Adjusting the HDI for Inequality1 - OECD.org · II PARTIALLY ADJUSTED HDI: The Income Adjusted HDI...

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Adjusting the HDI for Inequality: An Overview of Different Approaches, Data Issues, and Interpretations Milorad Kovacevic Statistical Unit, HDRO

HDRO Brown‐bag Seminar, September 28, 2009

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OBJECTIVES: • Review different ideas and proposals for HDI adjustments • Discuss the data issues at the international level: requirements and availability

• Interpret resulting indices • Stimulate the discussion

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OUTLINE: I INTRODUCTION II PARTIALLY ADJUSTED HDI III ADJUSTMENTS FOR INEQUALITIES IN MARGINAL

DISTRIBUTIONS IV HDI ADJUSTED FOR MULTIDIMENSIONAL INEQUALITY V SUMMARY

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I INTRODUCTION

The Human Development index (HDI) is overall well accepted as a summary measure of human development (HD) achievements: It is transparent, simple to calculate and interpret, and it serves the purpose ‐ to summarize the performance of the countries on the three dimensions of HD: standard of living, health, and education. Often criticized for:

Missing dimensions, wrong specifications, wrong measurements of dimensions and missing indicators, redundancy, and poor data quality.

Ignoring the extent of inequality in distribution of HD within a country is an example of wrong specification.

I INTRODUCTION

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“The worst form of inequality is to try to make unequal things equal.” Aristotle

The country‐average HDI conceals inequality in distribution of HD An equal distribution of HD within the country

Major obstacles for not accounting for inequality:

• lack of appropriate disaggregated data, • conceptual difficulties

Disaggregated analysis HDI adjusted for inequality

⇔

I INTRODUCTION

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Disaggregated analysis:

HDI disaggregated across different population subgroups: • assess disparities in achievements between the groups, • a more coherent view of human development in a particular country,

• no attempt to quantify inequality, • international comparison is generally difficult.

Examples: • HDI by gender (HDR 1991), by race and gender (HDR 1993:

U.S.A’s HDI by race and gender); • by provinces/municipalities (NHDR 2002: Mexico, Bulgaria); • by income quintiles (HDR 2006: 13 countries, Technical note 2).

I INTRODUCTION

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Inequality in dimensions of HDI “We differ in age, sex, physical and mental health, bodily prowess, intellectual abilities,

(…), and in many other respects (…) Such diversities, however, can be hard to accommodate adequately in the usual evaluative framework of inequality assessment.

As a consequence, this basic issue is often left substantially unaddressed in the evaluative literature.”

Amartya Sen, “Inequality Reexamined”, 1992, page 28

• Methods for measuring inequality were developed in relation to unequal concentration of income and other forms of material wealth

• Inequality in distribution of other characteristics is often recognized but rarely measured

• The inequality measures reported in the HDR were mainly about the income distribution

I INTRODUCTION

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HDR (1990): All three average measures of human development “conceal wide disparities in the overall population,” but that compared to income inequality, the “inequality possible in respect to life expectancy and literacy is much more limited: a person can be literate only once, and human life is finite.”

Hicks (1997) argues that

“there is significant life‐span inequality, ranging from infants who die at birth or before age one, to persons who die at ages over 100 years.”

I INTRODUCTION

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Welfare measure (standard) )(W x is the level of welfare (well‐being) associated with the income distribution )x,...,x( n1=x A number of axioms that )(W x has to satisfy

)(W x is maximized for the completely equal distribution (Atkinson, 1970; Kolm, 1969; Sen, 1973): )(μ))(μ),...,(μ(W xxx = A natural concept of inequality level (measure):

)(μ)(W

1)(Ixx

x -= ,

“the loss of welfare from inequality expressed as a percentage of the maximum achievable welfare (Atkinson, 1970).”

I INTRODUCTION

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Equivalently, ))I(1)((μ)(W xxx -= ,

“the welfare standard )(W x is the mean income discounted by the level of inequality in distribution x “ (ibid.)

