to

Life expectancy

Gastil’ s civil

GDP/

rate

improved water

at birth

liberties index

capita

Benin

113

117

117

20

131

Brazil

76

78

91

52

51

Djibouti

112

1

127

97

101

Oman

102

139

51

97

34

Saudi Arabia

95

55

53

143

31

South Africa

82

79

117

20

37

Standardized data for the attribute GDP making use of standardization methods Z2 and Z3 are in general

smaller than data for other attributes, which explains the close relation between Si and the GDP/capita attribute. 28

Note that there is nothing attractive on equal development across attributes, as such. Sierra Leona, for

instance, is very equally developed since it performs very badly on all attributes.

31 Figure 8 until Figure 10 depict the distribution functions of the aggregate SiIRUGLIIHUHQW 

values using the different standardization procedures. For largeYDOXHV RI WKH WUDQVODWHG normalization procedure (Z1) offers the most insightful results. The other two standardization procedures mimic the distribution of the attribute GDP/capita to a large extent, due to the smaller minimal value of GDP/capita compared to the other attributes, ZKLFKLVHVSHFLDOO\UHOHYDQWIRUODUJH YDOXHV. A remarkable result is the trimodal density function of Si which appears in most cases. The first mode (the high performing countries) consists of the West-European and North American countries, supplemented with Japan, Australia and New Zealand. The last mode (the low performing countries) consists of the Sub-Saharan countries and some poor Asian countries as Bangladesh, Pakistan and Nepal. The South American countries together with the transition countries and most of the Asian countries form the middle performing (and largest) mode of the density function.

Figure 8 Density function of Si (using standardization procedure Z1)

32

Figure 9 Density function of Si (using standardization procedure Z2)

Figure 10 Density function of Si (using standardization procedure Z3)

33

4.4 Second Step: Inequality Measurement of Si Once the Si’ s are derived, the calculation of the inequality measures is rather straightforward. As Maasoumi suggest we make use of the Generalized Entropy class of inHTXDOLW\PHDVXUHV,QHTXDOLW\LVFDOFXODWHGIRU GLIIHUHQW YDOXHV making use of formula (3.5). Results are reported in Table 629. The calculations are done for equal population shares ( pi = 1 N ), similarly to the existing applications of Maasoumi’ s approach in the literature. This one-country one-data point view has the disadvantage of giving the same weight to small countries (like Lesotho, for example) as to large, heavily populated ones, like China. Alternatively, pi could be set equally to the population share of country i, which results in a population weighted inequality measure. In his paper on global income inequality Salah-iMartin (2002) pointed at the importance of population-weighting for the results on income inequality.

Adopting

a

one-individual

one-datapoint

view

will

involve

some

interpretational problems, since a lot of the common well-being attributes are country attributes and not applicable to individuals (like life expectancy, literacy rate and access to water). It would be interesting in further research to examine the influence of populationweighting on the obtained results. A recent paper by Dutta, Pattanaik and Xu (2003) makes this line of inquiry only more interesting. Ideally the multidimensional inequality would be measured on individual data as suggested by the work of Dutta, Pattanaik and Xu. These authors investigate the multidimensional measurement of the standard of living on the basis of aggregate data. The exclusive availability of aggregate data for many global well-being attributes prevents researchers from following the conceptual framework of welfare economics that suggests measuring first the individual’ s achievements and then aggregating the overall standard of living amongst the individuals. Working with aggregate data leads only to the same results as working with individual data under some stringent conditions, which are generally not fulfilled in this study. 29

These figures are in line with the results of Hirschberg, Maasoumi and Slottje (1991), which is the paper

that shows the most similarity with mine. Their data are from the late 80’ s and are calculated for a restricted QXPEHURI ¶VDQGRQO\IRUVWDQGDUGL]DWLRQSURFHGXUH=1 and Z2.