• The Sen welfare standard: ))G(1)((μ)(S xxx -= )(G x : Gini index

• The Atkinson’s welfare standard: )(I x is the Atkinson’s inequality mesure )(I xε

The basic properties of a welfare functions: Symetry, replication invariance, monotonicity, the Pigou‐Dalton transfer principle, linear homogeneity, normalization, and continuity

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II PARTIALLY ADJUSTED HDI: The Income Adjusted HDI (HDR, 1991) An attempt to sensitize the HDI to the distributional inequality of the most unequal dimension ‐ income. The life expectancy and education indices remain unchanged, while the GDP index is modified as:

)100ln()000,40ln()100ln()GDPln()G1(

GDPI

*ind -

--=

and )GDPEDULE(31

HDI *indindind

* ++=

II PARTIALY ADJUSTED HDI

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• The argument: accounting for the disparity in income dimension would most likely cover the disparities in other two.

• A common misperception in the literature that income inequality is closely related to inequality in other dimensions.

• Inequality in different dimensions may be caused by different factors (e.g., Jensen and Nielsen, 1997):

‐ Income distribution is related to employment structure, minimum wage, social security provision, etc.

‐ School enrollment depends on the provision of public schools, legislation of child labour, labour markets

II PARTIALY ADJUSTED HDI

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IDAHDI was calculated only for a small number of countries (53) for the HDR 1991 to 1993. Gini index was provided for some countries (25) For other countries, Gini index was estimated from the 20/20 income ratio NB.: “The World Development Indicators” uses the same method to estimate the Gini index for some countries (WDI, 2009, page 75).

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III ADJUSTMENTS FOR INEQUALITIES IN MARGINAL DISTRIBUTIONS

Hicks (1997): Inequality‐Adjusted HDI (IAHDI) • The Gini index (G) for each dimension • Discounts the component indices by multiplication by (1‐G)

)]G1(GDP)G1(E)G1(LE[31

IAHDI Iind

Eind

LEind -+-+-=

• The adjustments were done to the component indices after normalization, and not to the indicators.

III ADJUSMENTS FOR MARGINAL INEQUALITY

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Illustration used aggregate and distributional data for 20 developing countries. • Aggregate data on literacy, life expectancy and GDP • The distributional data:

‐ age‐at‐death grouped by age into ten classes, the Gini index (0.15, 0.63) (mortality statistics from the U.N. Demographic Yearbook 1992);

- years of schooling classified in 6 categories, the Gini index was approximated ranging (0.32, 065) (from Ahuja and Filmer, 1995);

- income data in the form of income quintile shares with additional share for the top 10%, the Gini index (0.28, 0.60). (World Development Report, 1995)


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• The between‐group Gini underestimates the extent of inequality (the within‐group inequality is missing)

• The Hicks’ index requires the distributional data for all three dimensions but not necessarily across the same groups:

‐ the distribution of health dimension was across the age‐groups, ‐ the distribution of education was over education categories, and ‐ the income distribution was over the income quantiles.


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• Many basic properties for welfare function are satisfied by Hicks’ index

• It successfully incorporates sensitivity to distributional inequality • However it does not use the association between the dimension – it is not association sensitive

• Hicks’ index also violates the sub‐group consistency (Foster, Lopez‐Calva, Szekely, 2005).

The subgroup consistency (Foster and Schorrocks, 1991):

“the direction of the change in HDI of one group is transmitted into a change of the same direction of the overall HDI, assuming that the HDIs of other groups remain unaltered.”


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• Foster, Lopez‐Calva, Szekely (2005) used a general expression to emphasize the order of aggregation in Hicks’ index:

)](S),(S),(S[μIAHDI zyx=

‐ First aggregation is across persons, S(.) denotes the Sen welfare standard for the particular dimension: ))G(1)((μ)(S xxx -=

‐ Then across dimensions, [.]μ denotes an arithmetic mean ‐ Hicks’ index is the mean of Sen welfare levels across the three dimensions

‐ It is not possible to reverse the order of aggregation (path dependence)

))(),(),((G(HDI)](),(),([SIAHDI* zyxzyx μμμμμμ -1==


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Inequality Accounted By the General Mean of General Means A class of general means of order ε (Atkinson, 1970):

⎪⎩

⎪⎨⎧

=

++= 1∏ ,

1≠,)...(),(

)-1/(1-1-111

-1 εxεxaxa

μk

ak

εεnn

ε

ε kax

where a is a vector of weights which are all positive and sum to one. • These means are distribution sensitive:

- for ε>0 the mean puts more weight on the lower part of the distribution, so that )(μ xε-1 is smaller than the neutral arithmetic mean )(μ x1 ;

- for ε=0 it is neutral; - for ε<0 it is upper‐sensitive.


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• The order ε is interpreted as an “inequality aversion” towards inequality across persons.

• Atkinson defines the family of inequality measures as

)(μ)(μ

)(Ixx

x1

-1-1= εε

• Foster et al. (2005) used the Atkinson’s class of welfare functions,

))(I)((μ)( xxx εεμ -1-1=-1 ,

and interpreted it as the (arithmetic) mean of x discounted for the inequality in distribution measured by the Atkinson inequality measure with parameter ε.

• It satisfies all the basic properties of well defined welfare standard plus it satisfies the subgroup consistency .


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• In order to provide a combined adjustment to all three dimensions, some kind of averaging over dimensions is needed.

• The simple arithmetic mean over the dimensions, as in the case of the Hicks’ index,

))(),(),(( -1-1-1 zyx εεε μμμμ

results in an index that does not satisfy the subgroup consistency (Foster et al., 2005). The Gender Related Development Index (GDI) has the aforementioned form (ε=2), and thus, it is not subgroup consistent!


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• The general mean of the general means of the same order maintains all the properties and the subgroup consistency ((Foster et al., 2005):

))(),(),((H zyx εεεε μμμμε -1-1-1-1= , for ε>0

• The order of aggregation when using the general mean of the general means of the same order can be altered but the value of the index remains the same (path independence).


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• For illustration of the proposed method Foster et al. (2005) used data from Mexico:

- The income and education component came from a sample from the Population Census for the year 2000. The data were available at the individual level (over 10 million records).

- The infant survival rates were available at the municipality level.

• For each individual in the sample the income, education, and health indices were constructed.

• εH is computed for each state by aggregating first within each dimension, and then by aggregating across dimensions, for ε=0, and ε=3.


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• It is possible to apply the proposed method even when data are only available as group aggregates for groups that are different across dimensions.

• However, by using the group aggregates the within‐group inequality will not be accounted for.

• Proposed method is sensitive to inequality in marginal (dimensional) distributions, however it is not sensitive to correlation across dimensions that may alter overall inequality .

• Searching for correlation sensitive multidimensional indices.

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IV HDI ADJUSTED FOR MULTIDIMENSIONAL INEQUALITY • Atkinson and Bourguignon (1982) argue that a multidimensional social evaluation should be sensitive to both, the inequality in distribution of each dimension across population, as well as to the correlation between the dimensions of well‐being.

• The logic is that if there is a strong positive association between dimensions, the single‐dimensional and multidimensional comparisons are very much similar. On the other side if the association is loose the importance of assessing the inequality between dimensions increases.

IV MULTIDIMENSIONAL INEQUALITY

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• Multidimensional measures of inequality

• A new set of properties for these measures (Pigou‐Dalton Dominance Criterion, Uniform Dominance, Correlation‐Increasing Dominance)

• Maasoumi (1986, 1999), Tsui (1995, 1999), Bourguignon (1999), Seth (2009)

• Typically, a two‐stage aggregation is necessary


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• In the first stage aggregation over dimensions for each person ip , using the general mean

βββββ

/1]3/)[()( iiii zyxpμ ++= for all i

• Parameter β is related to the degree of substitutability between dimensions The elasticity of substitution is given by )/( β-11 : - When β=1, )/( β-11 →∞ (dimensions are perfect substitutes) - When β→‐∞, )/( β-11 →0 (dimensions are perfect complements)

- Common restriction is β≤1.