34 Table 6 Results of the two step procedure on global inequality. Standardization procedure Z1: translated normalization  -200

= -1

= 0.5

=0

=1

= 50

0

0.00300

0.00343

0.00346

0.00349

0.00356

0.00621

-0.5

0.00301

0.00344

0.00347

0.00350

0.00357

0.00625

-1

0.00301

0.00346

0.00349

0.00352

0.00359

0.00628

-2

0.00302

0.00350

0.00353

0.00356

0.00363

0.00638

Standardization procedure Z2: attribute share  -200

= -1

= 0.5

=0

=1

= 50

0

0.1123

0.0620

0.0655

0.0734

0.1027

0.2287

-0.5

0.1057

0.0623

0.0665

0.0753

0.1088

0.2521

-1

0.1014

0.0634

0.0683

0.0783

0.1177

0.2929

-2

0.0984

0.0680

0.0748

0.0884

0.1485

0.4832

Standardization procedure Z3: attribute share vis-à-vis maximum  -200

= -1

= 0.5

=0

=1

= 50

0

0.0125

0.0348

0.0468

0.0734

0.1910

0.4614

-0.5

0.0131

0.0356

0.0478

0.0753

0.2035

0.4794

-1

0.0137

0.0367

0.0493

0.0783

0.2261

0.5467

-2

0.0151

0.0400

0.0539

0.0884

0.3237

1.0019

Examining Table 6, one can see that the pattern in the values is rather robust with respect to WKH VWDQGDUGL]DWLRQ WHFKQLTXH WKH  DQG WKH  YDOXH 7KH DFWXDO values are quite variable and unfortunately they are very difficult to compare. Table 7 provides as benchmark the inequality measured in the GDP/capita and in the HDI for the same 145 countries.30

30

Since GE-class is scale independent and Z2 and Z3 are equal up to rescaling factor Xmax WKHLQHTXDOLW\

measures for Z2 and Z3 are equal.

35 Table 7 Results of the two step procedure based on HDI attributes GDP/capita

HDI

Z1

Z2

Z3

Z1

Z2

Z3

0

0.00477

0.44153

0.44153

0.00507

0.03789

0.03789

-0.5

0.00469

0.45720

0.45720

0.00513

0.03978

0.03978

-1

0.00460

0.51530

0.51530

0.00519

0.04213

0.04213

-2

0.00446

0.88743

0.88743

0.00534

0.04859

0.04859

The results of Table 6 for standardization procedure Z1IRUVPDOORUQHJDWLYHYDOXHVRI DUH smaller than the benchmarks in Table 7, which means that Si shows a more equal distribution than GDP/capita and HDI. The results of Table 6 calculated for standardization procedures Z2 and Z3DQGIRUPRGHUDWH YDOXHVDUHODUJHUWKDQWKHLQHTXDOLW\LQ+',EXW smaller than GDP/capita. This result points at the importance of the standardization procedure and the value of SDUDPHWHUV LQWKHFDOFXODWLRQRIWKHJOREDOLQHTXDOLW\$FOHDUYLHZRQPXOWLGLPHQVLRQDO inequality can not be achieved unless more theoretical insights are gathered about the exact influence of the standardization procedure and the degree of substitution between the DWWULEXWHVDVPHDVXUHGE\  Finally we return in Figure 11WRWKHLQIOXHQFHRISDUDPHWHU RQWKHLQHTXDOLW\Peasure.

2Q WKH KRUL]RQWDO D[LV WKH GLIIHUHQW  YDOXHV DUH SORWWHG 7KH YHUWLFDO D[LV VKRZV WKH corresponding GE, using the Z1 standardization procedure DQG D  RI -0.5. Other

VWDQGDUGL]DWLRQ SURFHGXUHV DQG  YDOXHV SURYLGH VLPLODU UHVXOWV and are therefore not

reported. The curve shows a stretched S-shape. Highest inequality is reached for large SRVLWLYH YDOXHVWUHDWLQJWKHDWWULEXWHVDVFRPSOHPHQWV7KHOLPLWLQJFDVHLVIRUPHGE\  approaching +’ZKHUH*(LVFDOFXODWHGIRUWKHVPDOOHVWDWWULEXWHYDOXHIor every country. /RZHVWLQHTXDOLW\LVUHDFKHGIRUODUJHQHJDWLYH YDOXHVZKHUHWKH*(LVFDOFXODWHGIRUWKH KLJKHVW DWWULEXWH YDOXH 6PDOO FKDQJHV LQ ’ s around 0 tend to have a relative relatively

large effect on the measured GE.

36

0.007 0.0065 0.006 0.0055 0.005 0.0045 0.004 0.0035 0.003 0.0025 0.002 -200

- 150

-100

- 50

0

50

100

150

200

B èt a

Figure 11,QIOXHQFHRISDUDPHWHU RQ GE inequality measure (using standardization procedure Z1 DQG

- 0.5)

This result can be interpreted as follows: A large number of countries perform relatively well on at least one attribute, therefore a largHQHJDWLYH VKRZVWKHOHDVWLQHTXDOLW\,IZH

VHW  WR EH SRVLWLYH DQG ODUJH ZH FRQFHQWUDWH RQ WKH ZRUVW SHUIRUPLQJ DWWULEXWH RI HYHU\ country, which is more unequally distributed among the countries, resulting in a higher GE measure. A relatively large DQGSRVLWLYH VHHPVDSSHDOLQJWRPHIURPSROLF\SHUVSHFWLYH thereby favoring more equal development across the attributes.

37

5. Conclusion Maasoumi’ s two step approach is an intuitively appealing and applicable method to the measurement of multidimensional inequality. Yet, some welfare theoretical issues have to be solved before the method can be presented as a benchmark for multidimensional distributional analysis. Until present there is no welfare justification found for the selection RI WKH ZHLJKWV

f

of the attributes in the aggregation procedure. Nor is there theoretical

clarity on the choice of the standardization procedure. Further theoretical and empirical research31 KDV WR EH GRQH RQ WKH LQIOXHQFH RI WKH  SDUDPHWHU FDSWXULQJ WKH GHJUHH RI substitutLRQ DPRQJVW WKH DWWULEXWHV (VWLPDWLRQV RI UHDOLVWLF  YDOXHV ZLOO not only be beneficial for the investigation of multidimensional inequality, but also for the construction of multi-attribute welfare indices. This brings me to the central question that I wanted to answer in this paper: can Maasoumi’ s approach enrich the concept of human development and vice versa? Let us first look at the contributions of the two step approach to the empirical application of the human development concept. The information theoretical roots of Maasoumi’ s approach support some arbitrary choices about the aggregation procedure taken by the UNDP in constructing indices like the HDI. Meanwhile the two step approach makes some implicit assumptions of the UNDP more explicit by confronting researchers with questions about the assumed degree of substitution, weighting and standardization procedures32. The concept of human development on its turn can enhance Maasoumi’ s method by offering a framework of thought for selecting and weighting the relevant attributes. Further enrichment by the human development framework can steer Maasoumi’ s approach away from the great reliance on statistical tools like principal components that are hard to interpret from a welfare theoretical perspective.

31

This research can be expected to have offshoots in consumption theory investigating the role and

interpretation of concave indifference curves. 32

As indicated before, not every question has already been answered adequately.

38

List of Figures Figure 1 Iso-LQGH[FXUYHVRIGLIIHUHQWSDUDPHWHUYDOXHVRI .............................................. 10 Figure 2 Dendrogram of the Cluster analysis....................................................................... 22 Figure 3 Density functions of Political rights and Civil liberties index............................... 24 Figure 4 Density functions of GDP/capita, internet users and mobile phone users............. 24 Figure 5 Density functions of adult literacy rate and female to male literacy rate .............. 25 Figure 6 Density functions of access to water and sanitation and the number of physicians ...................................................................................................................................... 25 Figure 7 Density functions of life expectancy, child mortality and underweight, fertility rate and urban population.................................................................................................... 26 Figure 8 Density function of Si (using standardization procedure Z1)................................. 31 Figure 9 Density function of Si (using standardization procedure Z2)................................. 32 Figure 10 Density function of Si (using standardization procedure Z3)............................... 32

)LJXUH  ,QIOXHQFH RI SDUDPHWHU  RQ *( LQHTXDOLW\ measure (using standardization procedure Z1 DQG - 0.5) ............................................................................................ 36

List of Tables Table 1 Goalposts for calculating the HDI (source: Human Development Report, 2003) .... 6 Table 2 Overview of Maasoumi’s aggregation procedure.................................................... 11 Table 3 attributes of well-being and their source................................................................. 17 Table 4 Principal components of distribution matrix (145X5) ............................................ 27 Table 5 Rankings of some unequal developed countries ..................................................... 30 Table 6 Results of the two step procedure on global inequality. ......................................... 34 Table 7 Results of the two step procedure based on HDI attributes .................................... 35

39

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