• The first stage aggregation yields individual overall achievements:

)p(μ),...,p(μ nββ 1


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• Aggregation in the second stage across persons using generalized means of order α :

))p(μ),...,p(μ( nββαμ 1

• Parameter α is the inequality “accros persons” aversion • Smaller α more emphasis on the lower end of the distribution • A distribution with more inequality is more punished

• The original HDI: α= β=1 • The Foster et al. (2005) class: α= β≤1

• Seth (2009): “Path independent index doesn’t satisfy all necessery dominance criteria for an association‐sensitive index”

• It is necessary that α , β≤1 and α≠ β


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• For illustration of the proposed method Seth used the same data as Foster et al. (2005) to calculate the HDI indices for 33 Mexican states:

• For each individual in the sample the income, education, and health indices were constructed.

• Three sets of parameters were applied: α= β=1 (HDI), α= β=‐2 (Foster et al.), α=‐3, β=‐1 (Seth)

• Rankings under both adjustments differ from the original HDI

• Seth’s method is the only one sensitive to changes in associations between dimensions


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A Simple Example Data from “Inequality in Human Development: An Empirical Assessment of 32 Countries” by Grimm, Harttgen, Klasen, Misselhorn, Munzi and Smeeding, Soc. Indicators Research, 2009 Mozambique (2002/2003)

Q1 Q2 Q3 Q4 Q5 alpha=1 -2 -3INC 0.115 0.242 0.325 0.412 0.639 EDU 0.436 0.463 0.464 0.468 0.528 HEALTH 0.266 0.295 0.282 0.322 0.341 beta=1 0.272 0.333 0.357 0.401 0.503 0.373 0.351 0.349beta=-2 0.178 0.300 0.335 0.386 0.453 0.331 0.281 0.268beta=-1 0.203 0.310 0.342 0.391 0.469 0.343 0.304 0.294


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Canada (2000) Q1 Q2 Q3 Q4 Q5 alpha=1 -2 -3INC 0.809 0.909 0.958 1 1 EDU 0.974 0.968 0.981 1 1 HEALTH 0.881 0.902 0.924 0.946 0.967 beta=1 0.888 0.926 0.954 0.982 0.989 0.948 0.946 0.945beta=-2 0.880 0.925 0.953 0.981 0.989 0.946 0.943 0.943beta=-1 0.883 0.925 0.954 0.981 0.989 0.946 0.944 0.944

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SUMMARY • More research is needed about the choice of parameters (α , β ) within the multidimensional framework

• Currently not more than 50 countries may have individual data available for international use (Klasen’s group)

• Use of grouped data may be possible for more than 50 countries (application of Foster et al., without accounting for association)

• GDI can (should) be revised to become subgroup consistent

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References Sen (1992). Inequality Reexamined Hicks (1997).World Development Atkinson (1970). J.l of Economic Theory Kolm (1969). Public Economics (Ed.) Sen (1973). On Economic Inequality Jensen and Nielsen (1997) J. of Population Economics The World Development Indicators (2009) Ahuja and Filmer (1995).World development report Foster, Lopez‐Calva, Szekely (2005). J. of Human Development. Foster and Schorrocks (1991). Econometrica Atkinson and Bourguignon (1982). Review of Economic Studies Maasoumi (1986). Econometrica Maasoumi (1999), Handbook of Income Inequality (Ed.) Tsui (1995). J. of Economic Theory Tsui (1999). Social Choice and Welfare Bourguignon (1999). Handbook of Income Inequality (Ed.) Seth (2009). J. of Human Development Grimm et al. (2009). Soc. Indicators Research, 2009

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